text:
\section{Introduction and background}
\label{Introduction}

\noindent A \textit{biorthogonal system} for an infinite-dimensional  (real or complex) separable Banach space
$(\ensuremath{\mathbb{X}}, \Vert\cdot\Vert)$
is a family $(\ensuremath{\mathbf{x}}_{j},\ensuremath{\mathbf{x}}_{j}^{\ast})_{j\in J}\subset \ensuremath{\mathbb{X}}\times \ensuremath{\mathbb{X}}^{\ast}$ verifying
\begin{itemize}
\item[(i)] $\ensuremath{\mathbb{X}}= \overline{\operatorname{span} \{ \ensuremath{\mathbf{x}}_{j} : j\in J\}}$, and
\item[(ii)] $\ensuremath{\mathbf{x}}_{j}^{\ast}(\ensuremath{\mathbf{x}}_{k})=1$ if $j=k$ and $\ensuremath{\mathbf{x}}_{j}^{\ast}(\ensuremath{\mathbf{x}}_{k})=0$ otherwise.
\end{itemize}
For brevity, we refer to $\mathcal{B}=(\ensuremath{\mathbf{x}}_j)_{j\in J}$ as a \textit{basis} and to the unequivocally determined $\mathcal{B}^*=(\ensuremath{\mathbf{x}}_j^*)_{j\in J}$ as its
\textit{orthogonal family.}
If the biorthogonal system fulfills the additional condition
\begin{itemize}
\item[(iii)] $\sup_{j\in J} \Vert \ensuremath{\mathbf{x}}_j\Vert \Vert \ensuremath{\mathbf{x}}^*_j\Vert <\infty$
\end{itemize}
we say that the basis and the biorthogonal system are \textit{bounded}. Finally, when the basis verifies
\begin{itemize}
\item[(iv)] $0<\inf_{j\in J} \Vert \ensuremath{\mathbf{x}}_j\Vert \le \sup_{j\in J} \Vert \ensuremath{\mathbf{x}}_j\Vert <\infty$
\end{itemize}
we say that the basis
is \textit{semi-normalized} (respectively \textit{normalized} if $\Vert \ensuremath{\mathbf{x}}_j\Vert=1$ for all $j\in J$).
Notice that a biorthogonal system fulfills simultaneously (iii) and (iv) if and only if
\[
\sup_{j \in J} \max\{ \Vert \ensuremath{\mathbf{x}}_j\Vert ,\Vert \ensuremath{\mathbf{x}}_j^*\Vert \}<\infty.
\]

Suppose $\mathcal{B}=(\ensuremath{\mathbf{x}}_j)_{j\in J}$ is a semi-normalized bounded basis in a Banach space $\ensuremath{\mathbb{X}}$ with orthogonal family
$\mathcal{B}^*=(\ensuremath{\mathbf{x}}_j^*)_{j\in J}$.
 Each $f\in \ensuremath{\mathbb{X}}$ has a unique formal series expansion in terms of the basis,
\begin{equation}
\label{formula:SeriesExpansion}
f=\sum_{j\in J} \ensuremath{\mathbf{x}}_j^*(f)\ensuremath{\mathbf{x}}_j.
\end{equation}
In order to try to make sense of the infinite sum in \eqref{formula:SeriesExpansion}, one can fix a bijective mapping $\pi\colon\ensuremath{\mathbb{N}}\to J$ and study the convergence of the formal series
$\sum_{n=1}^\infty \ensuremath{\mathbf{x}}_{\pi(n)}^*(f)\ensuremath{\mathbf{x}}_{\pi(n)}$. If this series converges to $f$ for every $f\in\ensuremath{\mathbb{X}}$ then $\mathcal{B}$ is a \textit{Schauder basis} for the bijection~$\pi$. Schauder  bases are very well-known and have been widely studied. They are characterized as those bases for which the partial sum operators $S_{\pi,m}\colon\ensuremath{\mathbb{X}}\to\ensuremath{\mathbb{X}}$, given by
\begin{equation}
\label{definition:PartialSum}
 f\mapsto S_{\pi,m}(f)= \sum_{n=1}^m \ensuremath{\mathbf{x}}_{\pi(n)}^*(f)\ensuremath{\mathbf{x}}_{\pi(n)}
\end{equation}
are uniformly bounded. The property that  $\sum_{n=1}^\infty \ensuremath{\mathbf{x}}_{\pi(n)}^*(f)\ensuremath{\mathbf{x}}_{\pi(n)}$ converges for any $f\in\ensuremath{\mathbb{X}}$ and any bijection $\pi$
 yields the more restrictive
class of \textit{unconditional bases}. Recall that, equivalently,  a basis is unconditional if and only if
for every choice of signs $\varepsilon=(\varepsilon_j)_{j\in J}\in\{-1,1\}^J$ the multiplier
\[
P_\varepsilon\colon \ensuremath{\mathbb{X}}\to \ensuremath{\mathbb{X}}, \quad f\mapsto \sum_{j\in J} \varepsilon_j\ensuremath{\mathbf{x}}_j^*(f) \ensuremath{\mathbf{x}}_j
\]
is well defined and the family of operators $(P_\varepsilon)_{\varepsilon\in \{\pm 1\}^J}$ is uniformly bounded.

An \textit{ordering} for an element $f\in\ensuremath{\mathbb{X}}$ (with respect to a basis $\mathcal{B}$) is a one-to-one map $\rho\colon\ensuremath{\mathbb{N}}\to J$ such that $\operatorname{supp}(f) := \{j\in J : \ensuremath{\mathbf{x}}_j^*(f)\not=0\} \subseteq \rho(\ensuremath{\mathbb{N}})$.
From the point of view of  approximation theory, given a function $f$ in $\ensuremath{\mathbb{X}}$ and an ordering $\rho$ for $f$, the sequence $(S_{\rho,m}(f))_{m=1}^\infty$
constructed as in \eqref{definition:PartialSum} defines an algorithm to approximate to~$f$. The minimal requirement we must impose to $\rho$ is that  $(S_{\rho,m}(f))_{m=1}^\infty$ converges to~$f$. In case $\mathcal{B}$ is a Schauder basis for some bijection $\pi$, the algorithm based on $\pi$ fulfills this requirement for any $f\in\ensuremath{\mathbb{X}}$. The independence of the ordering from the vector determines both the goodness and the limitations of this approximation algorithm for Schauder bases. The operators $S_{\pi,m}$ are linear and uniformly bounded,
but it is natural to wonder if by allowing the ordering to depend on each particular vector we can attain a higher rate of convergence.

The most important algorithm based on letting the ordering depend on the vector is the \textit{greedy algorithm}, also known as the
\textit{thresholding algorithm}. Since for each $f\in\ensuremath{\mathbb{X}}$ the sequence $(\ensuremath{\mathbf{x}}^*_j(f))_{j\in J}$ belongs to $c_0(J)$, there is an ordering $\rho$ for $f$ such that
\begin{equation}
\label{greedycondition}
|\ensuremath{\mathbf{x}}_{\rho(k)}^{\ast}(f)|\ge |\ensuremath{\mathbf{x}}_{\rho(n)}^{\ast}(f)| \quad\text{if}\quad k\le n.
\end{equation}
If the family $(\ensuremath{\mathbf{x}}_j^*(f))_{j\in J}$ contains several terms with the same absolute value then such an ordering for $f$ is not uniquely determined.
In order to get uniqueness, we fix a ``natural'' bijection $\tau \colon J \to \ensuremath{\mathbb{N}}$, and we impose the additional condition
\begin{equation}
\label{desempate}
\tau(\rho(k))\le\tau(\rho(n)) \quad\text{whenever}\quad |\ensuremath{\mathbf{x}}_{\rho(k)}^{\ast}(f)|= |\ensuremath{\mathbf{x}}_{\rho(n)}^{\ast}(f)|.
\end{equation}
If $f$ is infinitely supported, there is a unique ordering $\rho$ for $f$ which fulfills \eqref{greedycondition} and \eqref{desempate}, and such an ordering verifies $\rho(\ensuremath{\mathbb{N}})=\operatorname{supp} (f)$. In the case in which $f$ is finitely supported, there is a unique ordering $\rho$ for $f$ which fulfills \eqref{greedycondition}, \eqref{desempate} and the extra property $\rho(\ensuremath{\mathbb{N}})=J$. In any case, we will refer to such a unique ordering as the \textit{greedy ordering} for~$f$.
For each $m\in \ensuremath{\mathbb{N}}$, the \textit{$m$-term greedy approximation} to $f$
is given by
\[
  \mathcal{G}_{m}[\mathcal{B}, \ensuremath{\mathbb{X}}](f):=\mathcal{G}_{m}(f) =S_{\rho,m}(f)= \sum_{n=1}^{m}\ensuremath{\mathbf{x}}_{\rho(n)}^{\ast}(f)\ensuremath{\mathbf{x}}_{\rho(n)},
\]
where $\rho$ is the greedy ordering for $f$, and the sequence $(\mathcal{G}_m(f))_{m=1}^\infty$ is called the greedy algorithm for $f$ with respect to the basis~$\mathcal{B}$.

Konyagin and Temlyakov \cite{KonyaginTemlyakov1999} defined a basis to be \textit{quasi-greedy} if $\lim_{m\to \infty}\mathcal{G}_{m}(f)=f$ for $f\in \ensuremath{\mathbb{X}}$, that is, the
greedy algorithm with respect to the basis $\mathcal{B}$ converges in the Banach space~$\ensuremath{\mathbb{X}}$. Subsequently, Wojtaszczyk \cite{Wo2000} proved that these are precisely the bases for which the greedy operators $(\mathcal{G}_m)_{m=1}^{\infty}$ are uniformly bounded i.e., there exists a constant $C\ge 1$ such that, for all $f\in \ensuremath{\mathbb{X}}$ and $m\in \ensuremath{\mathbb{N}}$,
\begin{equation}
\label{result:WoCondition}
\Vert \mathcal{G}_m(f)\Vert\le C \Vert f\Vert.
\end{equation}
Notice the similarity between \eqref{result:WoCondition} and the  characterization of Schauder bases.  However,  the operators $\mathcal{G}_m$ are neither linear nor continuous.
We emphasize that, as Wojtaszczyk pointed out in \cite{Wo2000}, the choice of the bijection $\tau$ with respect to which we construct the greedy algorithm $(\mathcal{G}_{m})_{m=1}^{\infty}$ plays no relevant role in the theory.

Unconditional bases are a special kind of quasi-greedy bases. Although the converse is not true in general, quasi-greedy bases always retain in a certain sense a flavor of unconditionality. For example, they are \textit{unconditional for constant coefficients} \cite{Wo2000}, i.e., there is a constant $C$ (to be precise $C=2 C_{w}$, where $C_{w}$ is the least constant in \eqref{result:WoCondition}, works) such that
\begin{equation}
\label{result:UCCCondition}
 \frac{1}{C}\left\Vert \sum_{j\in A} \ensuremath{\mathbf{x}}_{i}\right\Vert\le \left\Vert \sum_{j\in A}\varepsilon_{i} \ensuremath{\mathbf{x}}_{i}\right\Vert\le C \left\Vert \sum_{j\in A} \ensuremath{\mathbf{x}}_{i}\right\Vert
\end{equation}
for any finite subset $A$ of $J$ and any choice of signs $\varepsilon_j\in\{\pm1\}$.

Before the concept of quasi-greedy basis was introduced in the literature, C\'ordoba and Fern\'andez \cite{CoFe1998} had studied the convergence of decreasing rearranged Fourier series. For $k\in\ensuremath{\mathbb{Z}}$ let us define $\tau_k\colon\ensuremath{\mathbb{R}}\to\ensuremath{\mathbb{C}}$ by $\tau_k(x)=e^{2\pi k x i}$. Let $1\le p<\infty$ and denote by $q$ its conjugate exponent, determined by
$1/p+1/q=1$. Then, with the usual identification of  $L_p^*(\ensuremath{\mathbb{T}})$ with $L_q(\ensuremath{\mathbb{T}})$, the double sequence $(\tau_k,\tau_{-k})_{k=-\infty}^\infty$ is a normalized bounded biorthogonal system for $L_p(\ensuremath{\mathbb{T}})$. The authors of \cite{CoFe1998} showed that for each $1<p<2$ there is a function $f\in L_p(\ensuremath{\mathbb{T}})$ whose decreasing rearranged Fourier series does not converge, which in our language can be stated as saying that  the trigonometric system $(\tau_k)_{k=-\infty}^\infty$ is not a quasi-greedy basis for $L_p(\ensuremath{\mathbb{T}})$.
Combining the condition characterizing quasi-greedy bases \eqref{result:WoCondition} with \cite{Telmyakov1998}*{Remark~2}, the result extends to the whole range of $p\in[1,\infty]\setminus\{2\}$ (replacing $L_p(\ensuremath{\mathbb{T}})$ with $\ensuremath{\mathcal{C}}(\ensuremath{\mathbb{T}})$ when $p=\infty$).
Wojtaszczyk gave a different proof of this result in \cite{Wo2000} that relies on~\eqref{result:UCCCondition}.

A natural way to continue this line of research is to consider Fourier series with respect to orthonormal bases. Let $(X,\Sigma,\mu)$ be a measure space such that the Hilbert space $L_2(\mu)$ is separable. Let
$(\ensuremath{\mathbf{x}}_j)_{j\in J}$ be an orthonormal basis of $L_2(\mu)$.
For $1\le p< \infty$, let $q$ be its conjugate exponent.
In case that  $\operatorname{span} \{ \ensuremath{\mathbf{x}}_j : j\in J\}$ is dense in $L_p(\mu)$ and
\[
\sup_{j\in J} \Vert \ensuremath{\mathbf{x}}_j \Vert_p \Vert \ensuremath{\mathbf{x}}_j \Vert_q<\infty,
\]
the identification of $L_p^*(\mu)$ with $L_q(\mu)$, yields that $(\ensuremath{\mathbf{x}}_j ,\overline{\ensuremath{\mathbf{x}}_j})_{j\in J}$ is a bounded biorthogonal system for $L_p(\mu)$. It therefore makes sense to investigate the convergence of the greedy algorithm with respect to the $L_p(\mu)$-normalized system
\[
\left( \Vert \ensuremath{\mathbf{x}}_j \Vert_p^{-1} {\ensuremath{\mathbf{x}}_j}, \Vert \ensuremath{\mathbf{x}}_j\Vert_p \overline{ \ensuremath{\mathbf{x}}_j } \right)_{j\in J}.
\]
Notice that if the measure $\mu$ is finite and the orthonormal basis $(\ensuremath{\mathbf{x}}_j)_{j\in J}$
 is \textit{uniformly bounded}, i.e.\ $\sup_{j\in J} \Vert \ensuremath{\mathbf{x}}_j\Vert_\infty<\infty$, then it is semi-normalized and bounded in $L_p(\mu)$ for any $1\le p<\infty$.
Nielsen \cite{Nielsen2007} proved that there is an uniformly bounded orthonormal basis of $L_2(\ensuremath{\mathbb{T}})$ which is quasi-greedy for $L_p(\ensuremath{\mathbb{T}})$ for any $1<p<\infty$, thus
exhibiting a behavior opposite to that of the trigonometric system.

In this paper we focus on Jacobi polynomials.
Recall that, for  scalars $\alpha$, $\beta>-1$, the  $L_2(\mu_{\alpha,\beta})$-normalized Jacobi polynomials
$(p_n^{(\alpha,\beta)})_{n=0}^\infty$ appear as the orthonormal polynomials associated to the measure $\mu_{\alpha,\beta}$ given by
\begin{equation}
\label{measureJacobi}
d\mu_{\alpha,\beta}(x)=(1-x)^\alpha (1+x)^\beta \chi_{(-1,1)} (x) \, dx.
\end{equation}
Since polynomials are dense in $L_p(\mu_{\alpha,\beta})$ for any $1\le p<\infty$, Jacobi polynomials of indices $\alpha$ and $\beta$ constitute an orthonormal basis of $L_2(\mu_{\alpha,\beta})$. Our main result on Jacobi polynomials establishes that the greedy algorithm for this kind of orthogonal polynomials follows the same pattern as the greedy algorithm for the trigonometric system.

\begin{theorem}
\label{result:MainTheorem}
Let $1\le p<\infty$ and $\min\{\alpha,\beta\}> -1/2$. The $L_p(\mu_{\alpha,\beta})$-normalized Jacobi polynomials of indices $\alpha$ and $\beta$ form a quasi-greedy basis for $L_p(\mu_{\alpha,\beta})$ if and only if $p=2$.
\end{theorem}

Section~\ref{Jacobi} is devoted to prove Theorem~\ref{result:MainTheorem}.
Before, in Section~\ref{UnconditionalLp} we develop
the functional analysis machinery that we will need in order to do that and
we show the following result on unconditional bases in $L_p(\mu)$-spaces.

\begin{theorem}
\label{NotSimultaneouslyBounded}Let $\mu$ be a finite measure and $p\in(1,\infty)\setminus\{2\}$. Suppose that $(\ensuremath{\mathbf{x}}_j)_{j\in J}$ is a semi-normalized unconditional basis of a non-Hilbertian Banach space $\ensuremath{\mathbb{X}}\subseteq L_p(\mu)$. Suppose also that $\ensuremath{\mathbb{X}}$ is complemented in $L_p(\mu)$. Then
\begin{itemize}
\item[(i)] $\limsup_{j\in J} \Vert \ensuremath{\mathbf{x}}_j\Vert_q=\infty$ for any $p<q$, and
\item[(ii)] $\liminf_{j\in J} \Vert \ensuremath{\mathbf{x}}_j\Vert_q=0$ whenever $\max\{q,2\}<p$.
\end{itemize}
\end{theorem}
Notice that Theorem~\ref{NotSimultaneouslyBounded} is relevant for its intrinsic importance within the framework of the theory of bases. Firstly, it extends to any $q$ a result that Kadets and Pe{\l}czy{\'n}ski proved only for $q=2$ (see \cite{KadPel1962}*{Corollary 9}). Secondly, it  generalizes the main result of Gapo\v{s}kin in \cite{Gaposkin1958}, where he shows that no normalized unconditional basis in
$L_p[0,1]$ can be uniformly bounded. Lastly, for finite measures, Theorem~\ref{NotSimultaneouslyBounded} overrides a recent result of the first two authors that says that if $\mu$ is a nonpurely atomic measure then there is no basis $\mathcal{B}$ that is simultaneously greedy (see the definition below) in two different $L_{p}(\mu)$ spaces, $1<p<\infty$ \cite{AA2015}*{Theorem~4.4}.

We end this preliminary section by singling out some notation and terminology that will be used heavily throughout.
Given families of positive real numbers $(\alpha_i)_{i\in I}$ and $(\beta_i)_{i\in I}$, the symbol $\alpha_i\lesssim \beta_i$ for $i\in I$ means that
$\sup_{i\in I}\alpha_i/\beta_i <\infty$, while $\alpha_i\approx \beta_i$ for $i\in I$ means that $\alpha_i\lesssim \beta_i$ and $\beta_i\lesssim \alpha_i$ for $i\in I$.

A basis $\mathcal{B}=(\ensuremath{\mathbf{x}}_j)_{j\in J}$ in a Banach space $\ensuremath{\mathbb{X}}$ is said to be \textit{democratic} if
there is a constant $D\ge 1$ such that
\[
\left\Vert \sum_{j\in A}\ensuremath{\mathbf{x}}_j \right\Vert \le D \left\Vert \sum_{j\in B} \ensuremath{\mathbf{x}}_j \right\Vert
\]
whenever $A$ and $B$ are finite subsets of $J$ with $|A|=|B|$.
To quantify the democracy of a basis $\mathcal{B}$ we consider the \textit{upper democracy function} of~$\mathcal{B}$ (also known as the \textit{fundamental function of~$\mathcal{B}$}) given by
\[
 \varphi_{u}[\mathcal{B},\ensuremath{\mathbb{X}}](N)=\sup_{|A|\le N}\left\Vert \sum_{j\in A}\ensuremath{\mathbf{x}}_j\right\Vert, \qquad N\in\ensuremath{\mathbb{N}},
\]
and the \textit{lower democracy function} of $\mathcal B$ in $\ensuremath{\mathbb{X}}$, defined as
\[
 \varphi_{l}[\mathcal{B},\ensuremath{\mathbb{X}}](N)=\inf_{|A|\ge N}\left\Vert \sum_{j\in A}\ensuremath{\mathbf{x}}_j\right\Vert, \qquad N\in\ensuremath{\mathbb{N}}.
\]
A quasi-greedy basis $\mathcal B$ is democratic if and only if $\varphi_{u}[\mathcal{B},\ensuremath{\mathbb{X}}](N)\approx \varphi_{l}[\mathcal{B},\ensuremath{\mathbb{X}}](N)$ for $N\in\ensuremath{\mathbb{N}}$.

A basis $(\ensuremath{\mathbf{x}}_n)_{n=1}^\infty$ is said to be \textit{almost greedy} if there is a constant $C\ge 1$ such that
\[
 \Vert x-\mathcal{G}_m(x)\Vert \le C \inf\left\{\left \Vert x-\sum_{j\in A} \ensuremath{\mathbf{x}}_j^*(x) \ensuremath{\mathbf{x}}_j\right\Vert : |A|=m\right\}
\]
for all $m\in \ensuremath{\mathbb{N}}$ and $x\in \ensuremath{\mathbb{X}}$.
Dilworth et al.\ \cite{DKKT2003} characterized almost greedy basis as those bases that are simultaneously quasi-greedy and democratic.

Finally, the best one can hope for in regards to the greedy algorithm is the existence of a constant $C\ge 1$ such that
\[
 \Vert x-\mathcal{G}_m(x)\Vert \le C \inf\left\{\left \Vert x-\sum_{j\in A} a_{j} \ensuremath{\mathbf{x}}_j\right\Vert : |A|=m,\; (a_{j})_{j\in A}\;\;\text{scalars}\right\}
 \]
for all $m\in \ensuremath{\mathbb{N}}$ and $x\in \ensuremath{\mathbb{X}}$. If this is the case, the basis is called \textit{greedy}. Konyagin and Temlyakov \cite{KonyaginTemlyakov1999} characterized greedy bases as those bases that are unconditional and democratic.

If necessary, the reader will find more background on Banach space theory in \cite{AlbiacKalton2006}
and on orthogonal polynomials  in~\cite{Szego}.

\section{Quasi-greedy and unconditional bases in $L_p(\mu)$-spaces}
\label{UnconditionalLp}

\noindent We start generalizing to quasi-greedy bases a fact which is standard for unconditional bases in $L_p(\mu)$-spaces.

\begin{lemma}
\label{result:Random}
Let $(X,\Sigma,\mu)$ be a finite measure space. Let $1\le p<\infty$ and $(\ensuremath{\mathbf{x}}_j)_{j\in J}$ be a quasi-greedy basis for
a separable subspace of $L_p(\mu)$. Then for $A\subseteq J$ finite,
\[
\left\Vert \sum_{j\in A}\ensuremath{\mathbf{x}}_j \right\Vert_p
\approx \left\Vert \left(\sum_{j\in A} |\ensuremath{\mathbf{x}}_j|^2\right)^{1/2}\right\Vert_p.
\]
\end{lemma}

\begin{proof}
Let $(\varepsilon_j)_{j\in J}$ be a Rademacher family defined on some probability space $(\Omega,P)$, and $A\subseteq J$ finite. Combining \eqref{result:UCCCondition}, Fubini's
theorem, and Khintchine's inequality yields
\begin{align*}
\left\Vert \sum_{j\in A}\ensuremath{\mathbf{x}}_j \right\Vert_p
&\approx \left( \int_\Omega \Big\Vert \sum_{j\in A}\varepsilon_j \ensuremath{\mathbf{x}}_j \Big\Vert_p^p\, dP\right)^{1/p}\\
&= \left(\int_X \int_\Omega \Big| \sum_{j\in A} \varepsilon_j \ensuremath{\mathbf{x}}_j\Big|^p \, dP \,d\mu\right)^{1/p}\\
&\approx \left\Vert \left(\sum_{j\in A} |\ensuremath{\mathbf{x}}_j|^2\right)^{1/2}\right\Vert_p.
\qedhere
\end{align*}
\end{proof}

Our next auxiliary result displays an estimate that is implied when a family of functions is simultaneously seminormalized in two different $L_{p}$ spaces.

\begin{lemma}
\label{result:PreviousDemocracyEstimates}
Let $1\le p< q\le 2$ (respectively, $2\le q <p\le\infty$) and let $(f_j)_{j\in J}$ be a family of measurable functions defined on a finite measure space $(X,\Sigma,\mu)$. Suppose that
$\Vert f_j\Vert_p\approx \Vert f_j \Vert_q\approx 1$ for $j\in J$. Then, for $A\subseteq J$ finite,
\[
|A|^{1/2} \lesssim \left\Vert \left( \sum_{j\in A}| f_j|^2\right)^{1/2}\right\Vert_p \lesssim |A|^{1/q}
\]
(respectively,
\[
|A|^{1/q} \lesssim \left\Vert \left( \sum_{j\in A}| f_j|^2\right)^{1/2}\right\Vert_p \lesssim |A|^{1/2}).
\]
\end{lemma}

\begin{proof}
Assume $1\le p<q\le 2$. Using the embeddings $\ell_q\subseteq\ell_2$ and $L_q(\mu)\subseteq L_p(\mu)$,
\begin{equation*}
\left\Vert \left(\sum_{j\in A} | f_j|^2\right)^{1/2}\right\Vert_p
\lesssim \left\Vert \left(\sum_{j\in A} | f_j|^q\right)^{1/q}\right\Vert_q
=\left(\sum_{j\in A} \Vert f_j\Vert_q^q\right)^{1/q}
\approx |A|^{1/q}.
\end{equation*}
Let $r=p/2<1$. Using that $\Vert f+g\Vert_r \ge \Vert f\Vert_r +\Vert g\Vert_r$ whenever $f$ and $g$ are measurable positive functions,
\begin{align*}
 \left\Vert \left(\sum_{j\in A} | f_j|^2\right)^{1/2}\right\Vert_p
&= \left\Vert \sum_{j\in A} | f_j|^2\right\Vert_r^{1/2}\\
&\ge \left( \sum_{j\in A} \Vert \,  |f_j|^2 \, \Vert_r \right)^{1/2}\\
&=\left( \sum_{j\in A} \Vert f_j \Vert_p^2 \right)^{1/2}\\
&\approx |A|^{1/2}.
\end{align*}

The case $2\le q<p\le\infty$ follows from a ``dual'' argument.
\end{proof}

\begin{lemma}
\label{result:DemocracyEstimates}
Let $(X,\Sigma,\mu)$ be a finite measure space. Suppose $1\le p< q\le 2$ (respectively, $2\le q <p<\infty$). Let  $(\ensuremath{\mathbf{x}}_j)_{j\in J}$ be a quasi-greedy basis for
a separable subspace $\ensuremath{\mathbb{X}}$ of $L_p(\mu)$ such that$\Vert \ensuremath{\mathbf{x}}_j \Vert_q\approx 1$ for $j\in J$. Then, for $N\in\ensuremath{\mathbb{N}}$,
\[
N^{1/2} \lesssim \varphi_{l}[\mathcal{B},\ensuremath{\mathbb{X}}](N) \le \varphi_{u}[\mathcal{B},\ensuremath{\mathbb{X}}](N) \lesssim N^{1/q}
\]
(respectively,
\[
N^{1/q} \lesssim \varphi_{l}[\mathcal{B},\ensuremath{\mathbb{X}}](N) \le \varphi_{u}[\mathcal{B},\ensuremath{\mathbb{X}}](N) \lesssim N^{1/2}).
\]
\end{lemma}

\begin{proof}
Quasi-greedy bases are semi-normalized, so $\Vert \ensuremath{\mathbf{x}}_j \Vert_p\approx 1$ for $j \in J$.
Then, just put together Lemma~\ref{result:Random} and Lemma~\ref{result:PreviousDemocracyEstimates}.
\end{proof}

The next two propositions are on-the-spot corollaries of Lemma~\ref{result:PreviousDemocracyEstimates} and Lemma~\ref{result:DemocracyEstimates}, respectively. We point out that a similar statement to Proposition~\ref{result:Democracy} with the stronger assumption that the basis be uniformly bounded was obtained by Dilworth et al.\ \cite{DST2012}*{Proposition~2.17}.

\begin{proposition}
\label{result:LdosSumEstimate}
Let $1\le p\le \infty$. Suppose $(f_j)_{j\in J}$ is a family of measurable functions defined on a finite measure space $(X,\Sigma,\mu)$ such that
$\Vert f_j\Vert_p\approx \Vert f_j \Vert_2\approx 1$ for $j\in J$. Then for $A\subseteq J$ finite,
\[
 \left\Vert \left( \sum_{j\in A}| f_j|^2\right)^{1/2}\right\Vert_p \approx |A|^{1/2}.
\]
\end{proposition}

\begin{proposition}
\label{result:Democracy}
Let $(X,\Sigma,\mu)$  be a finite measure space and let $1\le p<\infty$. Suppose  $\mathcal{B}=(\ensuremath{\mathbf{x}}_j)_{j\in J}$ is a quasi-greedy basis for
a separable subspace $\ensuremath{\mathbb{X}}$ of $L_p(\mu)$ with $\Vert \ensuremath{\mathbf{x}}_j \Vert_2\approx 1$ for $j\in J$.
Then $\mathcal{B}$ is democratic, hence almost greedy, and
its democracy functions verify
\[
\varphi_l[\mathcal{B},\ensuremath{\mathbb{X}}](N)\approx N^{1/2}\approx \varphi_u[\mathcal{B},\ensuremath{\mathbb{X}}](N),\quad N\in\ensuremath{\mathbb{N}}.
\]
\end{proposition}

We are now en route to completing the proof of Theorem~\ref{NotSimultaneouslyBounded}. Before we do so, we write down two classical results in the isomorphic theory of Banach spaces which are very well-known to the specialists. In order to make the paper as self-contained as possible we sketch their proofs.

\begin{theorem}
\label{result:KaPeSubbasis}
Let $(X,\Sigma,\mu)$ be a measure space. Suppose that $\mathcal{B}$ is a seminormalized unconditional basis of a non-Hilbertian Banach space $\ensuremath{\mathbb{X}}\subseteq L_p(\mu)$, $1<p<\infty$. Suppose also that $\ensuremath{\mathbb{X}}$ is complemented in $L_p(\mu)$. Then
$\mathcal{B}$ has a subbasis equivalent to the unit vector basis of~$\ell_p$.
\end{theorem}

\begin{proof}
 Without loss of generality we may and do assume that $L_p(\mu)$ is separable. Then $L_p(\mu)$ is isomorphic either to $L_p[0,1]$ or to~$\ell_p$
(see~\cite{JohnsonLindes2001}).
In the first case, the same argument used by Kadec and Pe{\l}czy{\'n}ski to prove \cite{KadPel1962}*{Theorem~4} leads to our goal. In the last case, by the Bessaga-Pe{\l}czy{\'n}ski selection principle (\cite{BePe1958}*{p.~214}), $\mathcal{B}$ has a subbasis equivalent to a block basic sequence of the unit vector basis of~$\ell_p$.
Since the unit vector basis of $\ell_p$ is perfectly homogeneous, this subbasis is equivalent to the unit vector basis of~$\ell_p$.
\end{proof}

\begin{lemma}
\label{result:EquivalentNorms}
Let $(X,\Sigma,\mu)$ be a finite measure space and let $0<p<q\le \infty$. Consider a subset $M\subseteq L_p(\mu)$ such that $\Vert f\Vert_p\approx\Vert f\Vert_q$ for $f\in M$. Then, for any $0<r\le q$, $\Vert f\Vert_r \approx\Vert f\Vert_p\approx\Vert f\Vert_q$ for $f\in M$.
\end{lemma}

\begin{proof}
The result is obvious for $p\le r\le q$, so we assume that $r<p$. Then, it is also obvious that $\Vert f\Vert_r\lesssim \Vert f\Vert_p$ for $f\in M$. To prove the reverse inequality, consider $0<a<p$ such that $a/q+(p-a)/r=1$.
Let $f\in M$. By H\"older's inequality,
\begin{align*}
\Vert f\Vert_p^p
&=\int_X |f|^a |f|^{p-a} \, d \mu\\
&\le \left( \int_X |f|^{q}\, d\mu \right)^{a/q} \left( \int_X |f|^{r}\, d\mu \right)^{(p-a)/r}\\
&=\Vert f\Vert_q^a \Vert f\Vert_r^{p-a}\\
&\lesssim \Vert f\Vert_p^a \Vert f\Vert_r^{p-a}.
\end{align*}
Simplifying, we get $\Vert f\Vert_p^{p-a}\lesssim \Vert f\Vert_r^{p-a}$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{NotSimultaneouslyBounded}]
Let $(\ensuremath{\mathbf{x}}_j)_{j\in J}$ be a semi-normalized unconditional basis of a non-Hilbertian Banach space $\ensuremath{\mathbb{X}}\subseteq L_p(\mu)$, where $p\in(1,\infty)\setminus\{2\}$ and $\mu$ is finite.
We divide the proof in three cases.

\smallskip

\noindent\textsc{Case 1:} $1<p<2$ and $p<q$. Assume that $\limsup_j \Vert x_j\Vert_q<\infty$.
We can suppose, without loss of generality, that $p<q\le 2$.
Let $J_0=\{ j : \Vert \ensuremath{\mathbf{x}}_j\Vert_q<\infty \}$, $\mathcal{B}_0=(\ensuremath{\mathbf{x}}_j)_{j\in J_0}$ and then define
$\ensuremath{\mathbb{X}}_0$ as the closed subspace spanned by $\mathcal{B}_0$ in $L_p(\mu)$.
We have that $J\setminus J_0$ is finite and that
$\Vert \ensuremath{\mathbf{x}}_j\Vert_q\approx 1$ for $j\in J_0$. By Lemma~\ref{result:DemocracyEstimates}, $\varphi_u[\mathcal{B}_0,\ensuremath{\mathbb{X}}_0](N)\lesssim N^{1/q}$. Furthermore, by
Theorem~\ref{result:KaPeSubbasis}, $\mathcal{B}_0$ has a subbasis equivalent to the unit vector basis of $\ell_p$. Therefore,
$N^{1/p}\lesssim \varphi_u[\mathcal{B}_0,\ensuremath{\mathbb{X}}_0](N)$. Combining, we obtain $N^{1/p}\lesssim N^{1/q}$. This absurdity proves the result.

\smallskip

\noindent\textsc{Case 2:} $2<p<\infty$ and $q<p$. This case is the ``dual'' of the previous one. Since  its proof is similar we leave it out to the reader.

\smallskip

\noindent\textsc{Case 3:} $2<p<q$. Suppose that $\limsup_j \Vert x_j\Vert_q<\infty$. Removing a finite set of terms from $\mathcal{B}$ we get
$\Vert \ensuremath{\mathbf{x}}_j\Vert_q\approx 1$. By Lemma~\ref{result:EquivalentNorms}, $\Vert \ensuremath{\mathbf{x}}_j\Vert_2 \approx 1$, contradicting the already proven Case~2.
\end{proof}

\begin{remark}
The proof of Theorem~\ref{NotSimultaneouslyBounded}
reinforces the role of  democracy as a hinge property in the study of unconditional bases in Banach spaces. This idea was already inferred in the work of Zippin~\cite{Zippin1966}, where he characterizes perfectly homogeneous bases.
\end{remark}

We close with the analogous result to Theorem~\ref{NotSimultaneouslyBounded} for the  case $p=1$.
To better understand the statement and its proof we recall that, given $1\le p\le \infty$, an infinite-dimensional Banach space $\ensuremath{\mathbb{X}}$ is said to be a \textit{$\mathscr{L}_p$-space} if there is $\lambda\ge 1$ such that for every finite-dimensional subspace $E\subseteq \ensuremath{\mathbb{X}}$ there is $d\in\ensuremath{\mathbb{N}}$ and a $d$-dimensional subspace
$E\subseteq F\subseteq \ensuremath{\mathbb{X}}$ that satisfies $d(F,\ell_p^d)\le \lambda$. For $1<p<\infty$, $\mathscr{L}_p$-spaces are characterized as non-Hilbertian complemented subspaces of $L_p(\mu)$-spaces, while $\ensuremath{\mathbb{X}}$ is an $\mathscr{L}_1$-space if and only if
$\ensuremath{\mathbb{X}}^{**}$ is isomorphic to a complemented subspace of an $L_1(\mu)$-space. Hence, since $\ensuremath{\mathbb{X}}$ embeds isometrically in $\ensuremath{\mathbb{X}}^{**}$, it is natural to regard
$\mathscr{L}_1$-spaces as (possibly non complemented) subspaces of $L_1(\mu)$-spaces.
A fundamental property is that any $\mathscr{L}_1$-space has the \textit{Grothendieck's Theorem property} (is a GT-space, for short).
 We refer to \cites{LinPel1968, LinRos1969} for details.

\begin{proposition}
\label{result:NotBoundedOne}
Let $\mu$ be a finite measure. Suppose that $\mathcal{B}=(\ensuremath{\mathbf{x}}_j)_{j\in J}$ is a quasi-greedy basis for a Banach space
$\ensuremath{\mathbb{X}}\subseteq L_1(\mu)$.
Assume also that $\ensuremath{\mathbb{X}}$ is a GT-space.
Then $\limsup_{j\in J} \Vert \ensuremath{\mathbf{x}}_j\Vert_q=\infty$ for any $1< q\le\infty$.
\end{proposition}

\begin{proof}
Appealing to \cite{DST2012}*{Theorem~4.2} we get
$\varphi_u[\mathcal{B},\ensuremath{\mathbb{X}}](N)\approx N$.
Assume that $\limsup_{j\in J} \Vert \ensuremath{\mathbf{x}}_j\Vert_q<\infty$ for some $1<q\le\infty$.
Then, without lost of generality we can suppose that $q\le 2$ and, by removing a finite set of terms from $\mathcal{B}$ if necessary, that $\Vert \ensuremath{\mathbf{x}}_j \Vert_q\approx 1$. Applying Lemma~\ref{result:DemocracyEstimates} we get $\varphi_u[\mathcal{B},\ensuremath{\mathbb{X}}](N)\lesssim N^{1/q}$, which leads to $N\lesssim N^{1/q}$, a contradiction.
\end{proof}

\begin{remark}
The subspace spanned by the Rademacher functions in $L_{p}(\mu)$ serves as an example to show that the assumption
``non-Hilbertian'' cannot be dropped from
Theorem~\ref{NotSimultaneouslyBounded} and that the assumption of
$\ensuremath{\mathbb{X}}$ being a GT-space cannot be dropped from
Proposition~\ref{result:NotBoundedOne}.
\end{remark}

\section{The greedy algorithm for Jacobi polynomials}
\label{Jacobi}

\noindent In this section, besides the orthonormal polynomials $(p_n^{(\alpha,\beta)})_{n=0}^\infty$ defined in Section~\ref{Introduction}, we consider
 the polynomials $(P_n^{(\alpha,\beta)})_{n=0}^\infty$ which are orthogonal for the measure
$\mu_{\alpha,\beta}$ defined in \eqref{measureJacobi} and verify the \textit{normalization condition}
\begin{equation}
\label{ValueInOne}
P_n^{(\alpha,\beta)} (1)=\binom{n+\alpha}{n}, \quad n\in \ensuremath{\mathbb{N}}\cup\{0\}.
\end{equation}
Of course, there are positive scalars $d_n$ such that $p_n^{(\alpha,\beta)}=d_n P_n^{(\alpha,\beta)}$, $n\ge 0$. It is well known that the normalization sequence
$(d_n)_{n=0}^\infty $ verifies
\begin{equation}
\label{eq:asym-dn}
d_n
= \Big(\frac {(2n+\alpha+\beta+1)n!\,\Gamma(n+\alpha+\beta+1)}
{2^{\alpha+\beta+1}\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}
 \Big)^{1/2}
\approx n^{1/2}, \, n\in\ensuremath{\mathbb{N}}.
\end{equation}
In what follows, with the aim of avoid cumbrous notations, we will denote by $(n^{1/2} P_n^{(\alpha,\beta)})_{n=0}^\infty$ the extension of $(n^{1/2} P_n^{(\alpha,\beta)})_{n=1}^\infty$ whose $0$-term
is the constant function~$1$.
In the light of \eqref{eq:asym-dn},
it is reasonable to expect that the sequences $(p_n^{(\alpha,\beta)})_{n=0}^\infty$ and $(n^{1/2} P_n^{(\alpha,\beta)})_{n=0}^\infty$ behave similarly.
 Wojtaszczyk \cite{Wo2000} confirmed this fact by showing that quasi-greedy bases verify the following perturbation principle.

\begin{theorem}[cf.~\cite{Wo2000}*{Proposition~3}]
\label{result:Perturbing}
Suppose that $(\ensuremath{\mathbf{x}}_j)_{j\in J}$ is a quasi-greedy basis for a Banach space~$\ensuremath{\mathbb{X}}$. Let $(\lambda_j)_{j\in J}$ be a family of scalars such that
\[
0<\inf_{j\in J} |\lambda_j|\le \sup_{j\in J} |\lambda_j|<\infty.
\]
Then $(\lambda_j \ensuremath{\mathbf{x}}_j)_{j\in J}$ is a quasi-greedy basis for $\ensuremath{\mathbb{X}}$.
\end{theorem}

A powerful tool to carry out estimates involving Jacobi polynomials is the so called \textit{Darboux formula}. The next theorem establishes an expression for the
error term associated to this formula which is accurate enough for our purposes.

\begin{theorem}[cf.~\cite{Szego}*{Theorem~8.21.13}]
\label{result:Darboux}
Let $\alpha$, $\beta>-1$ and $\delta>0$. Then
\begin{equation*}
n^{1/2} P_n^{(\alpha,\beta)}(\cos \theta)
= k(\theta)  \cos(n\theta+\phi(\theta))+ E_n(\theta)
\end{equation*}
with
\begin{align*}
\phi(\theta)
&=(\alpha+\beta+1)\theta/2-(2 \alpha+1)\pi/4, \\
k(\theta)
&= \pi^{-1/2}(\sin \tfrac{\theta}{2})^{-\alpha-1/2}
(\cos \tfrac{\theta}{2})^{-\beta-1/2},
\end{align*}
and the error term $E_n(\theta)$ verifies
\[
E_n(\theta) = \frac{k(\theta)}{n \sin \theta} \, O(1)
\]
for $n\in\ensuremath{\mathbb{N}}$, where the $O(1)$ holds uniformly in the interval
${\delta}/{n}\le \theta \le \pi-{\delta}/{n}$.
\end{theorem}

Darboux formula provides tight estimates for Jacobi polynomials when the variable is not too close to the endpoints  $-1$ and $1$. The technique to estimate Jacobi polynomials near $1$ is also well-known for experts. It is based on the formula
\begin{equation}
\label{derivative}
\left(P_n^{(\alpha,\beta)}\right)'= (1+\alpha+\beta+n) P_{n-1}^{(\alpha+1,\beta+1)}
\end{equation}
and the behavior of the roots of Jacobi polynomials. In the following lemma we reproduce this standard argument for the sake of completeness.

\begin{lemma}
\label{result:NearOneEstimate}
Let $\alpha>-1$ and $\beta>-1$. There is $d>0$ such that
\[
P_n^{(\alpha,\beta)}(x)\approx n^\alpha
\]
for $n\in\ensuremath{\mathbb{N}}$ and $1-d/n^2 \le x \le 1$.
\end{lemma}

\begin{proof}
Let  $z_n$ denote the largest root of $ P_n^{(\alpha,\beta)}$ and let $\gamma_n\in(0,\pi)$ be such that $\cos(\gamma_n)=z_n$.
It is known (see \cite{Szego}*{Theorem~8.9.1}) that $\gamma_n\approx 1/n$ and, consequently, $1-z_n\approx 1/n^2$.
Moreover, it is easy to deduce from \eqref{ValueInOne} and \eqref{derivative} that
\[
P_n^{(\alpha,\beta)}(1)\approx n^\alpha\quad\text{and}\quad(P_n^{(\alpha,\beta)})'(1)\approx n^{\alpha+2}.
\]
Choosing $d>0$ small enough we get
$z_n \le 1-d/n^2$ and
\[
n^\alpha\lesssim P_n^{(\alpha,\beta)}(1) - \frac{d}{n^2} (P_n^{(\alpha,\beta)})'(1).
\]
Let $1-d/n^2 \le x \le 1$.
Since $0\le (P_n^{(\alpha,\beta)})'(t)\le (P_n^{(\alpha,\beta)})'(1)$ for any $t\in[x,1]$,
\[
P_n^{(\alpha,\beta)}(x)
= P_n^{(\alpha,\beta)}(1) - \int_x^1 (P_n^{(\alpha,\beta)})'(t)\, dt
\ge P_n^{(\alpha,\beta)}(1) - \frac{d}{n^2} (P_n^{(\alpha,\beta)})'(1).
\]
For the reverse inequality we note that $P_n^{(\alpha,\beta)}(x)\le P_n^{(\alpha,\beta)}(1)\approx n^\alpha$.
\end{proof}

Darboux formula allows us to compute the $L_p(\mu_{\alpha,\beta})$-norms of Jacobi polynomials.
Let $\alpha,\beta$ be such that $\min\{\alpha,\beta\}> -1/2$ and denote
\[
p(\alpha,\beta)=\frac{4(\gamma+1)}{2\gamma+3}, \quad
q(\alpha,\beta)=\frac{4(\gamma+1)}{2 \gamma +1}, \quad\text{ where }\gamma=\max\{\alpha,\beta\}.
\]
Notice that $p(\alpha,\beta)$ and $q(\alpha,\beta)$ are conjugate exponents.
 We have (cf.~\cite{Leves-Kus}) that, for $n\ge 2$,
\begin{equation}
\label{result:JacobiBasisNorm}
\begin{aligned}
 \Vert p_n^{(\alpha,\beta)} \Vert_{L_p(\mu_{\alpha,\beta})}
&\approx n^{1/2}\Vert P_n^{(\alpha,\beta)} \Vert_{L_p(\mu_{\alpha,\beta})} \\
&\approx
 \begin{cases} 1, & \text{ if } 1\le p<q(\alpha,\beta), \\
 (\log n)^{1/p}, & \text{ if } p=q(\alpha,\beta),\\
 n^{(2 \gamma+1)/2-2( \gamma+1)/p}, & \text{ if } q(\alpha,\beta)<p<\infty.
\end{cases}
\end{aligned}
\end{equation}

An elementary consequence of \eqref{result:JacobiBasisNorm} is that Jacobi polynomials,
when $\min\{\alpha, \beta\}<-1/2$, are not uniformly bounded.
Using the terminology of bases, \eqref{result:JacobiBasisNorm} yields the following result.

\begin{lemma}
\label{result:NormalizationAndBoundedness}
Let $1\le p< \infty$ and  $\min\{\alpha, \beta\}>-1/2$.  Then
\begin{itemize}
\item[(a)] The Jacobi polynomials of indices $\alpha$ and $\beta$ form a bounded basis for $L_p(\mu_{\alpha,\beta})$ if an only if
$p(\alpha,\beta)<p<q(\alpha,\beta)$.
\item[(b)] If $p(\alpha,\beta)<p<q(\alpha,\beta)$, then both $(p_n^{(\alpha,\beta)})_{n=0}^\infty$ and $(n^{1/2} P_n^{(\alpha,\beta)})_{n=0}^\infty$
 form a semi-normalized bounded basis for $L_p(\mu_{\alpha,\beta})$.
\end{itemize}
\end{lemma}

\begin{remark}
Notice that the range of indices $p$ for which the Jacobi polynomials with $\min\{\alpha,\beta\}>-1/2$ form a bounded basis for $L_p(\mu_{\alpha,\beta})$ coincides with the range of indices for which they are a Schauder basis of $L_p(\mu_{\alpha,\beta})$ with the natural order (cf.~\cite{Pollard}).
\end{remark}

\begin{remark}
Lemma~\ref{result:NormalizationAndBoundedness}, combined with either Theorem~\ref{NotSimultaneouslyBounded} or~\cite{KadPel1962}*{Corollary~9}, gives that Jacobi polynomials with $\min\{\alpha,\beta\}>-1/2$ are not an unconditional basis for $L_p(\mu_{\alpha,\beta})$ unless $p=2$.
This result can also be obtained from \cite{KonigNielsen1994}*{Proposition~4}.
\end{remark}

Lemma~\ref{result:Random} provides a tool to check if a basis is a suitable candidate to be quasi-greedy, and leads us to compare norms of the form
$\Vert \sum_{j\in A} \ensuremath{\mathbf{x}}_j \Vert_p$ with norms of the form $\Vert (\sum_{j\in A} |\ensuremath{\mathbf{x}}_j|^2)^{1/2} \Vert_p$.
In this direction, we state the following result.

\begin{proposition}
\label{result:AverageJacobiNorm}
Let  $\alpha$, $\beta$ and $p$ be such that $\min\{\alpha, \beta\}>-1/2$ and
  $1\le p<q(\alpha,\beta)$.
Then, for $A\subseteq \ensuremath{\mathbb{N}}$ finite,
\[
\left\Vert\left(\sum_{n\in A} \left(n^{1/2} P_n^{(\alpha,\beta)}\right)^2\right)^{1/2}\right\Vert_{L_p(\mu_{\alpha,\beta})} \approx
\left\Vert\left(\sum_{n\in A} \left(p_n^{(\alpha,\beta)}\right)^2\right)^{1/2}\right\Vert_{L_p(\mu_{\alpha,\beta})} \approx |A|^{1/2}.
\]
\end{proposition}

\begin{proof}
Just combine Proposition~\ref{result:LdosSumEstimate} with~\eqref{result:JacobiBasisNorm}.
\end{proof}

Proposition~\ref{result:AverageJacobiNorm} says that the expected value of the $L_p(\mu_{\alpha,\beta})$-norms $\Vert \sum_{n\in A} \varepsilon_n n^{1/2} P_n^{(\alpha,\beta)}\Vert_{L_p(\mu_{\alpha,\beta})} $,
when $(\varepsilon_n)_{n\in A}$ runs over all possible signs $\{\pm 1\}^A$,
is of the order of $|A|^{1/2}$ (cf.\ \cite{AlbiacKalton2006}*{Theorem 6.2.13}). In next Proposition we find norms which deviate significantly from the average value for $p\not= 2$.

\begin{proposition}
\label{result:CCJacobiNorm}
Let $\alpha$, $\beta$ and $p$ be such $\min\{\alpha, \beta\} >-1/2$ and
$p(\alpha,\beta)<p<q(\alpha,\beta)$.
Then
\[
\left\Vert\sum_{n=0}^{N-1} (N+2n)^{1/2} P_{N+2n}^{(\alpha,\beta)}\right\Vert_{L_p(\mu_{\alpha,\beta})}\approx N^\omega, \quad N\in\ensuremath{\mathbb{N}},
\]
where $\omega=\max\{{(2\alpha+3)/2-2(\alpha+1)/p},{(2\beta+3)/2-2(\beta+1)/p}\}$.
\end{proposition}

\begin{proof}
 Since $P_n^{(\alpha,\beta)}(-x)=(-1)^n P_n^{(\beta,\alpha)}(x)$ it suffices to prove the estimate
\begin{equation*}
\label{eq:esti-alpha}
I_N:=\int_0^1\left|\sum_{n\in A_N} n^{1/2} P_{n}^{(\alpha,\beta)}(x)\right|^p \, d\mu_{\alpha,\beta}(x) \approx N^\sigma,
\end{equation*}
where $A_N=\{N+2n : 0\le n\le N-1\}$ and $\sigma=p(2\alpha+3)/2-2(\alpha+1)$. Notice that the hypothesis $p(\alpha,\beta)<p<q(\alpha,\beta)$ implies $0<\sigma<p$.

Consider $d>0$ as in Lemma~\ref{result:NearOneEstimate} and choose $\theta_N\in(0,\pi)$ such that
\[
\cos(\theta_N)=x_N=1-\frac{d}{(3N-2)^2}.
\]
Write $I_N=J_N+K_N$, where
\begin{align*}
J_N &= \int_{x_N}^1 \left|\sum_{n\in A_N} n^{1/2} P_{n}^{(\alpha,\beta)}(x)\right|^p \, d\mu_{\alpha,\beta}(x),\quad \text{and}\\
K_N &= \int_{0}^{x_N} \left|\sum_{n\in A_N} n^{1/2} P_{n}^{(\alpha,\beta)}(x) \right|^p \, d\mu_{\alpha,\beta}(x).
\end{align*}
Since $P_n(x)\approx n^\alpha$ for $N\in\ensuremath{\mathbb{N}}$, $n\in A_N$ and $x\in[x_N,1]$,
\[
J_N\approx \left(\sum_{n\in A_N} n^{\alpha+1/2} \right)^p \int_{x_N}^1 (1-x)^\alpha\, dx \approx N^{p(\alpha+3/2)} N^{-2(\alpha +1)} = N^{\sigma}.
\]
Let $k$, $\phi$, and $E_n$ be as is Theorem~\ref{result:Darboux}.
We have $K_N \le 2^{p-1} (L_N+ M_N)$, where
\begin{align*}
L_N &= \int_0^{x_N}\left| \sum_{n\in A_N} E_{n}(\arccos x) \right|^p \, d\mu_{\alpha,\beta}(x),\quad\text{and}\\
M_N &= \int_0^{x_N}\left| k(\arccos x) \sum_{n\in A_N} \cos(n \arccos x+\phi(\arccos x) )\right|^p \, d\mu_{\alpha,\beta}(x).
\end{align*}

Since $\theta_N\approx N^{-1}$, there exists $\delta>0$ such that $\delta/N\le \theta_N$. By Theorem~\ref{result:Darboux},
\[
|E_n(\theta)|\lesssim \frac{1}{n} (\sin \tfrac{\theta}{2})^{-\alpha-3/2}
\]
for $N\in\ensuremath{\mathbb{N}}$, $n\in A_N$ and $\theta\in[\theta_N,\pi/2]$.
The change of variable $x=\cos\theta$ yields
\[
L_N\lesssim\left( \sum_{n\in A_N} \frac{1}{n}\right)^p \int_{\theta_N}^{\pi/2} ( \sin\tfrac{\theta}{2})^ {-1-\sigma} \, d\theta
\approx \int_{\theta_N}^\infty \theta^{-1-\sigma}\, d\theta = \frac{1}{\sigma} \theta_N^{-\sigma}\approx N^\sigma.
\]
Using the change of variable $x=\cos\theta$ and the formula
\[
\left| \sum_{n\in A_N} \cos(n\theta+\phi(\theta))\right|
= \frac{|\sin(N\theta) \cos(2N\theta+\phi(\theta))|}{\sin \theta},
\]
which is obtained taking into account that we are adding the real part of a geometric sum, gives
\begin{align*}
M_N &\approx \int_{0}^{\pi/2} |\sin(N\theta) \cos(2N\theta+\phi(\theta))|^p (\sin\tfrac{\theta}{2})^{-1-\sigma}\,d\theta \\
&\approx \int_{0}^{\pi/2} |\sin(N\theta) \cos(2N\theta+\phi(\theta))|^p \theta^{-1-\sigma}\,d\theta \\
&= N^\sigma \int_{0}^{N\pi/2} |\sin(u) \cos(2u+\phi(u/N))|^p u^{-1-\sigma} \, du \\
& \approx N^\sigma.
\end{align*}
To deduce the last step in the estimate of $M_{N}$ we have used that
\[
 \int_0^\infty |\sin(u)|^p  u^{-1-\sigma} \, du<\infty
\]
and the Dominated Convergence Theorem.
\end{proof}

We are now in a position to complete the proof Theorem~\ref{result:MainTheorem} as advertised.

\begin{proof}[Proof of Theorem~\ref{result:MainTheorem}]
Assume that the $L_p(\mu_{\alpha,\beta})$-normalized sequence of Jacobi polynomials
is a quasi-greedy basis for $L_p(\mu_{\alpha,\beta})$.
Then, thanks to Lemma~\ref{result:NormalizationAndBoundedness}(a), $p(\alpha,\beta)<p<q(\alpha,\beta)$.

By Theorem~\ref{result:Perturbing} and Lemma~\ref{result:NormalizationAndBoundedness}(b), the basis
$(n^{1/2} P_n^{(\alpha,\beta)})_{n=0}^\infty$ is also quasi-greedy for $L_p(\mu_{\alpha,\beta})$. Combining Proposition~\ref{result:Democracy}
(or Lemma~\ref{result:Random} together with Proposition~\ref{result:AverageJacobiNorm}) with
Proposition~\ref{result:CCJacobiNorm} we obtain
$N^\omega \approx N^{1/2}$ for $N\in\ensuremath{\mathbb{N}}$.
Therefore, $\omega=1/2$, i.e.~$p=2$.
\end{proof}

\subsection*{Acknowledgements}
The first two authors were partially supported by the Spanish Research Grant \textit{An\'alisis Vectorial, Multilineal y Aplicaciones}, reference number MTM2014-53009-P, and the last two authors were partially supported by the Spanish Research Grant \textit{Ortogonalidad, Teor\'{\i}a de la Aproximaci\'on y Aplicaciones}, reference number MTM2012-36732-C03-02. The first-named author  also acknowledges the support of Spanish Research Grant \textit{ Operators, lattices, and structure of Banach spaces},
with reference MTM2012-31286.

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   author={Kushpel, A. K.},
   title={On the norm of the Fourier-Jacobi projection},
   journal={Numer. Funct. Anal. Optim.},
   volume={22},
   date={2001},

   pages={941--952},
}

\bib{LinPel1968}{article}{
   author={Lindenstrauss, J.},
   author={Pe{\l}czy{\'n}ski, A.},
   title={Absolutely summing operators in $\mathscr{L}_{p}$-spaces and their
   applications},
   journal={Studia Math.},
   volume={29},
   date={1968},
   pages={275--326},

}

\bib{LinRos1969}{article}{
   author={Lindenstrauss, J.},
   author={Rosenthal, H. P.},
   title={The $\mathscr{L}_p$ spaces},
   journal={Israel J. Math.},
   volume={7},
   date={1969},
   pages={325--349},

}

\bib{Nielsen2007}{article}{
   author={Nielsen, M.},
   title={An example of an almost greedy uniformly bounded orthonormal basis
   for $L_p(0,1)$},
   journal={J. Approx. Theory},
   volume={149},
   date={2007},
   number={2},
   pages={188--192},

}

\bib{Pollard}{article}{
    author={Pollard, H.},
    title={The mean convergence of orthogonal series III},
    journal={Duke Math. J.},
    volume={16},
    date={1949},
    pages={189--191},
}

\bib{Szego}{book}{
    author={Szeg\H{o}, G.},
    title={Orthogonal polynomials},
    series={Amer. Math. Soc. Colloq. Publ., vol. 23},
    publisher={Amer. Math. Soc.},
    address={Providence, R.I.},
    year={1967},
    edition={3rd edition}
}

\bib{Telmyakov1998}{article}{
   author={Temlyakov, V. N.},
   title={Greedy algorithm and $m$-term trigonometric approximation},
   journal={Constr. Approx.},
   volume={14},
   date={1998},
   number={4},
   pages={569--587},

}

\bib{Wo2000}{article}{
 author={Wojtaszczyk, P.},
 title={Greedy algorithm for general biorthogonal systems},
 journal={J. Approx. Theory},
 volume={107},
 date={2000},
 number={2},
 pages={293--314},

}

\bib{Zippin1966}{article}{
  author={Zippin, M.},
   title={On perfectly homogeneous bases in Banach spaces},
   journal={Israel J. Math.},
   volume={4},
   date={1966},
   pages={265--272},
}

\end{biblist}
\end{bibsection}

\end{document}

\begin{equation}
\label{Average}
\left\Vert \sum_{j\in A}\ensuremath{\mathbf{x}}_j \right\Vert_p
\approx\left( \int_\Omega \Vert \sum_{j\in A}\varepsilon_j \ensuremath{\mathbf{x}}_j \Vert_p^2\, dP\right)^{1/2}.
\end{equation}
for all $A\subseteq J$ finite.   Assume $1<p\le 2$. Using \eqref{Average} and that $L_p(\mu)$ is of cotype $2$,
\[
\left\Vert \sum_{j\in A}\ensuremath{\mathbf{x}}_j \right\Vert_p
\gtrsim \left( \sum_{j\in A} \Vert \ensuremath{\mathbf{x}}_j\Vert_p^2\right)^{1/2} \gtrsim |A|^{1/2}
\]
for all $A\subseteq J$ finite. For the converse inequality we use  \eqref{Average} and the embedding $L_2(\mu)\subseteq L_p(\mu)$.
\[
\left\Vert \sum_{j\in A}\ensuremath{\mathbf{x}}_j \right\Vert_p \lesssim \left( \int_\Omega \Vert \sum_{j\in A}\varepsilon_j \ensuremath{\mathbf{x}}_j \Vert_2^2\, dP\right)^{1/2}=
\left( \sum_{j\in A} \Vert \ensuremath{\mathbf{x}}_j\Vert_2^2\right)^{1/2} \approx |A|^{1/2}
\]
for all $A\subseteq J$ finite.

We begin  writting  out in a lemma the main argument used by Wojtaszczyk \cite{Wo2000} to prove that the trigonometric system is not a quasi-greedy basis for $L_p(\ensuremath{\mathbb{T}})$, $p\not=2$.

\begin{lemma}
\label{result:Averaging}
Let $(X,\Sigma,\mu)$ be a  measure space and $1\le p<\infty$. Suppose that $(\ensuremath{\mathbf{x}}_j)_{j\in J}$ is a quasi-greedy basis for  a subspace of $L_p(\mu)$. Then
\[
\left\Vert \sum_{j\in A} \ensuremath{\mathbf{x}}_j \right\Vert_p \approx \left\Vert \left(\sum_{j\in A} |\ensuremath{\mathbf{x}}_j|^2\right)^{1/2}\right\Vert_p
\]
for all $A\subseteq J$ finite.

\end{lemma}

\begin{proof}
Let $(\varepsilon_j)_{j\in J}$ be a  Rademacher family defined on some probability space $(\Omega,P)$. Appealing to \eqref{result:UCCCondition}, Fubini's theorem and Khintchine's inequality,
\begin{align*}
\Vert \sum_{j\in A}\ensuremath{\mathbf{x}}_j \Vert_p^p&\approx \int_\Omega \Vert \sum_{j\in A}\varepsilon_j \ensuremath{\mathbf{x}}_j \Vert_p^p\, dP\\
&=\int_X \int_\Omega | \sum_{j\in A} \varepsilon_j \ensuremath{\mathbf{x}}_j|^p \, dP \,d\mu\\
&\approx \int_X ( \sum_{n\in A} |\ensuremath{\mathbf{x}}_j|^2)^{p/2}\, d\mu
\end{align*}
for all $A\subseteq J$ finite.
\end{proof}

With the aim of continuing this line of research, we consider Fourier series with respect to systems of orthogonal polynomials. Denote by $\ensuremath{\mathcal{P}}$ the space of polynomic functions defined on $\ensuremath{\mathbb{R}}$.  Consider, for each $n\in\ensuremath{\mathbb{N}}\cup\{0\}$ the potential function $m_n$ given by $m_n(x)=x^n$.
Let $\mu$ be a positive measure on $\ensuremath{\mathbb{R}}$ such that $\ensuremath{\mathcal{P}}\subseteq L_1(\mu)$. Let $(p_n)_{n=0}^\infty$ be the sequence of orthonormal polynomials obtained by applying to
$(m_n)_{n=0}^\infty$ the Gram-Schmidt process in the Hilbert space $L_2(\mu)$. Let $1\le p< \infty$ and $q$ be its conjugate exponent. In the case in which
\[
\sup_{n\ge 0} \Vert p_n\Vert_p \Vert p_n \Vert_q<\infty,
\]
 identifying $L_p^*(\mu)$ with $L_q(\mu)$,  $(p_n,p_n)_{n=0}^\infty$ is a bounded biorthogonal system for the closed subspace of $L_p(\mu)$ generated by $\ensuremath{\mathcal{P}}$ and, consequently, it make sense to to investigate the greedy algorithm with respect to the  $L_p$-normalized system
\[
\left(   \Vert p_n\Vert_p^{-1} {p_n}, \Vert p_n\Vert_p p_n\right)_{n=0}^\infty.
\]

In this paper we  focus on Jacobi polynomials.  Recall that, for  $\alpha$, $\beta>-1$, the  $L_2$-normalized Jacobi polynomials
 $(p_n^{(\alpha,\beta)})_{n=0}^\infty$ arise as the orthonormal polynomials associated to the measure $\mu_{\alpha,\beta}$ given by
\[
d\mu_{\alpha,\beta}(x)=(1-x)^\alpha (1+x)^\beta \chi_{(-1,1)} (x) \, dx.
\]
It is well known that $\ensuremath{\mathcal{P}}$ is a dense subspace of $L_p(\mu_{\alpha,\beta})$  for every $1\le p<\infty$. So, Jacobi polynomials of indices $\alpha$ and $\beta$ constitute a basis for $L_p(\mu_{\alpha,\beta})$ for any $1\le p<\infty$. Our main result establishes that the greedy algorithm for this kind of orthogonal polynomials follows the same pattern as the greedy algorithm for the trigonometric system. We remark that K\"onig and Nielsen \cite{KonigNielsen1992} proved that the Jacobi polynomials of indices $\alpha$ and $\beta$  do not form an unconditional basis of $L_p(\mu_{\alpha,\beta})$, unless $p=2$.

\begin{theorem}
\label{result:MainTheorem}
Let $1\le p<\infty$ and $\alpha$, $\beta>-1$. The $L_p$-normalized Jacobi polynomials  of indices $\alpha$ and $\beta$ form a quasi-greedy basis for $L_p(\mu_{\alpha,\beta})$ if and only if $p=2$.
\end{theorem}

redpajama_set_name: RedPajamaArXiv


text:
Showdown: Gatekeeper vs. X-Flight
Bobbie May 2, 2018 4 Comments
There are only four wing coasters currently operating in the USA. X-Flight at Six Flags Great America was the second one to open and Gatekeeper at Cedar Point was the third. (The first was Wild Eagle at Dollywood and the fourth was Thunderbird at Holiday World.)
Layout, Theme and Setting
Gatekeeper is aptly named in that part of its track overhangs the entrance gates to Cedar Point. It's themed after a golden griffon and features a sprawling layout with 4, 164 feet of track. X-Flight, with an air force theme, occupies a much smaller area with 3,000 feet of track. The theming is nothing short of brilliant. The loading station is designed to look like an airplane hangar, ride ops are smartly attired in flight suits and one of the components of the ride area is a real control tower rescued from Chicago's O'Hare airport. Add to that a soundtrack purporting to give instructions for the mission ahead and you have a very nice setup. For theming X-Flight comes out way ahead.
Gatekeeper, travelling at a speed of 67 mph, boasts a 164-foot drop and six inversions (wingover, Immelmann, corkscrew, zero-g roll, inclined dive loop and inline twist) while X-Flight, flying at 55 mph, has a 120-foot drop and five inversions (wingover, zero-g roll, Immelmann, zero-g roll and inline twist). Both coasters feature keyhole elements which ratchet up the thrills. The elements are obviously quite similar but I would have to give Gatekeeper the edge here for greater diversity as well as greater height and speed.
A ride on Gatekeeper begins with an ascent up a 170-foot chain lift hill. This is followed by the wingover, which has the train rotating 180 degrees and dropping 164 feet in a half loop. Off to a good start. And Gatekeeper doesn't waste any time ascending directly into an Immelmann, followed by a camelback which provides some nice negative G's. From there the train goes through a corkscrew/flat spin and zero-g roll. What's interesting here is that riders are flipped into the zero-g roll while the train travels through the two keyhole elements near the park gates. The remainder of the course is notable for an inclined dive loop, inline twist and 360-degree helix. It's a good ride that not only offers variety but covers a lot of territory.
A ride on X-Flight likewise begins with a wingover, following an ascent up a 120-foot lift hill. (Because a wingover is an aerobatic maneuver, it seems particularly appropriate for X-Flight.) The train ascends into a zero-g roll, takes riders through billows of fog and goes up into an Immelmann. From there it makes a turn over a pond, at which point plumes of water shoot up. The next element is the second zero-g roll, after which the train makes a turn leading up to what is unquestionably the highlight of the ride. This is the inline twist through the control tower. It ranks among the most thrilling elements I've ever experienced on a coaster. The perception of a near collision is so overpowering as to be alarming. It was a case of omigod, we're going to hit that tower! I must have ridden X-Flight a dozen times and although I knew that we weren't really going to hit the tower, the sensation of a near miss was there on each and every ride. Oh, and there's more fog here to enhance the experience.
And the winner is…X-Flight. I suspect that I'm adopting an unpopular position but there's a good reason. Although I suppose that an argument could be made for Gatekeeper being the more complete coaster, I'm giving X-Flight the win for superior theming, special effects and superior thrill factor. The keyhole element on Gatekeeper didn't give me nearly as much of a thrill as the inline twist on X-Flight because I didn't get the same impression of an imminent collision. Don't get me wrong; Gatekeeper is excellent; it simply didn't blow me away the way X-Flight did. In short, the ride experience on X-Flight is awesome. It's a flight that's not to be missed.
What's your take? Have you ridden Gatekeeper and X-Flight and if so, how do you think they match up?
Eric	 May 2, 2018, 1:47 pm
Good analysis. I'm surprised that I've been on 75% of the wing coasters in the USA (Thunderbird was built 2 years after I visited Holiday World). The elements on GateKeeper were supercharged versions of those on X-Flight, but all the damn mayflies at Cedar Point kinda ruined my experience.
I still maintain that X-Flight is the #1 coaster at SFGAm, given its pacing, size, and overall ride duration. Wild Eagle was graceful fun, but not as good as either of these.
Bobbie	 May 2, 2018, 2:28 pm
I likewise found X-Flight to be the best coaster at SFGAM. It was certainly the only one I liked enough to ride a dozen times. Not that I didn't like others – Viper in particular was a standout – but X-Flight just had that extra kick.
manuwino	 May 4, 2018, 11:52 pm
Bobbie, thank you for this post. Its very inspiring.http://metropoliscomix.com/
Brian MacDonald	 May 11, 2018, 7:56 am
Great comparison, Bobbie. I haven't ridden either of these yet (if I get very lucky with business travel this summer, I may get them both), but from what I've read, a lot of enthusiasts agree with you. The consensus is that while Gatekeeper is larger, size alone doesn't make a good wing coaster. The only wing coaster I've ridden so far is Wild Eagle, which was a disappointment. It uses the terrain, but not effectively, and there are no near-miss elements at all, unless you count some trees. I think the lesson is that a wing coaster needs good theming and interaction to be effective.

redpajama_set_name: RedPajamaCommonCrawl


text:
Transport, Airports
Chicago Midway to Get Improvements From $784 Million Bond Sale
Elizabeth Campbell, Bloomberg
- May 26, 2014 12:00 pm
Midway has always seemed minor league — both in its on-time numbers and its facilities — to O'Hare. Perhaps this will improve matters a bit.
Chicago plans to offer about $784 million of airport-revenue bonds next week for projects including work on runways at city-run Midway International.
The borrowing for Chicago's second-busiest airport, which began operating in 1927, will price as soon as May 29, according to data compiled by Bloomberg. Moody's Investors Service ranks the securities A3, the seventh-highest level.
Proceeds of the deal will also go toward mitigating noise for local neighborhoods and buying land.
The bonds are backed by revenue generated at the airport, according to offering documents. Midway served about 10.2 million passengers last year, making it the 24th-busiest airport in the U.S., the documents show. Southwest Airlines Co. accounted for about 91 percent of passengers last year and more than half of operating revenue.
Moody's has a stable outlook on the bonds, noting that the airport is "largely insulated" from the financial strains of the third-most populous U.S. city.
Midway bonds are benefiting from a rally in the $3.7 trillion municipal market. A portion of the airport's November deal maturing in January 2035 traded at 4.37 percent on May 22, down from 5.21 percent when the federally tax-exempt bonds were issued.
Municipal airport bonds have earned 7.1 percent this year, outpacing the 6.2 percent gain for the broader municipal market, Bank of America Merrill Lynch data show.
Kelley Quinn, spokeswoman for Mayor Rahm Emanuel, didn't have an immediate comment on the sale.
–With assistance from Brian Chappatta in New York.
To contact the reporter on this story: Elizabeth Campbell in Chicago at ecampbell14@bloomberg.net. To contact the editors responsible for this story: Stephen Merelman at smerelman@bloomberg.net.
Serious about airlines? Subscribe to Skift Airline Weekly to get the strongest business insights in the world of commercial aviation
Tags: chicago, mdw, midway
Photo Credit: FILE - This March 12, 2013 file photo shows the air traffic control tower at Chicago's Midway International Airport. M. Spencer Green / Associated Press
Brian Sumers, Skift
Chicago Is Becoming the Center of the Growing Conflict Between United and American
G Adventures + Skift
The Ripple Effect: Redefining Community Tourism and its Power to Transform Lives
Colin Nagy, Skift
Understanding Virgin Hotel Chicago's Quick Rise to Consumer Favorite
These Hilton Hotels Have Meeting Group Sales Down to a Science in Chicago
Chicago Tourism CEO Fights For Funds Amid Illinois Budget Impasse
Trump Travel Ban Expands to New Countries: Report https://t.co/1lOKTc6Ucr


More than 1,000 companies competed in local Challenge Cup competitions in 45 cities around the world.
Companies at the global round will compete for $1 million in prizes. GenZ CEO Grant Thompson said the Challenge Cup has already given his company inroads with investors.
The company, founded in 2013, has seven employees, including co-founder Chase Newman and Thompson. Its office is at South Featherly Way near the intersection of West Victory and South Maple Grove roads.
The local competition was hosted by Trailhead, the Downtown working space and nonprofit that helps startups develop business plans and court investors. Trailhead Director of Engagement Jeff Reynolds said GenZ's success bodes well for the Treasure Valley and for Trailhead's 200 entrepreneur members.
"Having an impressive startup like GenZ on the global stage shows potential and current company founders, as well as the rest of the world, how much innovation is happening in Idaho," Reynolds said.


Features: writing practice, write books, explanations.
MOD adds decorative machine of high quality.
The original name of the Ferrari 458 Spider.
Now for the Sims 4 6 iPhone appeared. New smartphone replacement from Apple Inc. enjoyable game!


text:
What's the Deal With Parler and its Rising Popularity?
Examining the "free speech"–centric, far right–friendly alternative to Twitter.
Parler app logo.
Chloe Hadavas
This article has been republished on January 10, 2021, in light of Amazon S3, one of the most powerful web hosting services in the industry, cutting server access to Parler. Parler is now looking to move to a new service provider.
The basic idea of Parler is an awful lot like Twitter. But instead of tweets, users post "Parleys". Instead of retweets, there are "echoes." And upon registering, the suggested accounts to follow include new outlets such as Breitbart, the Epoch Times, and the Daily Caller, as well as the political accounts for Rand Paul, Mark Levin, and Team Trump.
In June, right-wing users started flocking to this alt-Twitter, whose main selling point is that it vows to champion free speech. As mainstream platforms banned more far-right accounts, removed hate speech with newfound vigour, and attached warning labels to a few of President Donald Trump's tweets, Parler became, for many, an attractive solution to Twitter's supposed ills.
Now, it's the second most popular app in the App Store, and last week it was estimated to have reached more than 1.5 million daily users, snagging some high-profile newbies: Senator Ted Cruz, Representative Elise Stefanik, Representative Jim Jordan, Donald Trump Jr., and Eric Trump. What led to Parler's founding in August 2018 was, predictably, disillusionment with the likes of the Silicon Valley giants. Henderson, Nevada–based software engineers Jared Thomson and John Matze created the platform, according to Parler's website, "[a]fter being exhausted with a lack of transparency in big tech, ideological suppresssion [sic] and privacy abuse."
Yet while the platform is being billed as the big "free speech" alternative to Twitter, it isn't exactly unique. Nor is it as uncensored as it claims to be. Parler is just the latest in a long line of rival social networks that have appeared (and, often, disappeared) in the past decade as alternatives to Big Tech. And, if the past is any indicator, it's unlikely that Parler will become anything more than a fringe platform in the near future.
Some of the platforms to emerge as alternatives to the major social networks have taken a hard line on data privacy. Ello, for example, was founded in 2014 as an ad-free network that promised never to sell user data to advertisers. (After being dubbed a "Facebook killer," the site was overwhelmed with new users and crashed frequently; it could never scale up and instead became a community for digital artists.) MeWe, another Facebook rival, offers the industry's first Privacy Bill of Rights. (It also takes a laissez-faire approach to content moderation.) And while its 8 million users are dwarfed by Facebook's 2.6 billion, MeWe is one of the few successful alternative networks in that it's continued to grow since its founding in 2016.
Also read: Understanding Right-Wing Resurgence in the US and India
Matze, Parler's CEO who counts Ayn Rand and conservative economist Thomas Sowell among his influences, fancies his platform a sort of free-speech utopia: "We're a community town square, an open town square, with no censorship," Matze told CNBC. "If you can say it on the street of New York, you can say it on Parler." And while Parler says it is unbiased—Matze is offering a $20,000 "progressive bounty" for a popular liberal pundit to join—it's evidently become an unofficial home to the far right, which has long claimed to be mistreated by mainstream platforms. When alt-right "celebrities," such as Milo Yiannopoulos and Laura Loomer, are banned from Twitter, Parler is their next step. (Loomer announced last week that she has become the first person whose Parler following—572,000—exceeds her pre-ban Twitter following.)
In this regard, Parler is most similar to Gab, the free speech–driven platform launched in 2017 that's known as a haven for extremists. "[F]ar angrier and uglier" than Parler, Gab quickly became a breeding ground for anti-Semitism and neo-Nazism, where posts calling for terrorist attacks and violence against minorities circulate.
Gab's fate, however, represents one iteration of the circle of life for platforms of its ilk: After it was connected to an instance of terrorism in 2018, when the suspect in the Pittsburgh synagogue shooting posted about his intentions to act just before he killed 11 people, Gab never quite recovered. Its server, GoDaddy, dropped it, and though it eventually found another home online, its popularity waned following the shooting and the period offline. In 2019, a software engineer for Gab's web hosting company said that the platform probably had a few tens of thousands of users at most—rather than the 835,000 that Gab claimed—though the hosting company later denied that.
But Parler doesn't quite have Gab's teeth. (Andrew Torba, Gab's founder, has referred to Parler as a network for "Z-list Maga celebrities.") While even Gab has limits to free speech, since its content policy purports to ban extremism, Parler is stricter. It goes far beyond what you might expect from a platform whose entire ethos is freedom of expression. Matze listed a few of the basic rules in a "Parley" on Tuesday:
This is how the Parler CEO describes their content policies pic.twitter.com/fiPnNCpxgd
— Cristiano Lima (@viaCristiano) June 30, 2020
As the top Twitter comment points out, "Twitter allows four of the five things that Parler censors." Parler's thorough community guidelines also prohibit spam, terrorist activity, defamation, "fighting words," and obscenity, among other kinds of speech. And Parler's user agreement includes clauses that may seem antithetical to its mission.
Also read: As the Far-Right Culture War Escalates in Germany, Concerns Grow
The platform "may remove any content and terminate your access to the Services at any time and for any reason or no reason," it states. But perhaps most surprising is this:
17. You agree to defend and indemnify Parler, as well as any of its officers, directors, employees, and agents, from and against any and all claims, actions, damages, obligations, losses, liabilities, costs or debt, and expenses (including but not limited to all attorneys fees) arising from or relating to your access to and use of the Services. Parler will have the right to conduct its own defence, at your expense, in any action or proceeding covered by this indemnity.
The indemnity provision means that if Parler faces a lawsuit for something you post, you pay. Basically, you're free to say whatever you want—as long as it falls within the community guidelines, and as long as you're willing to take the risk.
That Parler has been reportedly banning users en masse this week only further illuminates the façade of free speech on the platform; but regardless of the extent to which one can or cannot "Parley" whatever they want, the fact remains that the platform is becoming an important space for the American far right.
It's worth considering, then, what its members might do with it. Part of the concern over polarised platforms is that they can lead to radicalisation: In general, they're seen as part of the pipeline to extremism. First, extremist movements find a foothold in mainstream platforms, where they present their norms in a slightly more palatable way, explained Jeremy Blackburn, a computer science professor at Binghamton University who researches fringe and extremist web communities. Then they gain ground in platforms like Parler that straddle the fringe and mainstream.
"Once you remove any question of there being an echo chamber, there's just obvious consequences," Blackburn said.
While this may be cause for concern, Amarnath Amarasingam, an extremism researcher and professor at Queen's University, is skeptical that Parler will really galvanise the right. "I think part of what animates the right—and the left to some extent—and particularly the far right, is the ability to argue with 'the other,'" Amarasingam said.
Interacting (and fighting) with the left reinforces the far right's identity, giving it meaning and purpose, he said, and from studying similar platforms like Gab, Amarasingam has found that "talking to yourself in the dark corners of the internet is actually not that satisfying." And while he believes it might lead to the radicalization of certain individuals within the far right, the platform itself won't necessarily further the ideologies of extremist right-wing groups.
What Parler could do, Amarasingam believes, is serve as a kind of sounding board for the far right, a place for fringe movements to try out and refine different arguments. Essentially, it could be a "factory of sorts," churning out ideas before they're deployed into the mainstream. Maybe one day, at least—for now, a good portion of the conversation of Parler is about how fantastic the platform is and how dumb the old tech giants are. Amarasingam acknowledged this.
"[W]hat that indicates to me is that they actually are just using Parler to vent their anger of being suspended from what really matters, which has been more mainstream platform," he said. "And so I think they'll very much try to get back into wherever the conversation is happening."
Also read: Why Regulating Social Media Will Not Solve Online Hate Speech
There's also the matter of growth. Normally, these networks just don't get that big. They're considered "fringe" platforms for a reason, and there's rarely a solid business model behind them.
In Parler's case, the network was started with angel funding, and Matze hasn't devised a clear business plan since. Currently, his tentative model is to match conservative influencers with advertisers, and have Parler take a cut of the influencer fee. But given brands' recent reluctance to advertise on Facebook, this plan seems far from foolproof. With only 30 employees, Parler's ability to handle more users will be tested.
It might grow—especially if Trump does decide to join after all—but, as Amarasingam put it, "if you're not in the mainstream, you're not in the mainstream."
"Generally speaking, what I expect to see in these sites is they hit a certain threshold of users, just like any other social networking platform," said Blackburn. "And then for these types of platforms that are explicitly attracting these certain types of users, probably one of them will do something stupid, then they get shut down or deplatformed, and the next one pops up."
Chloe Hadavas is a writer based in Washington.
This piece was originally published on Future Tense, a partnership between Slate magazine, Arizona State University, and New America.


text:
Get Started Control Panel
Sarrah Hussain
Why do you need User Roles?
Administration, control and management of major/minor options in a Portal plays a key role in application management with respect to an organization. The level of granularity with which the functions, settings, rules and procedures can be defined eases the process of streamlining how an organization uses the system. This can only be achieved by varying role definitions of each user to equip them with different levels of authoritative and contributive options within the Portal.
User Roles determine the access level and permissions of a person authorized to use VIDIZMO. In a default VIDIZMO installation, there are certain predefined roles assigned to users with a default set of permissions. These roles can also be configured as per the need of the organization.
About VIDIZMO User Roles:
Roles are granted permissions which enable users of that specific role to perform various functions within the application. The assigned permissions define a particular function within the Portal. For instance, a Manager is likely to have permissions relevant to managing VIDIZMO Portals.
VIDIZMO's default user roles include:
Note: Every role has certain capabilities, access levels, and permissions that allow the user to complete various tasks. These roles can be customized to meet various business needs, please contact VIDIZMO Support for any customization.
Administrators are Account (main portal) owners. An Administrator has full control over the application and can access all functionalities. They have complete control over Media Management, Users and Groups Management, Account Settings, Analytics, Control Panel, and Billing Reports. An Administrator also has the permissions to manage Email Templates, Streaming Servers, Audit Logs, Transcoding, Workflow Queue and Login Settings etc.
An Administrator is typically an IT Admin responsible for configuration, monitoring, and security of your systems.
Managers are specific portal's owners who are typically responsible for regular operations such as managing users, media, Portal Settings, monitoring usage via Analytics etc. They can create, edit, delete, or brand Portals as per business needs. Managers have full rights to control and manage media content such as publish or delete any media, schedule a live webcast, change access permissions etc for all media uploaded in their designated Portal. Moreover, Managers are also responsible for managing User Registrations and can also moderate comments and content uploaded by Contributors.
Managers have access to Media Management, Users and Groups, Portal Settings, Analytics, and Control Panel.
A Moderator performs the support role for Managers. They can publish, edit settings and moderate media and Comments on a Portal which are pending approval. However, any content or comment submitted by Moderator or above roles do not require moderation.
Moderators have access to Media Management.
A Contributor is basically a Viewer with the additional capability to publish and edit their uploaded content. However, the content submitted by Contributors might require approval from the Moderator+ roles depending on Portal Settings.
Note: Once the content is approved and published, if Contributors edit their content such as its basic information, the media will be unpublished and resubmitted for moderation.
A Viewer is a regular end user of your portal who views media content published in a Portal. A Viewer can either be anonymous or authenticated and only has permission to view content that he/she has been given access to. In addition, Viewers can update their Profile, view media assigned to them, view their Personal Viewing Reports.
Anonymous Viewer
Anonymous Viewer is a non-authenticated user who has restricted access to only view public media on any Account/Portal.
Note: Administrators can choose to require a simple password before anonymous viewers can view any public media. However, this is still considered as anonymous viewing since there is no user identification or authentication involved.
To learn how a user Role can be modified, click here on How to Change a User Role.
Sarrah is the author of this solution article.
user roles Administrator Manager Moderator Contributor viewers Anonymous Access Management


Jamie talks about here so I thought I better join up and say hullo to u all!
I drive a super cool DelSol hehe!!
I is good! How is my jim jam?
yes! proper JDM and i love it!
welcome there on no not more jap crap ko how do ya do there?
dnt listen to him... he drives a lupo!


Get insider access to iconic Texas sporting events with state-of-the-art amenities.
There's nothing like walking onto a football field or into a performance arena to get your adrenaline flowing.
In addition to providing that game-day feeling, the following three venues, all newly opened or planned to open in 2019, offer an array of meeting spaces and amenities for planners and guests. From expansive indoor and outdoor venues to exclusive meeting spaces, groups large and small can tap into the energy and history of some of Texas' most beloved teams and athletic events in state-ofthe-art surroundings.
Dickies Arena in Fort Worth, scheduled to open in November 2019, has been decades in the making. The 14,000-seat arena is the culmination of thoughtful planning to design a venue that will fit the iconic style of the nearby Will Rogers Memorial Center Campus.
Dickies Arena will be the new home for the Fort Worth Stock Show and Rodeo starting in 2020, an event the Will Rogers Memorial Center has hosted since the 1940s. Dickies Arena, which is being built through a private-public partnership, will cost about $540 million.
In addition to the arena's main event floor, groups will be able to use a 209,000-squarefoot outdoor exhibit plaza, Cunningham says. The plaza can be closed off for private and preshow events and can hold up to 3,000 people.
From the plaza, guests will be able to look down into a 90,000-square-foot exhibit space, which is connected to the arena below. The windows at the plaza level, above the exhibit space, provide ample natural lighting. There are also two additional multipurpose rooms, each at 1,750 square feet, Cunningham says.
The arena is starting to book events for late 2019 and 2020. The arena has already confirmed it will begin hosting the NCAA women's gymnastics championships in 2022. Dickies Arena will also host the first and second rounds of the NCAA men's basketball tournament in 2022.
Kyle Field at Texas A&M University, with a seating capacity of more than 102,000 people, is considered one of the most intimidating venues in college sports. The fan base at Bryan College Station gets so loud at home football games that the fans are considered the team's 12th man, says Bishop Turon, sports marketing manager for Bryan College Station Sports and Events.
Planners can immerse their guests in the Aggie spirit in the Hall of Champions inside Kyle Field. The hall provides 32,800 square feet of column-free space, with a maroon ceil - ing adorned with a gigantic Texas A&M logo. The Hall of Champions features video screens and ribbon boards, making it a versatile venue, Turon says. The hall has hosted events ranging from championship dinners and banquets to the launch of Tesla's Hyperloop initiative.
Texas A&M University takes advantage of the flexible space for its use, too. The Hall of Champions is used regularly to showcase exhibits celebrating championship teams from Aggie history.
The hall has a 60-foot-tall ceiling, which provides flexibility for staging and draping, and the option to roll in a big stage or screen, Turon says. Kyle Field has a variety of rooms near the Hall of Champions, including two large clubs and smaller breakout rooms. "You can almost turn this entire stadium into a conference center," Turon says.
The Star, a 91-acre development anchored by The Dallas Cowboys World Headquarters and Training Facility, is a dream for Dallas Cowboys fans.
The Star includes the Ford Center, a 510,000-square-foot indoor athletic facility that offers three football fields (one indoor and two outdoor) used by both the Cowboys and the high school football teams of the Frisco Independent School District. The Ford Center seats 12,000 people and can be rented out for private events for up to 15,000 people.
Hosting private events at the space gives guests an insider's look at the franchise. In addition to offering a VIP guided tour of The Star during their meetings, planners can also host meetings and events in the same spaces used by Dallas Cowboys players and coaches. These include the team auditorium and dining hall and The War Room, where the Cowboys complete all of their draft work each season. Another popular space is the Grand Atrium, which displays all five Super Bowl trophies and rings. The atrium provides 3,500 square feet of space with marble floors, a five-story glass wall and a view of artist Leo Villareal's art installation Volume.
The Baylor Scott & White Sports Therapy & Research Center at The Star provides a unique opportunity for guests to directly gain insight on their own athletic performance. Groups will be able to host private events at the institute, according to Ryan Callison, director of marketing and communications at Visit Frisco.
The walkability of the Star has been a big hit with groups, Callison says. The Gatorade Sports Science Institute is located across the street from the 300-room Omni Frisco Hotel, and dining, shopping and entertainment are all located within walking distance. With the Omni connected to the Ford Center, guests don't even need to walk outside for their events.


text:
Bronze Age Levant
By Levi Clancy for Student Reader on Monday 11th May, 2009
Aram Damascus
Chalcolithic Levant
Domestication, agriculture and horticulture in the Levant
Iron Age Levant
Levant terrain
Neolithic Levant
Pilgrim flasks
Syro-Palestinian corridor in the 9th century BC
Early Bronze Age
Early Bronze Age I
The Early Bronze Age I (3500 - 3050 BC) was a transition era from the Chalcolithic. The EB I was contemporary with Late Predynastic and Early First Dynasty in Egypt.
The Early Bronze Age I is marked by rural proto-urban (unwalled) cities. The EB I is continuous with the Chalcolithic, especially in the north. Houses are mostly either: apsidal, with a curve at one end; ellipsoidal, with both ends curving; or, occasionally, caves. Burial customs included: caves, most commonly; shaft tombs; charnel houses, where bodies were laid out like in a tomb; nawamis, field stone chambers which still stand in the Sinai today; and cremation. The Canaanean Blade first appeared in the EB I (no sooner), a blade with a trapezoidal cross-section. Later in the Early Bronze Age I, Egypt begins interacting with the Negev, South Coast and Sinai. There were regional ceramic families. A notable site is Bab edh-Dhra, inhabited from 3,1000-2,000 BC.
There are four Early Bronze Age I ceramic assemblages in the Levant: Proto-Urban A, B, C and D. These do not have clear temporal bonds and we will explore from north to south:
Decription
Proto-Urban C
In Esdraelon. Located in northern Israel (Jezreel Valey and Galilee). Very shiny, which is a new feature. Coonnected with northern levantine and Anatolian ceramic assemblages (ie, Dark Finish Burnished Ware or DFBW). Gray-Burnished ware. Largely in the Esdraelon culture (Jezreel Valley and North).
Proto-Urban B
Characterized by Line Group Painted Ware (LGPW), central hil country. Line group painted ware. Maybe the lines are meant to resemble netting. They have handles which could be used to carry on a string. Netting to avoid scratching etc.
Proto-Urban A
Characterized by Red Burnished Ware (RBW). Found in north and south. Jericho, Azor (coast), T. el-Far'ah North, Bab edh-Dhra. Clay turns red in presence of oxygen or black in absence of oxygen while firing. Similar form with looped handle to proto-urban B, but this is burnished and shiny.
Proto-Urban D
Impressed slashed Ware (ISW). Central and southern Israel. Largely in Umm Hammad, a Transjordan (modern Jordan) site and probably earliest EB I assemblage. May be among earliest of EB I cultures, thus dislaying similarities to Chalcolithic ceramics.
Early Bronze Age II
Urban life begins in the Early Bronze Age II (3050 - 2700 BC). The EB II was contemporary with Late I and II Dynasties of Egypt.
The Early Bronze Age II is the Southern Levant's first urban period. Notable Early Bronze Age II finds include the Tell Kineret Tomb, Arad and et-Tell (Biblical Ai). As the trade infrastructure developed, Egypt no longer required a permanent presence on the coastal plains. However, Egyptians had a huge presence in the Negev and Sinai to exploit the region's valued metals, especially copper. Egyptian presence in the Levant is confirmed by Abydos Ware, a valuable pottery native to the Negev and Sinai that has been found in tombs in Abydos and other Egyptian towns.
Early Bronze Age III
The Early Bronze Age III (2700 - 2300 BC) is contemporary with III-VI Dynasties in Egypt.
The primary distinction between Early Bronze Ages II and III is the emergence of Khirbet Kerak Ware (KKW). Construction of temples continued, with a highlight being the enormous altar in EB III layer at Megiddo. The crescentric axe (aka epsilon axe) is an unusually shaped tool that first appears in the EB III.
The Southern Levant's relationship with Egypt evolved.
Egyptians were present in EB I; Egyptians left in EB II but continued strong trade; and in EB III, Egypt and Byblos (access point for Lebanon's timber) began direct maritime trade and bypassed the Southern Levant. The Early Bronze Age III brought violence between Egypt and the Southern Levant, clearly evidenced in: the Wadi Mighar inscriptions; the 5th dynasty tomb of Ante at Deshahe in Egypt; other reliefs; and Autobiography of Weni, a text by a royal palace worker named Weni (2,373-2,296 BC).
Early Bronze Age IV
The Early Bronze Age IV (2300 to 1900 BC) was contemporary with VII-XI Dynasties in Egypt.
Also known as the Intermediate Bronze Age, the Early Bronze Age IV was markedly different between the Northern and Sourthern Levant. The Northern Levant continued urbanizing, and the particularly sophisticated kingdom of Ebla even developed its own form of writing. Meanwhile, the Southern Levant underwent a gradual urban collapse over hundreds of years, leaving behind few artifacts and no hints of writing. Unmaintained fortification walls eroded away, leaving cities unwalled. Some urban centers were abandoned, leading to a boom in rural settlement and pastoral nomadism. Pastoralism worked well because wool could be produced (eating the animal itself was rare) and there was a high demand for wool in Syria and Egypt. Below are possibilities why the Southern Levant de-urbanized during the Early Bronze Age IV:
Pollen cores do not indicate a localized ecological catastrophe.
If Egypt stopped trading, this could destroy the markets.
Summed disasters (diseases, etc) may have dealt a death blow.
Burial customs included: shaft tombs (Jericho); megalithic dolments, made of a horizontal stone atop upright stones (Golan Heights; Upper Galilee); tumuli (aka cairns), mounds of stone and debris (central Negev); cists, rectangular burials; burial caves; and pits.
A notable Early Bronze Age IV structure is Beer Resisim, an encampment for an extended family or small clan. Beer Resisim was occupied over a ~400 year period for ~30-50 years at a time. Similar to Beer Reisisim is Ein Ziq, also located in an arid region that is now so dry that modern bedouins dare not breach it due to its utter inability to support life. Another important site is Ain Samiya, where a silver goblet was found that depicts Tiamat, Narnuk, other Mesopotamian icons, and even Mesopotamians themselves holding up a sun disk. This shows a strong Syrian and Mesopotamian influence.
While Early Bronze Age IV ceramics fit well into the overall Early Bronze Age's assemblage, there are some deviations: the wheel is limitedly used, indicating ceramics are purely practical rather than an art form; caliciform wares, a chalice arising from a North Levantine influence; painted or incised parallel lines; and a return of regionalism. One regional item found only in the EB IV is the four-spouted lamp. Four wicks together burn oil four times faster than a single wick, and one explanation is that people began using fish oil, which burns dimmer than olive oil and required more wicks. Also, presence of the fenestrated axe continues from the EB III.
Early to Middle Bronze Age
After urbanism's rise in the EB II-III and collapse in EB III, it returns in MB IIA-C.
EB/MB differences
Settlement patterns; de-urbanization; a lack of rectilinear houses; radical changes in ceramic forms and quality.
EB/MB similarities
Metallurgy; continuous (albeit evolving) ceramic traditions.
Middle Bronze Age
Middle & Late Bronze Political Organization
Middle and Late Bronze Age (1900-1200 BC) political organization was limited to city-states with a high degree of complexity. The kingdom of Ashkelon has yielded textual sources attesting to several Amorite kings. Mechanisms driving Amorite spread included merchantile activity (trade ports developed into civilizations) and mercenary activity. Amorites moved into the highlands, which were perfect for cultivating olives and orchards; these highlands became the epicenter of power.
Middle Bronze Age I
The Middle Bronze Age I (1900 - 1700 BC) is contemporary with XII and even early XIII Dynasties in Egypt.
The Middle Bronze Age I (aka Middle Bronze Age IIA, with EB IV → MB I) was marked by an emergence of northern states, and the arrival of the new Amorite ethnic group (originating ~2,000 BC in Syria and later coalescing into an ethnic group). Material culture peaks ~1,700 BC, although fortified cities already existed: Tel-Ashkalon (~ 50 hectares, largest site north of Egypt); Tel-Burgah; and Tel-Kabri. A distinction of the Middle Bronze Age IIA is red burnishing, likely an attempt to imitate metal vessels.
Egypt provides all major textual sources, including: execration texts; Tale of Sinuhe and Beni Hasan reliefs. The Beni Hasan reliefs are from an Egyptian monarch's tomb, and depict: asiatics (Amorites) arriving with all their belongings; Semites and Egyptians peacefully interacting during the Middle Bronze Age; and a caravan of asiatic donkeys.
Middle Bronze Age II
The Middle Bronze Age II (1700 - 1640 BC) was contemporary with the Late XIII Dynasty of Egypt.
The Middle Bronze Age II was marked by emergence of southern states. As Egypt began to unravel, so did hegemony by its centralized government. Levantine Middle Bronze Age II developments include: fortification and enclosure of the water system in Jerusalem; and consistency amongst MB II/III gates, which are the weakest parts of a fortification. These two developments indicate that communities systematically addressed a common threat. Major textual sources include: the Mari texts, preserved by Hammurabi's campaign in Mesopotamia; Alalakh VII texts in the northern Levant; many isolated tablets in the southern Levant, including Hazor, Shechem, Hebron and other. The 66 ha town of Hazor: had an upper and lower town over a 66 ha region; was mentioned in the Mari texts, differentiating it from Ashkalon; and yielded a handful of MB and LB cuneiform tablets. Timnah (Tell Batash), a town on the southern coastal plane, is another notable Middle Bronze Age II site.
Middle Bronze Age III
The Middle Bronze Age III (1640 - 1530 BC) was contemporary with XV Dynasty (Hyksos Period) of Egypt.
The Middle Bronze Age III was marked by the Hyksos kingdom emerging in Egypt and terminates when the Hyksos are expelled from Egypt. Amorites ruled much of the Levant until Hurrians from modern central Syria (mostly east of the Euphrates) pushed the Amorites into the nothern Levant. Long-distance trade was extensive, including maritime transport of opium from Cyprus to the Levant. Unique to the MB III is: chocolate-on-white ware (~1,500 BC); discontinuation of red burnish; and Tell el-Yehudiyeh Ware, a conglomeration of ceramic forms featuring extensive anthropomorphism. Major textual sources include: Hebron tablet; Manethos' account of Hyksos' rule in the Egyptian town of Avaris, preserved by Josephus in ~100 AD.
Late Bronze Age
General textual sources for the Levantine Late Bronze Age include: temple and tomb reliefs and inscriptions from Egypt's New Kingdom; and Late Bronze Age tablets, from Alalakh, Qatna, Hazor, Kumidi, Megiddo, Beth-Shean, Aphek, Ta'anach and more.
Late Bronze Age IA
The Late Bronze Age (1530 - 1470 BC) was contemporary with Dynasty XVII to Thutmose III in Egypt.
The Late Bronze Age IA: begins with Hyksos' conquest of Egypt; continues through expulsion of the Hyksos from Egypt; and ends with Thutmose III's annual campaigns. The Late Bronze Age IA was highly destructive to the Levant, with over 30 sites identified as obliterated, and led to a population decline. The extent of Egypt's fortresses in this region remains unclear, although by the Late Bronze Age II the Egyptians had built fortress way-stations and depots. Major textual sources include: the Alalakh archive from ~1,500-1,430 BC; Egyptian temple and tomb reliefs and inscriptions; and the Carnarvan Tablet. The Carnarvan Tablet describes how Levantine Amorite princes had divided up the land, pestering Egyptian settlements and causing Egypt to aggressively smite the asiatics.
The Battle of Megiddo occured c 1470 BC.
During the ~60 years after Hatshepsu's death, Canaanites mustered a coalition in an attempt to end centuries of Egyptian attack. The Canaanites met at Megiddo to fight the Egyptians, but the Egyptians discreetly took the Aruna pass and successfully sprung upon Megiddo from behind. After this, the Canaanites failed to coalesce again and resistance was on a mere local level.
Late Bronze Age IB
The Late Bronze Age IB (1470 - 1400 BC) was contemporary with the reign of Thutmose III to the Amarna Period.
Late Bronze Age IIA
The Late Bronze Age IIA (1400 - 1300 BC) was contemporary with Amarna period and aftermath in Egypt.
Fortifications continued through 1,470 BC, but were rare by the Late Bronze Age II A (Hazor is an exception). Egypt balkanized the region, as it was easier to control vassals than to interact with the capital city of a territory. This decentralization of power led to de-urbanization. Mud brick fortifications quickly dissolved after a few winters without maintenance. Pastoral nomadism rose, as indicated by cemeteries found without settlements.
Egypt's Amarna Period (c 1350 BC)
Egypt had solid control over the Levant during the Amarna Period (aka Pax Aegyptiaca), signified by the reigns of Amenhotep III and Amenhotep IV (Akhenaton). However, Egypt neglected to maintain the infrastructure of the region and city-states began to fight. This is evidenced by the highly regarded Amarna Letters. Egypt neglects infrastructure maintenance in the region and the city-states start fighting. There were international exchange networks, evidenced by ceramics in tomb contexts (from the late MBA onward), and also the Amarna Letters. The Amarna Letters consist of cuneiform tablet letters from c 1,350 BC from the court of Amenhotep III and IV at Amarna, with over 300 letters from rulers at Levantine towns to the Egyptian pharaoh. -- Mycenaean ceramics and Cypriot ceramics. Further evidence for international contacts involves shipwrecks such as Ulu Buruun, c 1,305 BC, whose cargo is hugely diverse from cobalt ingots to gold; and Cape Gelidonya (c 1,200 BC).
Also, alphabetic writing developed at this time. Phoenician alphabetic script (exemplified by the Lachish Ewer) arose after Proto-Sinaitic (alphabet developed from Egyptian hieroglyphs) and Proto-Canaanite (22 inscriptions dating from 17th to 12th century BC, very protracted). This was a major development in an age of booming internationalism, as more people can write and work with 22 letters than 200/300 Egyptian signs. Also emerging at the same time in the north is Ugaritic, which adapts cuneiform that chooses 22 signs but still looks like cuneiform. Additionally, this era had a very high level of artistic prowess, evidenced by the Megiddo ivories -- tiny furniture inlays and box components.
Late Bronze Age IIB
The Late Bronze Age IIB (1300 - 1200 BC) was contemporary with Egypt's Dynasty 19.
Bronze Age → Iron Age
Culprits for destruction at the so-called collapse of the Bronze Age: Aramaeans; 'Apiru; other unknown groups; general social unrest; Israelites; Sea Peoples. The Philistines in Biblical Tradition: Bile (originate in Crete?) Arrive with Sea Peoples around 1,180 BC. Anachronistic references to Philistines in Gen 21:32-34; 26:1, 8, 14-15; and in Exodus 13:17; 15:14; 23:3. Story of Samson (Judges 13-16).
Alternate chronologies
A common difference is attributed the Early Bronze Age IV as a transitional, initial epoch of the Middle Bronze Age. The three major Middle Bronze Ages are then labeled as IIA, IIB and IIC.
Conventional name
(Intermediate Bronze Age)
Middle Bronze Age IIA
Middle Bronze Age IIB
Middle Bronze Age IIC


OK, so I'm not that old to remember when people treated air travel like it was a big deal. But doesn't it all look cool now?
A few minutes ago this looked like just another frozen dinner. So yes, I'll have another drink!
Like so many other things, I sometimes feel like I was born too late. It would've been awesome to work for Pan Am back in the glory days.
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