text:
\section{Introduction}
Recently, there has been an expanding interest in utilizing  the {\it spectral expansion} based methodology that enabled progress in data-driven analysis of high-dimensional nonlinear systems. The research was  initiated  in \cite{MezicandBanaszuk:2004,Mezic:2005}, using composition (Koopman) operator representation  originally defined in~\cite{Koopman:1931}. The framework is being used for  model reduction, identification, prediction, data assimilation and control of deterministic (e.g., \cite{Budisicetal:2012,mauroy2017koopman,rowley2009,Williamsetal:2015,Bruntonetal:2016,Giannakisetal:2015,korda2016linear}) as well as stochastic dynamical systems~(e.g.,~\cite{takeishi2017subspace,riseth2017operator,wu2017variational}). This has propelled the theory to wide use on a diverse set of applications  such as fluid dynamics \cite{SharmaMezicM16,Glazetal:2016}, power grid dynamics \cite{Susukietal:2016}, neurodynamics \cite{brunton2016extracting}, energy efficiency \cite{GeorgescuandMezic:2015},  molecular dynamics \cite{wuetal:2016} and data fusion \cite{Williamsetal:2015}.

Numerical  methods for approximation of the spectral properties of the Koopman operator have been considered since the inception of the  data-driven analysis of dissipative dynamical systems  \cite{MezicandBanaszuk:2004,Mezic:2005}. These belong to the class of Generalized Laplace Analysis (GLA) methods \cite{Mezic:2013}. An alternative line of algorithms, called the Dynamic Mode Decomposition (DMD) algorithm \cite{schmid:2010,rowley2009} have also been advanced, enabling concurrent data-driven determination of approximate eigenvalues and eigenvectors of the underlying DMD operator.
The examples of  DMD-type algorithms are 1) the companion-matrix method proposed by Rowley et al. \cite{rowley2009}, 2) the SVD-based DMD developed by Schmid \cite{schmid:2010},  3) the Exact DMD method introduced by Tu et al. \cite{tu2014dynamic}, 4) the Extended DMD~\cite{Williamsetal:2015}.
The relationship between these methods and the spectral operator properties of the Koopman operator was first noticed in \cite{rowley2009},  based on the spectral expansion developed in \cite{Mezic:2005}.  However, rigorous results in this direction are sparse. Williams et al. \cite{Williamsetal:2015} provided a result the corollary of which is that the spectrum of the EDMD approximation is contained in the spectrum of the Koopman operator provided the observables belong to a finite-dimensional invariant subspace of the Koopman operator and the data matrix is of the same rank. The work of Arbabi and Mezi\'c \cite{ArbabiandMezic:2016} suggested that an alternative assumption to the finite rank is that the number of sampling points $M$ goes to infinity even though the results of~\cite{ArbabiandMezic:2016} still implicitly rely on a finite-dimensional assumption.

The work \cite{klus2015numerical} showed that, assuming either independent identically distributed (iid) or ergodic sampling from a measure $\mu$, the EDMD operator on $N$ observables constructed using $M$ samples, $\mathcal{K}_{N,M}$, converges as $M\to\infty$ to $\mathcal{K}_N$, where $\mathcal{K}_N$ is the $L_2(\mu)$-orthogonal projection of the action of the Koopman operator on the finite-dimensional subspace of observables. In this work, we show that $\mathcal{K}_N$ converges to $\mathcal{K}$ in the strong operator topology. As a result, predictions of a given observable obtained using $\mathcal{K}_N$ or $\mathcal{K}_{N,M}$ over any finite prediction horizon converge in the $L_2(\mu)$ norm (as $N$ or $N$ and $M$ tend to infinity) to its true values. In addition, we show that, as $N\rightarrow \infty$, accumulation points of the spectra of $\mathcal{K}_N$ correspond to eigenvalues of the Koopman operator, and the associated eigenfunctions converge weakly to an eigenfunction of the Koopman operator, provided that the weak limit of the eigenfunctions is nonzero. The results hold in a very general setting with minimal underlying assumptions. In particular, we {\it do not} not assume that the finite-dimensional subspace of observables is invariant under the action of the Koopman operator or that the dynamics is measure preserving.

As a by-product of our results, we propose an analytic version of the EDMD algorithm which allows one to construct  $\mathcal{K}_N$ directly, without the need for sampling, under the assumption that the transition mapping of the dynamical system is known analytically and the $N$-dimensional subspace of observables is such that the integrals of the products of the observables precomposed with the transition mapping can be evaluated in closed form. This method is not immediately useful for large-scale data-driven applications that the EDMD was originally designed for but it may prove useful in control applications (e.g.,~\cite{korda2016linear}), where model is often known, or for numerical studies of Koopman operator approximations on classical examples, eliminating the sampling error in both cases.

Finally, we analyze convergence of $\mathcal{K}_{N,N}$, i.e., the situation where the number of samples $M$ and the number of observables $N$ are equal. Under the additional assumptions that the sample points lie on the same trajectory and the mapping $T$ is a homeomorphism, we prove convergence along a subsequence to weak eigenfunctions (or eigendistributions in the sense of~\cite{GelfandandShilov:1964}) of the Koopman operator, which also turn out to be eigenmeasures of the Perron-Frobenius operator.

The paper is organized as follows: in Section~\ref{EDMD} we introduce the setting of EDMD. In Section~\ref{EDMDl2} we show that EDMD is an orthogonal projection of the action of the Koopman operator on a finite subspace of observables with respect to the empirical measure supported on sample points drawn from a measure $\mu$. In Section~\ref{EDMDconv} we show that this projection converges to the $L_2(\mu)$-projection of the action of the Koopman operator. In Section \ref{KNconv} we analyze the convergence of the EDMD approximations as the dimension of the subspace $N$ goes to infinity, showing convergence in strong operator topology and convergence of the eigenvalues along a subsequence plus weak convergence of the associated eigenfunctions. In Section~\ref{sec:finiteHor} we show convergence of finite-horizon predictions. Section~\ref{analyEDMD} describes the analytic construction of~$\mathcal{K}_N$. Section~\ref{KNNconv} contains results for the case when $M=N$ and only convergence to weak eigenfunctions, along a subsequence, is  proven. We conclude in Section~\ref{conc}.

\new{
\subsection*{Notation}
The spaces of real and complex numbers are denoted by $\mathbb{R}$ and $\mathbb{C}$ with $\mathbb{R}^{n\times k}$ and $\mathbb{C}^{n\times k}$ denoting the corresponding real and complex $n\times k$ matrices; the real and complex column vectors are denoted by $\mathbb{R}^n := \mathbb{R}^{n\times 1}$ and  $\mathbb{C}^n := \mathbb{C}^{n\times 1}$. The complex conjugate of $a\in \mathbb{C}$ is denoted by $\overline{a}$. Given a matrix $A\in \mathbb{C}^{n\times k}$, $A^\top$ denotes its transpose and $A^{\mathsf{H}}$ denotes its Hermitian transpose (i.e., $A_{i,j}^\top = A_{j,i}$ and $A^{\mathsf{H}}_{i,j} = \overline{A_{j,i}}$). The Moore-Penrose pseudoinverse of a matrix $A$ is denoted by~$A^\dagger$. The Frobenius norm of a matrix $A$ is denoted by $\| A \|_F = \sqrt{ \sum_{i,j}{ |A_{i,j}|^2 }}$. Given a vector $c\in \mathbb{C}^n$, the symbol $\|c\|_2 := \sqrt{\sum_{i}|c_i|^2}$ denotes its Euclidean norm.
}

\section{Extended Dynamic Mode Decomposition}
\label{EDMD}
We consider a discrete time dynamical system
\begin{equation}\label{eq:sys}
x^+ = T(x)
\end{equation}
with $T:\mathcal{M} \to \mathcal{M}$, where $\mathcal{M}$ is a topological space\footnote{We choose to work in the general setting of dynamical systems on arbitrary topological spaces which encompasses dynamical systems on finite-dimensional manifolds (in which case one can regard $\mathcal{M}$ as a subset of $\mathbb{R}^n$), as well as infinite dimensional dynamical systems, arising, for example, from the study of partial differential equations or dynamical systems with control inputs~\cite{korda2016linear}.}, and we assume that we are given snapshots of data
\begin{equation}\label{eq:data}
\boldsymbol  X = [x_1,\ldots, x_M],\quad \boldsymbol Y = [y_1,\ldots, y_M]
\end{equation}
satisfying $y_i = T(x_i)$. We do not assume that the data points line on a single trajectory of~(\ref{eq:sys}).

Given a vector space of observables $\mathcal{F}$ such that $\psi: \mathcal{M} \to \mathbb{C}$ and $\psi \circ T \in \mathcal{F}$ for every $\psi \in \mathcal{F}$, we define the Koopman $\mathcal{K}: \mathcal{F}\to \mathcal{F}$ by
\[
\mathcal{K} \psi = \psi \circ T,
\]
where $\circ$ denotes the pointwise function composition. Given a set of linearly independent basis functions $\psi_i \in \mathcal{F}$, $i=1,\ldots,N$, and defining
\begin{equation}\label{eq:FNdef}
\mathcal{F}_N := \mr{span}\{\psi_1,\ldots,\psi_N\},
\end{equation}
the EDMD constructs a finite-dimensional approximation $\mathcal{K}_{N,M}:\mathcal{F}_N\to \mathcal{F}_N$ of the Koopman operator by solving the least-squares problem
\begin{equation}\label{eq:ls}
\min_{A\in \mathbb{C}^{N\times N}} \| A\boldsymbol \psi(\boldsymbol X) - \boldsymbol\psi(\boldsymbol  Y)  \|_F^2 = \min_{A\in \mathbb{C}^{N\times N}} \sum_{i=1}^M \| A\boldsymbol \psi(x_i) - \boldsymbol\psi(y_i)  \|_2^2,
\end{equation}
where
\[
\boldsymbol\psi(\boldsymbol  X) = [\boldsymbol \psi(x_1),\ldots, \boldsymbol \psi(x_M)],\quad \boldsymbol\psi(\boldsymbol  Y) = [\boldsymbol \psi(y_1),\ldots, \boldsymbol \psi(y_M)]
\]
and
\[
\boldsymbol \psi(x) = [\psi_1(x),\ldots,\psi_N(x)]^\top.
\]
Denoting \begin{equation}\label{eq:ANM}
A_{N,M} = \boldsymbol\psi(\boldsymbol Y)\boldsymbol\psi(\boldsymbol X)^\dagger,
\end{equation}
a solution\footnote{In general, the solution to~(\ref{eq:ls}) may not be unique; however, the matrix $A_{N,M} = \boldsymbol\psi(\boldsymbol Y)\boldsymbol\psi(\boldsymbol X)^\dagger$, where $\cdot^
\dagger$ denotes the Moore-Penrose pseudoinverse, is always uniquely defined and $A_{N,M}$ is always a minimizer in~(\ref{eq:ls}).  } to~(\ref{eq:ls}), the finite-dimensional approximation of the Koopman operator
\[
\mathcal{K}_{N,M} : \mathcal{F}_N \to \mathcal{F}_N
\]
is then defined by
\begin{equation}\label{eq:KNM_def}
\mathcal{K}_{N,M} \phi = c_\phi^{\mathsf{H}} A_{N,M}\boldsymbol\psi
\end{equation}
for any $\phi = c_\phi^{\mathsf{H}} \boldsymbol\psi$, $c_\phi\in\mathbb{C}^N$ (i.e., for any $\phi \in \mathcal{F}_N$). The operator $\mathcal{K}_{N,M}$ will be referred to as the EDMD operator.

\section{EDMD as $L_2$ projection}
\label{EDMDl2}
To the best of our knowledge, the results of this section and Section~\ref{EDMDconv} were first obtained in~\cite[Section~3.4]{klus2015numerical} and hinted at already in~\cite{Williamsetal:2015}. Here we rephrase these results in a form more suitable for our purposes.

\new{

From here on we assume\footnote{\new{Since $\mathcal{K}:\mathcal{F}\to\mathcal{F}$, the assumption of $\mathcal{F} = L_2(\mu)$ implies that the composition relation $\phi\circ T$, $\phi\in L_2(\mu)$, gives rise to a well-defined operator from $L_2(\mu)$ to $L_2(\mu)$. In particular this implies that $\| \phi_1\circ T - \phi_2\circ T\|_{L_2(\mu)} = 0$ whenever $\| \phi_1-\phi_2 \|_{L_2(\mu)} = 0$ and that $\int_{\mathcal{M}} |\phi\circ T|^2 \, d\mu < \infty$ for all $\phi \in L_2(\mu)$.}} that $\mathcal{F} = L_2(\mu)$, where $\mu$ is a given positive measure on $\mathcal{M}$. This assumption in particular implies that the basis functions $\psi_i$ in~(\ref{eq:FNdef}) belong to $L_2(\mu)$ and hence $\mathcal{F}_N$ is a closed (but not necessarily invariant) subspace of $L_2(\mu)$. Note that the measure $\mu$ is \emph{not} required to be invariant for~(\ref{eq:sys}) and hence the Koopman operator $\mathcal{K}$ is not necessarily unitary. In practical applications, the measure $\mu$ will typically be the uniform measure on $\mathcal{M}$ or other measure from which samples can be drawn efficiently.

We recall that given an arbitrary positive measure $\nu$ on $\mathcal{M}$, the space $L_2(\nu)$ is the Hilbert space of all measurable functions $\phi:\mathcal{M}\to \mathbb{C}$ satisfying
\[
\| \phi\|_{L_2(\nu)} := \sqrt{\int_{\mathcal{M}} |\phi(x)|^2\,d\nu(x)} < \infty.
\]
Assuming $\mathcal{F}_N $ is a closed subspace of $L_2(\nu)$, the $L_2(\nu)$-projection of a function $\phi\in L_2(\nu)$ onto  $\mathcal{F}_N \subset L_2(\nu)$ is defined by
\begin{equation}\label{eq:projDef}
P_N^\nu \phi = \operatornamewithlimits{arg\,min}_{f\in \mathcal{F}_N} \|f - \phi\|_{L_2(\nu)}= \operatornamewithlimits{arg\,min}_{f\in \mathcal{F}_N} \int_{\mathcal{M}} |f - \phi|^2\,d\nu =\operatornamewithlimits{arg\,min}_{c\in \mathbb{C}^N} \int_{\mathcal{M}} |c^{\mathsf{H}} \boldsymbol\psi - \phi|^2\,d\nu.
\end{equation}

Now, given the data points $x_1,\ldots,x_M$ from~(\ref{eq:data}), we define the empirical measure $\hat{\mu}_M$ by
\begin{equation}\label{eq:emp_meas_def}
\hat\mu_M = \frac{1}{M}\sum_{i=1}^M \delta_{x_i},
\end{equation}
where $\delta_{x_i}$ is the Dirac measure at $x_i$. In particular, the integral of a function $\phi$ with respect to $\hat{\mu}_M$ is given by
\[
\int_\mathcal{M} \phi(x)d\hat\mu_M(x) = \frac{1}{M}\sum_{i=1}^M \phi(x_i).
\]

We remark that the EDMD subspace $\mathcal{F}_N$ defined in~(\ref{eq:FNdef}) is a closed subspace of both $L_2(\hat{\mu}_M)$ and $L_2(\mu)$ and hence the projections $P_N^{\mu}$ and $P_N^{\hat\mu_M}$ are well defined.
}

Now we are ready to state the  following characterization of $\mathcal{K}_{N,M} \phi$:

\begin{theorem}\label{thm:L2proj}
Let $\hat \mu_M$ denote the empirical measure ~(\ref{eq:emp_meas_def}) associated to the sample points $x_1,\ldots, x_M$ and assume that the $N\times N$ matrix
\begin{equation}\label{eq:Mmuhat_def}
M_{\hat\mu_M}  =\frac{1}{M}\sum_{i=1}^M \boldsymbol\psi(x_i)\boldsymbol\psi(x_i)^{\mathsf{H}} = \int_{\mathcal{M}} \boldsymbol\psi \boldsymbol\psi^{\mathsf{H}}\,d\hat\mu_M
\end{equation}
is invertible. Then for any $\phi \in \mathcal{F}_N$
\begin{equation}\label{eq:KoopL2proj}
\mathcal{K}_{N,M}\phi = P_{N}^{\hat{\mu}_M} \mathcal{K} \phi = \operatornamewithlimits{arg\,min}_{f \in \mathcal{F}_N} \| f - \mathcal{K} \phi  \|_{L_2(\hat\mu_M )} ,
\end{equation}
i.e.,
\begin{equation}\label{eq:KoopL2_altern}
\mathcal{K}_{N,M} = P_N^{\hat\mu_M} \mathcal{K}_{|\mathcal{F}_N},
\end{equation}
where $\mathcal{K}_{|\mathcal{F}_N}:\mathcal{F}_N\to\mathcal{F}$ is the restriction of the Koopman operator to $\mathcal{F}_N$.
\end{theorem}
\begin{proof}
Since the matrix $M_{\hat\mu_M}$ is invertible, the least-squares problem~(\ref{eq:ls}) has a unique solution given~by
\[
a_i = \Big(\sum_{j=1}^M \boldsymbol\psi(x_j)\boldsymbol\psi(x_j)^{\mathsf{H}}\Big)^{-1}\sum_{j=1}^M \boldsymbol\psi(x_j)\overline{ \psi_i(y_j)},
\]
where $a_i^{\mathsf{H}} \in \mathbb{C}^{1\times N}$ is the $i^{\mr{th}}$ row of $A_{N,M}$. Therefore
\[
A_{N,M}^{\mathsf{H}} = \Big(\sum_{j=1}^M \boldsymbol\psi(x_j)\boldsymbol\psi(x_j)^{\mathsf{H}}\Big)^{-1}\sum_{j=1}^M \boldsymbol\psi(x_j)\boldsymbol\psi(y_j)^{\mathsf{H}}.
\]
On the other hand, analyzing the minimization problem on the right-hand side of~(\ref{eq:KoopL2proj}), we get for any $\phi = c_\phi^{\mathsf{H}} \boldsymbol \psi$
\[
 \operatornamewithlimits{arg\,min}_{f \in \mathcal{F}_N} \| f - \mathcal{K} \phi  \|_{L_2(\hat\mu_M )}  = \operatornamewithlimits{arg\,min}_{c\in \mathbb{C}^N}  \frac{1}{M}\sum_{i=1}^M ( c^{\mathsf{H}} \boldsymbol\psi(x_i) - c_\phi^{\mathsf{H}}\boldsymbol\psi(y_i))^2
\]
with the unique minimizer (since the minimized functions is strictly convex in $c$)
\[
c = \Big(\sum_{j=1}^M \boldsymbol\psi(x_j)\boldsymbol\psi(x_j)^{\mathsf{H}}\Big)^{-1}\sum_{j=1}^M \boldsymbol\psi(x_j)\boldsymbol\psi(y_j)^{\mathsf{H}} c_\phi = A_{N,M}^{\mathsf{H}} c_\phi.
\]
Hence,  $\operatornamewithlimits{arg\,min}_{f \in \mathcal{F}_N} \| f - \mathcal{K} \phi  \|_{L_2(\hat\mu_M )} = c^{\mathsf{H}}\boldsymbol\psi= c_\phi^{\mathsf{H}} A_{N,M}\boldsymbol\psi = \mathcal{K}_{N,M}\phi $ as desired, where we used~(\ref{eq:KNM_def}) in the last equality.
\end{proof}

Theorem~\ref{thm:L2proj} says that for any function $\phi\in\mathcal{F}_N$, the EDMD operator $\mathcal{K}_{N,M}$ computes the $L_2(\hat\mu_M)$-orthogonal projection of $\mathcal{K}\phi$ on the subspace spanned by $\psi_1,\ldots,\psi_N$.

\begin{remark}If the assumption that the matrix $M_{\hat\mu_M}$ is invertible does not hold, then the solution to the projection problem on the right-hand side of~(\ref{eq:KoopL2proj}) may not be unique\footnote{To be more specific, if the matrix $M_{\hat\mu_M}$ is not invertible, the solution to (\ref{eq:KoopL2proj}) may not be unique when viewed as a member of $ L_2(\mu)$. When viewed as a member of $L_2(\hat\mu_M)$, the solution is unique (since it is a projection onto a closed subspace of a Hilbert space). This is because in this case two functions from $\mathcal{F}_N$ belonging to distinct $L_2(\mu)$ equivalence classes may fall into the same $L_2(\hat\mu_M)$ equivalence class.}. The action of the EDMD operator $\mathcal{K}_{N,M}$ (which is defined uniquely by~(\ref{eq:ANM}) and (\ref{eq:KNM_def})) then selects one solution to the projection problem. In concrete terms, we have
\[
\mathcal{K}_{N,M}\phi\in \Argmin_{f\in\mathcal{F}_N}\| f - \mathcal{K} \phi\|_{L_2(\hat\mu_M)},
\]
where $\Argmin_{f\in\mathcal{F}_N}\| f - \mathcal{K} \phi\|_{L_2(\hat\mu_M)}$ denotes the set of all minimizers of $\|f - \mathcal{K}\phi \|_{L_2(\hat\mu_M)}$ among $f\in\mathcal{F}_N$.
\end{remark}

\section{Convergence of $\mathcal{K}_{N,M}$ as $M\to\infty$}
\label{EDMDconv}
The first step in understanding convergence of EDMD is to understand the convergence of $\mathcal{K}_{N,M}$ as the number of samples $M$ tends to infinity. In this section we prove that \[
\mathcal{K}_{N,M} \to \mathcal{K}_N,
\]
 where
\begin{equation}
\label{eq:KNdef}
\mathcal{K}_N = P_N^\mu \mathcal{K}_{|\mathcal{F}_N},
\end{equation}
provided that the samples $x_1,\ldots,x_M$ are drawn independently from a given probability distribution $\mu$ (e.g., uniform distribution for compact $\mathcal{M}$ or Gaussian for if $\mathcal{M} = \mathbb{R}^n$).

\begin{assumption}[$\mu$ independence]\label{as:indep}
The basis functions $\psi_1,\ldots,\psi_N$ are such that \[\mu\{x\in \mathcal{M} \mid c^{\mathsf{H}} \boldsymbol \psi(x)=0 \} = 0\] for all nonzero $c
\in\mathbb{C}^N$.
\end{assumption}
This is a natural assumption ensuring that the measure $\mu$ is not supported on a zero level set of a linear combination of the basis functions used. It is satisfied if $\mu$ is any measure with the support equal to $\mathcal{M}$ in conjunction with the majority of most commonly used basis functions such as polynomials, radial basis functions with unbounded support (e.g., Gaussian, thin plate splines) etc. This assumption in particular implies that the matrix $M_{\hat\mu_M}$ defined in~(\ref{eq:Mmuhat_def}) is invertible with probability one for $M \ge N$ if $x_j$'s are iid samples from~$\mu$.

\begin{lemma}\label{lem:projConv}
If Assumption~\ref{as:indep} holds, then for any $\phi \in \mathcal{F}$ we have with probability one
\begin{equation}
\lim_{M\to\infty}\|P_N^{\hat\mu_M} \phi -  P_N^{\mu} \phi\| = 0,
\end{equation}
where $\|\cdot \|$ is any norm on $\mathcal{F}_N$ (which are all equivalent since $\mathcal{F}_N$ is finite dimensional).
\end{lemma}
\begin{proof}
We have
\begin{align*}
P_N^{\mu} \phi &= \operatornamewithlimits{arg\,min}_{f\in\mathcal{F}_N} \int_{\mathcal{M}} |f - \phi|^2\, d\mu= \boldsymbol\psi^{\mathsf{H}}\operatornamewithlimits{arg\,min}_{c\in\mathbb{C}^N} \int_{\mathcal{M}} |c^{\mathsf{H}} \boldsymbol \psi - \phi|^2\, d\mu \\ &=\boldsymbol\psi^{\mathsf{H}}\operatornamewithlimits{arg\,min}_{c\in\mathbb{C}^N} \big[c^{\mathsf{H}} M_{\mu} c - 2\Re\{c^{\mathsf{H}} b_{\mu,\phi}\}   \big],
\end{align*}
where
\[
M_\mu  = \int_{\mathcal{M}} \boldsymbol\psi \boldsymbol\psi^{\mathsf{H}}\,d\mu \in \mathbb{C}^{N\times N},\quad b_{\mu,\phi} = \int_{\mathcal{M}}\boldsymbol\psi\overline{\phi}\,d\mu \in \mathbb{C}^N
\]
and we dropped the constant term in the last equality which does not influence the minimizer. By Assumption~\ref{as:indep}, the matrix $M_{\mu}$ is invertible and hence Hermitian positive definite. Therefore the unique minimizer is $c = M_{\mu}^{-1}b_{\mu,\phi}$. Hence
\[
P_N^{\mu}\phi = b_{\mu,\phi}^{\mathsf{H}} M_{\mu}^{-1}\boldsymbol\psi.
\]
On the other hand, the same computation shows that
\[
P_N^{\hat\mu_M}\phi = b_{\hat\mu_M,\phi}^{\mathsf{H}} M_{\hat\mu_M}^{-1}\boldsymbol\psi
\]
with
\[
 b_{\hat\mu_M,\phi} = \int_{\mathcal{M}}\boldsymbol\psi \overline{\phi}\,d\hat\mu_M = \frac{1}{M}\sum_{i=1}^M \boldsymbol\psi(x_i)\overline{\phi(x_i)}
\]
and with the matrix $M_{\hat\mu_M}$ defined in~(\ref{eq:Mmuhat_def}) guaranteed to be Hermitian positive definite by Assumption~\ref{as:indep} with probability one for $M\ge N$. The result then follows by the strong law of large numbers which ensures that \[\lim_{M\to\infty} (b_{\hat\mu_M,\phi}^{\mathsf{H}} M_{\hat\mu_M}^{-1}) = b_{\mu,\phi}^{\mathsf{H}} M_{\mu}^{-1}\]
with probability one since the matrix function $A\mapsto A^{-1}$ is continuous and the samples $x_i$ are iid.
\end{proof}

\begin{theorem}\label{thm:convFin}
If Assumption~\ref{as:indep} holds, then we have with probability one for all $\phi \in \mathcal{F}_N$
\begin{equation}\label{eq:opConvStrongFin}
\lim_{M\to\infty}\|\mathcal{K}_{N,M} \phi -  \mathcal{K}_N \phi\| = 0,
\end{equation}
where $\| \cdot\|$ is any norm on $\mathcal{F}_N$. In particular
\begin{equation}\label{eq:opConvNormFin}
\lim_{M\to\infty}\|\mathcal{K}_{N,M} -  \mathcal{K}_N \| = 0,
\end{equation}
where $\| \cdot\|$ is any operator norm and
\begin{equation}\label{eq:opConvSpecFin}
\lim_{M\to\infty}\mr{dist}\big(\sigma(\mathcal{K}_{N,M}),\sigma(\mathcal{K}_N)\big) = 0,
\end{equation}
where $\sigma(\cdot) \subset \mathbb{C}$ denotes the spectrum of an operator and $\mr{dist}(\cdot,\cdot)$ the Hausdorff metric on subsets of $\mathbb{C}$.
\end{theorem}
\begin{proof}
For any fixed $\phi \in\mathcal{F}_N$ we have by Theorem~\ref{thm:L2proj}
\[
\mathcal{K}_{N,M} \phi = P_{N}^{\hat\mu_M} \mathcal{K} \phi = P_{N}^{\hat\mu_M} (\phi\circ T). \]
By definition of $\mathcal{K}_N \phi$ we have $\mathcal{K}_N \phi =P_N^\mu (\phi\circ T) $ and therefore~(\ref{eq:opConvStrongFin}) holds from Lemma~\ref{lem:projConv} with probability one. Since $\mathcal{F}_N$ is finite dimensional, (\ref{eq:opConvStrongFin}) holds with probability one for all basis functions of $\mathcal{F}_N$ and hence by linearity for all $\phi\in\mathcal{F}_N$. Convergence in the operator norm~(\ref{eq:opConvNormFin}) and spectral convergence~(\ref{eq:opConvSpecFin}) follows from~(\ref{eq:opConvStrongFin}) since the operators $\mathcal{K}_{N,M}$ and $\mathcal{K}_N$ are finite dimensional.
\end{proof}

Theorem~\ref{thm:convFin} tells us that in order to understand the convergence of $\mathcal{K}_{N,M}$ to $\mathcal{K}$ it is sufficient to understand the convergence of $\mathcal{K}_N$ to $\mathcal{K}$. This convergence is analyzed in Section~\ref{KNconv}.

\subsection{Ergodic sampling}\label{sec:ErgodSamp}
The assumption that the samples $x_1,\ldots,x_M$ are drawn independently from the distribution~$\mu$ can be replaced by the assumption that $(T,\mathcal{M},\mu)$ is ergodic and the samples $x_1,\ldots,x_M$ are the iterates of the the dynamical system starting from some initial condition $x\in\mathcal{M}$, i.e., $x_i = T^i(x)$. Provided that Assumption~\ref{as:indep} holds, both Lemma~\ref{lem:projConv} and Theorem~\ref{thm:convFin} hold without change; the statement ``with probability one'' is now interpreted with respect to drawing the initial condition $x$ from the distribution $\mu$. The proofs follow exactly the same argument, only the strong law of large numbers is replaced by the Birkhoff's ergodic theorem in Lemma~\ref{lem:projConv}.

\section{Convergence of $\mathcal{K}_N$ to $\mathcal{K}$}
\label{KNconv}
In this section we investigate convergence of $\mathcal{K}_N$ to $\mathcal{K}$ in the limit as $N$ goes to infinity. Since the operator $\mathcal{K}_N$ is defined on $\mathcal{F}_N$ rather than $\mathcal{F}$, we extend the operator to all of $\mathcal{F}$ by precomposing with $P_N^\mu$, i.e., we study the convergence of $\mathcal{K}_N P_N^\mu = P_N^\mu\mathcal{K} P_N^\mu:\mathcal{F}\to\mathcal{F} $ to $\mathcal{K}:\mathcal{F}\to\mathcal{F}$ as $N\to\infty$. Note that as far as spectrum is concerned, precomposing with $P_N^\mu$ just adds a zero to the spectrum.

\new{
To simplify notation, in what follows we denote the $L_2(\mu)$ norm of a function $f$ by
 \[
\| f\| := \| f\|_{L_2(\mu)} = \sqrt{\int_{\mathcal{M}} |f|^2\,d\mu}
\]
and
the usual $L_2(\mu)$ inner product by
\[
\langle f,g\rangle := \int_{\mathcal{M}} f\overline g\,d\mu.
\]

\subsection{Preliminaries}

Before stating our results, we recall several concepts from functional analysis and operator theory.

\begin{definition}[Bounded operator]
An operator $\mathcal{A}:\mathcal{F}\to\mathcal{F}$ defined on a Hilbert space $\mathcal{F}$ is bounded if
\[
\| \mathcal{A} \| := \sup\limits_{f \in \mathcal{F},\; \| f \|=1} \| \mathcal{A} f \| < \infty.
\]
The quantity $\| \mathcal{A} \| $ is referred as the norm of $\mathcal{A}$.
\end{definition}

\begin{definition}[Convergence in strong operator topology]\label{def:strongConv}
A sequence of bounded operators $\mathcal{A}_i: \mathcal{F} \to \mathcal{F}$  defined on a Hilbert space $\mathcal{F}$ convergences strongly (or in the strong operator topology) to an operator $\mathcal{A} : \mathcal{F}\to \mathcal{F}$ if
\begin{equation}\label{eq:strongConvDef}
 \lim_{i\to\infty}  \| \mathcal{A}_i g - \mathcal{A} g\|
\end{equation}
for all $g \in \mathcal{F}$.
\end{definition}

\begin{definition}[Weak convergence]\label{def:weakConv}
A sequence of elements $f_i \in \mathcal{F}$ of a Hilbert space $\mathcal{F}$ converges weakly to $f \in \mathcal{F}$, denoted $f_i \xrightarrow{w} f$, if
\begin{equation}\label{eq:weakConvDef}
 \lim_{i\to\infty}  \langle f_i, g\rangle  = \langle f,g\rangle
\end{equation}
for all $g \in \mathcal{F}$.
\end{definition}
We emphasize that Definition~\ref{def:strongConv} pertains to convergence of operators defined on $\mathcal{F}$ whereas Definition~\ref{def:weakConv} pertains to convergence of elements of $\mathcal{F}$. We also remark that strong convergence of $f_i \to f$ (i.e., $\|f_i - f\|\to 0$) implies weak convergence but not vice versa. Similarly, convergence in the strong operator topology implies convergence in the weak operator topology (i.e, $\mathcal{A}_i g \xrightarrow{w} \mathcal{A}_g$ for all $g$) but does not imply convergence in the operator norm (i.e., $\|\mathcal{A}_i - \mathcal{A} \| \to 0$).

In our setting of $\mathcal{F} = L_2(\mu)$, the statements~(\ref{eq:strongConvDef}) and~(\ref{eq:weakConvDef}) translate to the requirements that, respectively,
\[
\lim_{i\to\infty}  \sqrt{\int_{\mathcal{M}} | \mathcal{A}_i g - \mathcal{A} g |^2\,d\mu} = 0\qquad \mr{and}\qquad  \lim_{i\to\infty}  \int_{\mathcal{M}} f_i\overline g \,d\mu = \int_{\mathcal{M}} f\overline g \,d\mu
\]
for all $g\in L_2(\mu)$.
}

For the remainder of this work, we invoke the following assumption:
\begin{assumption}\label{as:bounded}
The following conditions hold:
\begin{enumerate}
\item The Koopman operator $\mathcal{K}:\mathcal{F} \to \mathcal{F}$ is bounded.
\item The observables $\psi_1,\ldots,\psi_N$ defining $\mathcal{F}_N$ are selected from a given orthonormal basis of $\mathcal{F}$, i.e., $(\psi_i)_{i=1}^\infty$ is an orthonormal basis of $\mathcal{F}$.
\end{enumerate}
\end{assumption}
The first part of the assumption holds for instance when $T$ is invertible, Lipschitz with Lipschitz inverse and $\mu$ is the Lebesgue measure on $\mathcal{M}$ (or any measure absolutely continuous w.r.t. the Lebesgue measure with bounded and strictly positive density). The second part of the assumption is non-restrictive as any countable dense subset of $\mathcal{F}$ can be turned into an orthonormal basis using the Gram-Schmidt  process.

\subsection{Convergence in strong operator topology}
In this section we prove convergence in the strong operator topology (Definition~\ref{def:strongConv}) of $\mathcal{K}_N P_N^\mu$ to $\mathcal{K}$. First, we need the following immediate lemma:

\begin{lemma}\label{lem:projStrong}
If $(\psi_i)_{i=1}^\infty$ form an orthonormal basis of $\mathcal{F} = L_2(\mu)$, then $P_N^\mu$ converge strongly to the identity operator $I$ and in addition $\| I - P_N\| \le 1 $ for all $N$.
\end{lemma}
\begin{proof}
Let $\phi = \sum_{i=1}^\infty c_i\psi_i$ with $\| \phi\| = 1$. Then by Parseval's identity $\sum_{i=1}^\infty |c_i|^2 = 1$ and
 \[\|P_N^\mu \phi - \phi \| = \Bigg\| \sum_{i=N+1}^\infty c_i\psi_i \Bigg\| =  \sum_{i=N+1}^\infty |c_i|^2 \to 0\]
with $\sum_{i=N+1}^\infty |c_i|^2 \le 1$ for all $N$.
\end{proof}

Now we are ready to prove strong convergence of $P_N^\mu\mathcal{K} P_N^\mu$ to $\mathcal{K}$.
\begin{theorem}\label{thm:strongConv}
If Assumption~\ref{as:bounded} holds, then the sequence of operators $\mathcal{K}_NP_N^\mu=P_N^\mu \mathcal{K} P_N^\mu $ converges strongly to $\mathcal{K}$ as $N\to \infty$, i.e.,
\[
\lim_{N\to\infty} \int_{\mathcal{M}} | \mathcal{K}_N P_N^\mu \phi - \mathcal{K}\phi |^2\,d\mu  = 0
\]
for all $\phi \in \mathcal{F}$.
\end{theorem}
\begin{proof}
Let $\phi \in \mathcal{F}$ be given. Then, writing $\phi=P_N^\mu \phi + (I -P_N^\mu)\phi$ we have
\begin{align*}
\| P_N^\mu \mathcal{K} P_N^\mu \phi - \mathcal{K} \phi \| &= \| (P_N^\mu -I)\mathcal{K} P_N^\mu \phi + \mathcal{K}(P_N^\mu-I)\phi    \| \le \| (P_N^\mu -I)\mathcal{K} P_N^\mu \phi  \|  + \|\mathcal{K} \|  \|( I - P_N^\mu)\phi    \|  \\
& \le \| (P_N^\mu -I)\mathcal{K} \phi  \| + \| (P_N^\mu -I)\| \| \mathcal{K} P_N^\mu \phi -\mathcal{K}  \phi  \| +  \|\mathcal{K} \|  \|( I - P_N^\mu)\phi    \| \to 0
\end{align*}
by Lemma~\ref{lem:projStrong} and by the fact that $\mathcal{K} P_N^\mu\phi \to \mathcal{K}\phi$ since $\mathcal{K}$ is continuous by Assumption~\ref{as:bounded}.
\end{proof}

\subsection{Weak spectral convergence}

Unfortunately, strong converge does not in general guarantee convergence of the spectra of the operators. This is guaranteed only if the operators converge in the operator norm\footnote{A sequence of operators $\mathcal{A}_i$ converges in the operator norm to an operator $\mathcal{A}$ if $\lim_{i\to\infty}\|\mathcal{A}_i -\mathcal{A}\| = 0 $.}. \new{An important exception to this is the case of $\mathcal{F}_N$ being an invariant subspace, i.e., $\mathcal{K} f \in \mathcal{F}_N$ for all $f\in\mathcal{F}_N$ in which case the spectra of $\mathcal{K}_N$ and $\mathcal{K}_{|\mathcal{F}_N}$ coincide. Here, however, we \emph{do not assume} that $\mathcal{F}_N$ is invariant and  prove certain spectral convergence results in a weak sense}. In particular, we prove convergence of the eigenvalues of $\mathcal{K}_N$ along a subsequence and weak convergence of the associated eigenfunctions (see Definition~\ref{def:weakConv}), provided that the weak limit of the eigenfunctions is nonzero.

\begin{theorem}\label{thm:weakConv}
If Assumption~\ref{as:bounded} holds and $\lambda_N$ is a sequence of eigenvalues of $\mathcal{K}_N$ with the associated normalized eigenfunctions $\phi_N \in \mathcal{F}_N$, $\|\phi_N  \|=1$, then there exists a subsequence $(\lambda_{N_i},\phi_{N_i})$ such that
\[
\lim_{i\to\infty}\lambda_{N_i} = \lambda,\quad \phi_{N_i} \xrightarrow{w} \phi,
\]
where $\lambda \in \mathbb{C}$  and $\phi\in \mathcal{F}$ are such that $\mathcal{K}\phi = \lambda \phi$. In particular if $\| \phi\| \ne 0$, then $\lambda$ is an eigenvalue of $\mathcal{K}$ with eigenfunction $\phi$.
\end{theorem}
\begin{proof}
First, observe that since $\mathcal{K}_N \phi_N = \lambda_N \phi_N$ with $\phi_N \in \mathcal{F}_N$, we also also have  $P_N^\mu \mathcal{K} P_N^\mu \phi_N =  \lambda_N \phi_N $. Hence $|\lambda_N|  \le  \| P_N^\mu \mathcal{K} P_N^\mu   \| \le \| \mathcal{K}\| < \infty$ by Assumption~\ref{as:bounded} and the fact that $\| P_N^\mu\| \le 1 $. Therefore the sequence $\lambda_N$ is bounded. Since $\phi_N$ is normalized and hence bounded, by weak sequential compactness of the unit ball of a Hilbert space (which follows from the Banach-Alaoglu theorem~\cite[Theorems 3.15]{rudinFA} and Eberlein-\v Smulian theorem~\cite{dunford_schwartz}), there exists a subsequence $(\lambda_{N_i},\phi_{N_i})$ such that $\lambda_{N_i}\to\lambda $ and $\phi_{N_i}\xrightarrow{w} \phi $.

It remains to prove that $(\lambda,\phi)$ is an eigenvalue-eigenfunction pair of $\mathcal{K}$. For ease of notation, set $\lambda_{i} = \lambda_{N_i}$ and $\phi_i  = \phi_{N_i}$. Denote $\hat\mathcal{K}_i = \mathcal{K}_{N_i} P_{N_i}^\mu = P_{N_i}^\mu \mathcal{K} P_{N_i}^\mu$ and observe that $\hat\mathcal{K}_i \phi_i = \lambda_i\phi_i$ for all $i$. Then we have
\begin{align*}
  \mathcal{K}\phi =  \hat\mathcal{K}_i(\phi-\phi_i) + (\mathcal{K}-\hat\mathcal{K}_i)\phi + \hat\mathcal{K}_i\phi_i.
\end{align*}
Taking the inner product with an arbitrary $f\in\mathcal{F}$ and using the fact that $\hat\mathcal{K}_i\phi_i = \lambda_i\phi_i$, we get
\[
 \langle \mathcal{K}\phi,f \rangle =  \langle\hat\mathcal{K}_i(\phi-\phi_i),f \rangle+\langle(\mathcal{K}-\hat\mathcal{K}_i)\phi,f\rangle + \langle\lambda_i\phi_i ,f \rangle.
\]
Now, the second term on the right hand side $\langle(\mathcal{K}-\hat\mathcal{K}_i)\phi,f\rangle\to 0$ since $\hat\mathcal{K}_i$ converges strongly to $\mathcal{K}$ by Theorem~\ref{thm:strongConv}. The last term $\langle\lambda_i\phi_i ,f \rangle \to \langle \lambda\phi,f\rangle$ since $\lambda_i\to\lambda$ and $\phi_i \xrightarrow{w} \phi$. It remains to show that the first term converges to zero. We have
\[\langle \hat\mathcal{K}_i(\phi-\phi_i),f \rangle = \langle P_{N_i}^\mu\mathcal{K} P_{N_i}^\mu(\phi-\phi_i),f \rangle = \langle \mathcal{K} (P_{N_i}^\mu\phi-\phi_i), P_{N_i}^\mu f \rangle,    \]
where we used the fact that $P_{N_i}^\mu$ is self-adjoint and $\phi_i \in \mathcal{F}_{N_i}$ and hence $P_{N_i}^\mu \phi_i = \phi_i$. Denote $h_i := \mathcal{K} (P_{N_i}^\mu\phi-\phi_i)$. We will show that $h_i\xrightarrow{w} 0$. Indeed, denoting $\mathcal{K}^\star$ the adjoint of $\mathcal{K}$, we have
\[
\langle \mathcal{K} (P_{N_i}^\mu\phi-\phi_i),f\rangle = \langle (P_{N_i}^\mu\phi-\phi + \phi -\phi_i),\mathcal{K}^\star f\rangle = \langle P_{N_i}^\mu\phi-\phi ,\,\mathcal{K}^\star f\rangle + \langle \phi -\phi_i,\,\mathcal{K}^\star f\rangle \to 0,
\]
since $P_{N_i}^\mu$ converges strongly to the identity (Lemma~\ref{lem:projStrong}) and $\phi_i \xrightarrow{w} \phi$. Finally, we show that $\langle h_i, P_{N_i}^\mu f\rangle \to 0$. We have
\[
\langle h_i, P_{N_i}^\mu f\rangle  = \langle h_i, P_{N_i}^\mu f - f\rangle + \langle h_i, f\rangle.
\]
The second term goes to zero since $h_i\xrightarrow{w} 0$. For the first term we have
\[
\langle h_i, P_{N_i}^\mu f - f\rangle \le \| h_i\| \|P_{N_i}^\mu f - f  \| \to 0
\]
since $P_{N_i}^\mu$ converges strongly to the identity operator (Lemma~\ref{lem:projStrong}) and $h_i$ is bounded since $\mathcal{K}$ is bounded by Assumption~\ref{as:bounded}, $\| P_{N_i}^\mu\| \le 1$ and $\| \phi_i \| \le 1$.
Therefore we conclude that
\[
 \langle \mathcal{K}\phi,f \rangle = \lim_{i\to\infty} \langle \lambda_i \phi_i,f\rangle =  \langle \lambda \phi ,f \rangle
\]
for all $f \in \mathcal{F}$. Therefore $\mathcal{K} \phi = \lambda \phi$.
\end{proof}

\textbf{Example}\;\; {\it As an example demonstrating that the assumption that the weak limit of $\phi_N$ is nonzero is important, consider $\mathcal{M} = [0,1]$, $T(x) = x$ and $\mu$ the Lebesgue measure on $[0,1]$. In this setting, the Koopman operator $\mathcal{K}: L_2(\mu)\to L_2(\mu)$ is the identity operator with the spectrum being the singleton $\sigma(\mathcal{K})=\{1\}$. However, given \emph{any} $\lambda \in \mathbb{C}$ and the sequence of functions $\phi_N = \sqrt{2}\sin(2\pi N x)$, we have \[\mathcal{K}\phi_N - \lambda\phi_N = \phi_N - \lambda\phi_N  = (1-\lambda)\sqrt{2}\sin(2\pi N x)\xrightarrow{w} 0\]
with $\| \phi_N\|^2 = \int_{0}^1 2\sin^2(2\pi N x)\,dx = 1$. Therefore if $\phi_N$ were the eigenfunctions of $\mathcal{K}_N$ with eigenvalues $\lambda_N\to \lambda \ne 1$, then the sequence $\lambda_N$ would converge to a spurious eigenvalue $\lambda$. Fortunately, in this case, we have $\sigma(\mathcal{K}_N) = \{ 1\}$ and hence no spurious eigenvalues exist; however, in general, we cannot rue out this possibility, at least not as far as the statement of  Theorem~\ref{thm:weakConv} goes. See Figure~\ref{fig:sinus} for illustration.

This example, with the highly oscillatory functions $\phi_N$, may motivate practical considerations in detecting spurious eigenvalues, e.g., using Sobolev type (pseudo) norms $\int_{\mathcal{M}} \| \nabla \phi_N\|^2\,d\mu$ or other metrics of oscillatoriness. See, e.g., \cite{giannakis2016data} for the use of Sobolev norms in the context of Koopman data analysis and forecasting.}

\begin{figure*}[th]
\begin{picture}(140,170)
\put(130,0){\includegraphics[width=80mm]{./Figures/SinusNx.pdf}}
\put(245,0){$x$}
\put(105,90){$\phi_N(x)$}
\end{picture}
\caption{Graph of the functions $\phi_N(x) = \sqrt{2}\sin(2\pi Nx)$ for $N = 1$ and $N=10$. These functions satisfy $\| \phi_N\|_{L_2}=1$ and $\phi_N\xrightarrow{w} 0$ as $N\to\infty$. If such $\phi_N$ happen to be eigenfunctions of $\mathcal{K}_N$ with eigenvalues $\lambda_N$, then the accumulation points of the sequence $(\lambda_N)_{N=1}^\infty$ need not be eigenvalues of the Koopman operator.}
\label{fig:sinus}
\end{figure*}

As an immediate corollary of Theorem~\ref{thm:weakConv}, we get:
\begin{corollary}\label{thm:weakConv_NM}
If Assumption~\ref{as:bounded} holds and $\lambda_{N,M}$ is a sequence of eigenvalues of $\mathcal{K}_{N,M}$ with the associated normalized eigenfunctions $\phi_{N,M} \in \mathcal{F}_N$, $\|\phi_{N,M}  \|=1$, then there exists a subsequence $(\lambda_{N_i,M_j},\phi_{N_i,M_j})$ such that with probability one
\[
\lim_{i\to\infty}\lim_{j\to\infty}\lambda_{N_i,M_j} = \lambda,\quad  \lim_{i\to\infty}\lim_{j\to\infty}\langle \phi_{N_i,M_j},f\rangle = \langle \phi,f\rangle,
\]
for all $f \in \mathcal{F}$, where $\lambda \in \mathbb{C}$  and $\phi\in \mathcal{F}$ are such that $\mathcal{K}\phi = \lambda \phi$. In particular if $\| \phi\| \ne 0$, then $\lambda$ is an eigenvalue of $\mathcal{K}$ with eigenfunction $\phi$.

\end{corollary}
\begin{proof}
First notice that since $\mathcal{K}_{N,M} \to \mathcal{K}_N$ in the operator norm (Theorem~\ref{thm:convFin}) and $\|\mathcal{K}_N\| \le \|\mathcal{K} \| < \infty$, the sequence $\lambda_{N,M}$ is bounded. Since $\phi_{N,M}$ are normalized and hence bounded,  we can extract a subsequence $(\lambda_{N,M_j},\phi_{N,M_j})$ such that $\lim_{j\to\infty} \lambda_{N,M_j}=\lambda_N \in \mathbb{C}$ and $\lim_{j\to\infty} \phi_{N,M_j} = \phi_N \in \mathcal{F}_N$ (strong convergence as $\mathcal{F}_N$ is finite dimensional). Then
\[
\mathcal{K}_N\phi_N = (\mathcal{K}_N - \mathcal{K}_{N,M_j})\phi_N + \mathcal{K}_{N,M_j}(\phi_N-\phi_{N,M_j}) + \mathcal{K}_{N,M_j}\phi_{N,M_j}.
\]
Since $\mathcal{K}_{N,M_j}$ converges strongly to $\mathcal{K}_N$ with probability one (Theorem~\ref{thm:convFin}) and since $\phi_{N,M_j}$ converges strongly to $\phi_N$, the first two terms go to zero with probability one as $j$ tends to infinity. The last term is equal to $\lambda_{N,M_j} \phi_{N,M_j}$ and hence necessarily $\mathcal{K}\phi_N = \lambda_N \phi_N$, $\| \phi_N\| =1$, with probability one. The result then follows from Theorem~\ref{thm:weakConv}.
\end{proof}

\section{Implications for finite-horizon predictions}\label{sec:finiteHor}
One of the main roles of an approximation to the Koopman operator is to provide a \emph{prediction} of the evolution of a given observable. Whereas obtaining accurate predictions over an infinite-time horizon cannot be expected in general from the EDMD approximation of the Koopman operator, a prediction over any \emph{finite horizon} is asymptotically, as $N\to\infty$, exact when the prediction error is measured in the $L_2(\mu)$ norm:
\begin{theorem}\label{thm:predFin}
Let $f\in \mathcal{F}^n$ be a given (vector) observable\footnote{We choose to state the theorem for vector observables as this is the form of prediction typically encountered in practice. For a vector observable $f\in \mathcal{F}^n$, the norm $\| f\|$ is defined by $\sum_{i=1}^n \|f_i \|_{L_2(\mu)}, $ where $f_i\in\mathcal{F}$ is the $i^{\mr{th}}$ component of $f$.} and let Assumption~\ref{as:bounded} hold. Then for any $\Omega \in \mathbb{N}$ we have
\begin{equation}\label{eq:predFin_gen}
\lim_{N\to\infty} \sup_{i\in\{1,\ldots,\Omega\}} \| (\mathcal{K}_N)^i P_N^\mu f - \mathcal{K}^i f  \| = 0.
\end{equation}
In particular, if $f\in \mathcal{F}_{N_0}^n$ for some $N_0 \in \mathbb{N}$, then \begin{equation}\label{eq:predFin_spec}
\lim_{N\to\infty} \sup_{i\in\{1,\ldots,\Omega\}} \| (\mathcal{K}_N)^i  f - \mathcal{K}^i f  \| = 0.
\end{equation}
\end{theorem}
\begin{proof}
We proceed by induction. Let $f\in \mathcal{F}$. For $\Omega=1$, the result is exactly Theorem~\ref{thm:strongConv}. Let the result hold form some $\Omega \in \mathbb{\mathbb{N}}$. It is sufficient to prove that $\| (\mathcal{K}_N)^{\Omega+1} P_N^\mu f - \mathcal{K}^{\Omega+1} f  \| \to 0$ as $N\to \infty$. We have
\begin{align*}
\| (\mathcal{K}_N)^{\Omega+1} P_N^\mu f - \mathcal{K}^{\Omega+1} f  \| & = \| \mathcal{K}_N(\mathcal{K}_N)^{\Omega} P_N^\mu f - \mathcal{K}\Kc^{\Omega} f  \|  = \| \mathcal{K}_N g_N - \mathcal{K} g  \| \\
&\le \| \mathcal{K}_N g- \mathcal{K} g \| + \| \mathcal{K}_N(g_N - g) \| \le  \| \mathcal{K}_N g- \mathcal{K} g \| + \| \mathcal{K}\| \| g_N - g \|.
\end{align*}
where $g_N = (\mathcal{K}_N)^{\Omega} P_N^\mu f$ and $g = \mathcal{K}^{\Omega} f$; in the last inequality we used the fact that $\|\mathcal{K}_N\| \le \| K\|$. The term  $\| \mathcal{K}_N g- \mathcal{K} g \| $ tends to zero by Theorem~\ref{thm:strongConv}, whereas the term $\| g_N - g \| \to 0$ by the induction hypothesis. This proves~(\ref{eq:predFin_gen}) for a scalar observable $f\in \mathcal{F}$. The general result with a vector valued observable $f\in \mathcal{F}^n$ follows by applying the the same reasoning to each component of $f$. The result~(\ref{eq:predFin_spec}) follows from (\ref{eq:predFin_gen}) since if $f\in \mathcal{F}_{N_0}^n$ for some $N_0\in\mathbb{N}$, then $P_N^\mu f = f$ for $N\ge N_0$.
\end{proof}

To be more specific on practical use of $\mathcal{K}_N$ for prediction, assume that $f\in \mathcal{F}_{N_0}^n$ for some $N_0\in\mathbb{N}$. Then for all $N\ge N_0$ there exists a matrix $C_N\in \mathbb{R}^{n
\times N}$ such that $f = C_N\boldsymbol\psi_N$, where $\boldsymbol\psi_N = [\psi_1,\ldots,\psi_N]^\top$ are the observables used in EDMD. Assume that an initial state $x_0$ is given and the values of the observables $\boldsymbol \psi_N(x_0)$ are known and we wish to predict the value of the observable $f$ at a state $x_i = T^i(x_0)$, i.e., $i$ steps ahead in the future. Using $\mathcal{K}_N$, this prediction is given by $C_N A_N^i \boldsymbol\psi_N(x_0)$, where \[A_N = \lim_{M\to\infty} A_{N,M}\] with $A_{N,M}$ defined\footnote{In Section~\ref{analyEDMD}, we show how the matrix $A_N$ can be constructed analytically.} in~(\ref{eq:ANM}). Theorem~\ref{thm:predFin} then says that
\begin{equation}\label{eq:finitePred_specpec}
\lim_{N\to\infty} \int_{\mathcal{M}} \| C_N A_N^i \boldsymbol\psi_N   - f\circ T^i \|^2_2\,d\mu = 0\qquad \forall\, i\in\mathbb{N}.
\end{equation}
A typical application of Theorem~\ref{thm:predFin} is the prediction of the future state $x$ of the dynamical system~(\ref{eq:sys}) with a finite-dimensional state-space $\mathcal{M}\subset \mathbb{R}^n$. In this case, one simply sets $f(x) = x$. A crucial feature of the predictor obtained in this way is its \emph{linearity} in the ``lifted state'' $z=\boldsymbol\psi_N(x)$, allowing linear tools to address a nonlinear problem. This concept was succesfully applied to model predictive control in~\cite{korda2016linear} and to state estimation in~\cite{surana_estim}.

\begin{remark}
If $A_{N,M}$ is used instead of $A_N$ in Theorem~\ref{thm:predFin} and Equation~(\ref{eq:finitePred_specpec}), then the same convergence results hold with a double limit, first taking the number of samples $M$ to infinity and then the number of basis functions $N$. In particular, we get
\begin{equation}\label{eq:finitePred_specpec_NM}
\lim_{N\to\infty} \lim_{M\to\infty} \int_{\mathcal{M}} \| C_N A_{N,M}^i \boldsymbol\psi_N   - f\circ T^i \|^2_2\,d\mu = 0\qquad \forall\, i\in\mathbb{N}.
\end{equation}
\end{remark}

\iffalse
{\color{blue}
\subsection{The case of an unbounded operator}
In this section we show that convergence of finite-horizon predictions can be proven without the assumption of the operator $\mathcal{K}$ being bounded. As before we shall assume that $(\psi_i)_{i=1}^N$ is an orthonormal basis of $L_2(\mu)$ and in addition we assume that $\psi_i$'s are uniformly bounded, i.e., there exists a constant $m < \infty$ such that $|\psi_i(x)| \le m$ for all $i\in\mathbb{N}$ and for all $x \in \mathcal{M}$. We shall also assume that the measure $\mu$ is finite, i.e., $\mu(\mathcal{M}) < \infty$.

First we note that in this setting Theorems~\ref{thm:L2proj} and \ref{thm:convFin} are still valid. Hence, given $f\in \mathcal{F}_{N_0}$ such that $f = c_N^\top \boldsymbol\psi$ for any $N\ge N_0$, we have
\begin{equation}\label{eq:unboundAux}
c_{N}^\top A_N\boldsymbol\psi_N = P_N^\mu (f\circ T).
\end{equation}
Since $f\in \mathcal{F}_{N_0}$ and the $\psi_i$'s are bounded,  $f \circ T \in L_2(\mu)$ and hence
\begin{equation}\label{eq:indstep1}
\lim_{N\to\infty} (c_{N}^\top A_N\boldsymbol\psi_N - f\circ T)^2\, d
\mu.
\end{equation}

\begin{theorem}\label{thm:unbounded}
\[
\lim_{N\to\infty} (c_{N}^\top A_N^i\boldsymbol\psi_N - f\circ T^i)^2\, d
\mu,\quad \forall\, i\ge 1,
\]
where $f\circ T^i$ denotes repeated composition, i.e., $f\circ T^i = f\underbrace{\circ T\circ T\circ \ldots \circ T}_{i\, \mr{times}}$.
\end{theorem}
\begin{proof}
We proceed by induction. For $i=1$, the results is~(\ref{eq:indstep1}). Let the result hold for $i \ge 1$. We want to prove that $(c_{N}^\top A_N^{i+1}\boldsymbol\psi_N - f\circ T^{i+1})^2 \to 0$.
Using (\ref{eq:unboundAux})  with $f$ replaced by $c_N^\top A_N^i\boldsymbol\psi_N \in  \mathcal{F}_N$ we get
\[
c_{N}^\top A_N^{i+1}\boldsymbol\psi_N = c_{N}^\top A_N^{i} A_N\boldsymbol\psi_N = P_N^\mu (c_N^\top A_N^i\boldsymbol\psi_N\circ T).
\]
Proceeding we obtain
\[
c_{N}^\top A_N^{i+1}\boldsymbol\psi_N = P_N^\mu (c_N^\top A_N^i\boldsymbol\psi_N\circ T) = P_N^\mu ( (P_N^\mu(c_N^\top A_N^{i-1} \boldsymbol\psi_N\circ T))\circ T ) = P_N^\mu\mathcal{K} P_N^\mu\mathcal{K}.
\]
In summary

\[
c_{N}^\top A_N^{i}\boldsymbol\psi_N = \underbrace{P_N \mathcal{K}  \ldots P_N\mathcal{K}}_{i\;\mr{times}}.
\]
{\color{red} Probably need to assume that $P_N \psi_i\circ T$ converges pointwise or prove that if $f = \sum_{i=1}^\infty\alpha_i \psi_i$, then $f\circ T = \sum_{i=1}^\infty \alpha_i \psi_i\circ T$}

\end{proof}
}
\fi

\section{Analytic EDMD}
\label{analyEDMD}
The results of the previous sections suggests a variation of the EDMD algorithm provided that the mapping $T$ is known in closed form and provided that the basis functions $\psi_i$ are such that the integrals of $\int_{\mathcal{M}} \psi_i \overline{\psi_j}\,d\mu$ and $\int_{\mathcal{M}} (\psi_i\circ T)\overline{\psi_j}\,d\mu$ can be computed analytically. This is the case in particular if $T$ and $\psi_i$'s are simple functions such as multivariate polynomials or trigonometric functions and $\mu$ is the uniform distribution over a simple domain $\mathcal{M}$ such as a box or a ball, or, e.g., a Gaussian distribution over $\mathbb{R}^n$.

Provided that such analytical evaluation is possible, one can circumvent the sampling step of EDMD and construct directly $\mathcal{K}_N$ rather than $\mathcal{K}_{N,M}$. Indeed, define
\begin{equation}
A_N = M_{T,\mu} M_{\mu}^{-1},
\end{equation}
where
\[
M_{\mu} = \int_{\mathcal{M}} \boldsymbol\psi \boldsymbol\psi^{\mathsf{H}}\,d\mu,\quad M_{T,\mu} = \int_{
\mathcal{M}}(\boldsymbol\psi\circ T)\boldsymbol\psi^{\mathsf{H}}\,d\mu.
\]
Then the operator from $\mathcal{F}_N$ to $\mathcal{F}_N$ defined by $c^{\mathsf{H}} \boldsymbol \psi\mapsto c^{\mathsf{H}} A_N\boldsymbol\psi$ is exactly $\mathcal{K}_N = P_N^\mu \mathcal{K}_{|\mathcal{F}_N}$.
\begin{theorem}
If the matrix $M_\mu$ is invertible, then for any $\phi = c_\phi^{\mathsf{H}} \boldsymbol\psi \in \mathcal{F}_N$ we have
\[
c_\phi^{\mathsf{H}} A_N\boldsymbol\psi = \mathcal{K}_N\phi.
\]
\end{theorem}
\begin{proof}
Given $\phi = c_\phi^{\mathsf{H}} \boldsymbol\psi$ we get
\[
\mathcal{K}_N\phi = P_N^\mu \mathcal{K} \phi = \boldsymbol\psi^{\mathsf{H}}\operatornamewithlimits{arg\,min}_{c\in\mathbb{C}^N}\int_{\mathcal{M}}[c^{\mathsf{H}}\boldsymbol\psi - c_\phi^{\mathsf{H}} (\boldsymbol \psi\circ T)]^2\,d\mu = \boldsymbol\psi^{\mathsf{H}}\operatornamewithlimits{arg\,min}_{c\in\mathbb{C}^N}\big[ c^{\mathsf{H}} M_{\mu} c - 2\Re\{c^{\mathsf{H}} M_{T,\mu}^{\mathsf{H}} c_{\phi}\}\big]
\]
with the unique minimizer $c = M_{\mu}^{-1}M_{T,\mu}^{\mathsf{H}} c_\phi$ (since $M_\mu$ is invertible, therefore Hermitian positive definite, and hence the minimized function is strictly convex). Therefore as desired \[\mathcal{K}_N\phi = c^{\mathsf{H}}\boldsymbol\psi = c_\phi^{\mathsf{H}} M_{T,\mu} M_{\mu}^{-1}\boldsymbol\psi = c_\phi^{\mathsf{H}} A_N \boldsymbol\psi. \]
\end{proof}

\paragraph{Example} {\it In order to demonstrate the use of Analytic DMD, we compare the spectra of $\mathcal{K}_N$ and $\mathcal{K}_{N,M}$ for various values of $M$. The system considered is the logistic map
\[
x^+ = 2x^2 - 1,\quad x \in [-1,1].
\]
The measure $\mu$ is taken to be the uniform distribution on $[-1,1]$, which is not invariant and hence the dynamics is not measure preserving. The finite-dimensional subspace $\mathcal{F}_N$ is the space of all polynomials of degree no more than eight. For numerical stability we chose a basis of this subspace to be the Laguerre polynomials scaled such that they are orthonormal with respect to the uniform measure on $[-1,1]$. Spectra of $\mathcal{K}_N$ and $\mathcal{K}_{N,M}$ for $M=10^2$, $M=10^3$ and $M=10^5$ are depicted in Figure~\ref{fig:analDMDspec}. We observe that a relatively large number of samples $M$ is required to obtain an accurate approximation of the spectrum of $\mathcal{K}_{N}$. This example demonstrates that a special care must be taken in practice when drawing conclusions about spectral quantities based on a computation with a small number of samples. On the other hand, Figure~\ref{fig:analDMDpred} suggests that, at least on this example, predictions generated by $\mathcal{K}_{N,M}$ are less affected by sampling. Indeed, even for $M=100$ the prediction accuracy of $\mathcal{K}_{N,M}$ is comparable to that of $\mathcal{K}_N$ and for $M=1000$ the two predictions almost coincide.
\begin{figure*}[th]
\begin{picture}(140,135)
\put(-10,0){\includegraphics[width=60mm]{./Figures/AnalyticDMD_M100.pdf}}
\put(155,0){\includegraphics[width=60mm]{./Figures/AnalyticDMD_M1000.pdf}}
\put(320,0){\includegraphics[width=60mm]{./Figures/AnalyticDMD_M100000.pdf}}

\put(230,0){\footnotesize $\Re\{\lambda\}$}
\put(-25,60){\footnotesize $\Im\{\lambda\}$}
\put(60,105){\footnotesize $ M = 10^2$}
\put(225,105){\footnotesize $ M = 10^3$}
\put(390,105){\footnotesize $ M = 10^5$}

\end{picture}
\caption{Comparison of the spectra of $\mathcal{K}_N$ (blue circles) computed using Analytic DMD and the spectra of $\mathcal{K}_{N,M}$ for different values of $M$ (red crosses).}
\label{fig:analDMDspec}
\end{figure*}

\begin{figure*}[th]
\begin{picture}(140,180)
\put(130,0){\includegraphics[width=80mm]{./Figures/AnalyticDMD_predComp.pdf}}
\put(235,0){time}
\put(120,90){ $x$}
\end{picture}
\caption{Comparison of predictions generated using $\mathcal{K}_N$ and using $\mathcal{K}_{N,M}$ for different values of $M$.}
\label{fig:analDMDpred}
\end{figure*}

}

\section{Convergence of $\mathcal{K}_{N,N}$}
\label{KNNconv}
In this section we investigate what happens when we simultaneously increase the number of basis functions $N$ and the number of samples $M$. We treat the special case of $M = N$ for which interesting conclusions can be drawn. Set therefore $M = N$ and denote $\lambda_N = \lambda_{N,N}$ any eigenvalue of $\mathcal{K}_{N,N}$ and $\phi_N = \phi_{N,N} \in \mathcal{F}_N$, $\|\phi_N\|_{C(\mathcal{M})} =1$, the associated eigenfunction, where $\|\phi \|_{C(\mathcal{M})} = \sup_{x\in\mathcal{M}} |\phi(x)|$; such normalization is possible if the basis functions $\psi_i$ are continuous and $\mathcal{M}$ compact, which we assume in this section. First notice that, assuming $M_{\hat\mu_N}$ invertible, for $N=M$ the system of equations
\[
\boldsymbol\psi(\boldsymbol Y) = A\boldsymbol\psi(\boldsymbol X)
\]
with the unknown $A\in \mathbb{R}^{N\times N}$ has a solution and hence the minimum in the least squares problem~(\ref{eq:ls}) is zero. In other words, for any $f\in\mathcal{F}_N$, the EDMD operator $\mathcal{K}_{N,N}$ applied to $f$ matches the value of the Koopman operator applied to $f$ on the samples points $x_1,\ldots,x_N$:
\[
(\mathcal{K} f)(x_i) = (\mathcal{K}_{N,N}f)(x_i)
\]
for all $f\in \mathcal{F}_N$. This relation is in particular satisfied for the eigenfunctions $\phi_N$ of $\mathcal{K}_{N,N}$, obtaining
\[
(\phi_N\circ T)(x_i) = \lambda_N \phi_N(x_i).
\]
Multiplying by an arbitrary $h\in\mathcal{F}$ and integrating with respect to the empirical measure supported on the sample points~(\ref{eq:emp_meas_def}), we get
\begin{equation}\label{eq:eig_sample}
\int_{\mathcal{M}} h\cdot (\phi_{N}\circ T)\, d\hat\mu_{N}  = \lambda_{N} \int_{\mathcal{M}} h \phi_{N} \, d\hat\mu_{N}.
\end{equation}
Define the linear functional $L_{N} : C(\mathcal{M})\to \mathbb{C}$ by
\[
L_N(h) = \int_{\mathcal{M}} h\, \phi_{N}\, d\hat{\mu}_{N},
\]
and
\[
(\mathcal{K} L_N)(h) = \int_{\mathcal{M}} h\cdot (\phi_{N}\circ T)\, d\hat\mu_{N}.
\]
With this notation, the relationship~(\ref{eq:eig_sample}) becomes
\[
\mathcal{K} L_N = \lambda_{N} L_N
\]
Since $\| \phi_{N}\|_{C(\mathcal{M})} = 1$, we have $\| L_N \| = \sup_{h\in C(\mathcal{M})} \frac{|L_N(h)|}{\|h \|_{C(\mathcal{M})}} \le 1$ and $\| \mathcal{K} L_N \| \le 1$. Therefore, assuming separability\footnote{A sufficient condition for $C(\mathcal{M})$ to be separable is $\mathcal{M}$ compact and metrizable.} of $C(\mathcal{M})$, by the Banach-Alaoglu theorem (e.g., \cite[Theorems 3.15, 3.17]{rudinFA}) there exists a subsequence, along which these functionals converge in the weak$^\star$ topology\footnote{A sequence of functionals $L_i\in C(\mathcal{M})^\star$ converges in the weak$^\star$ topology if $\lim_{i\to\infty}L_i(f) = L(f)$ for all $f\in C(\mathcal{M})$.} to some functionals $L \in C(\mathcal{M})^\star$ and $\mathcal{K} L\in C(\mathcal{M})^\star$ satisfying
\[
\mathcal{K} L = \lambda L,
\]
where $\lambda$ is an accumulation point of $\lambda_N$. Furthermore by the Riesz representation theorem the bounded linear functionals $L$ and $\mathcal{K} L$ can be represented by complex-valued measures $\nu$ and $\mathcal{K}\nu$ on $\mathcal{M}$ satisfying
\[
\mathcal{K}\nu = \lambda\nu.
\]

We remark that $\mathcal{K} L$ and $\mathcal{K}\nu$ are here merely symbols for the weak$^\star$ limit of $\mathcal{K} L_N$ and its representation as a measure; in particular the functional $\mathcal{K} L$ is not necessarily of the form $(\mathcal{K} L)(h)=\int_{\mathcal{M}} h\cdot (\rho\circ T)\,d\mu$ for some function $\rho$.

In order to get more understanding of $\mathcal{K}\nu$ (and hence $\mathcal{K} L$), we need to impose additional assumptions on the structure of the problem. In particular we assume that the mapping $T:\mathcal{M}\to\mathcal{M}$ is a homeomorphism and that the points $x_1,\ldots,x_N$ lie on a single trajectory, i.e., $x_{i+1} = T(x_i)$. With this assumption, Equation~(\ref{eq:eig_sample}) reads
\begin{equation}\label{eq:eig_sample_1}
\frac{1}{N} \sum_{i=1}^N h(x_i)\phi_N(x_{i+1})  = \lambda_{N} \frac{1}{N} \sum_{i=1}^N h(x_i)\phi_N(x_i),
\end{equation}
where we set $x_{N+1}:= T(x_N)$. The left-hand side of~(\ref{eq:eig_sample_1}) is
\begin{align}
\frac{1}{N} \sum_{i=1}^N h(x_i)\phi_N(x_{i+1}) &= \frac{1}{N} \sum_{i=1}^N h(T^{-1} x_i)\phi_N(x_i) + \frac{1}{N}(h(x_N)\phi_N(x_{N+1}) - h(T^{-1}x_1)\phi_N(x_1))\nonumber \\
&= \int_{\mathcal{M}} h\circ T^{-1} \, d\nu_N + \frac{1}{N}(h(x_N)\phi_N(x_{N+1}) - h(T^{-1}x_1)\phi_N(x_1)), \label{eq:1Ncomp}
\end{align}
where $\nu_N$ is the measure $\phi_N d\hat\mu_N$. Setting $\xi_N:=h(x_N)\phi_N(x_{N+1}) - h(T^{-1}x_1)\phi_N(x_1)$, the relation~(\ref{eq:eig_sample_1}) becomes
\[
\int_{\mathcal{M}} h\circ T^{-1} \, d\nu_N + \frac{1}{N}\xi_N = \lambda_N\int_{\mathcal{M}} h\,d\nu_N.
\]
Since $h$ is bounded on $\mathcal{M}$ ($h$ is continuous and $\mathcal{M}$ compact) and $\| \phi_N\|_{C(\mathcal{M})} = 1$, the term $\xi_N$ is bounded; in addition $h\circ T^{-1}$ is continuous since $T$ is a homeomorphism by assumption. Therefore, taking a limit on both sides, along a subsequence such that $\nu_{N_i}\to \nu$ weakly\footnote{A sequence of Borel measures $\mu_i$ converges weakly to a measure $\mu$ if $\lim_{i\to\infty}\int f\,d\mu_i = \int f\,d\mu $ for all continuous bounded functions $f$. This convergence is also referred to as narrow convergence and it coincides with convergence in the weak$^\star$ topology if the underlying space is compact (which is the case in our setting).}, $\hat\mu_{N_i}\to \mu$ weakly and $\lambda_{N_i}\to\lambda$, we obtain
\begin{equation}\label{eq:eigFunctional}
\int_{\mathcal{M}} h\circ T^{-1} \, d\nu = \lambda\int_{\mathcal{M}} h\,d\nu
\end{equation}
for all $h\in C(\mathcal{M})$.

\subsection{Weak eigenfunctions / eigendistributions}
To understand relation~(\ref{eq:eigFunctional}), note that a completely analogous computation to~(\ref{eq:1Ncomp}) shows that the measure $\mu$ is invariant\footnote{A measure $
\mu$ on $\mathcal{M}$ is invariant if $\mu(T^{-1}(A)) = \mu(A)$ for all Borel sets $A\subset \mathcal{M}$ or equivalently if $\int_{\mathcal{M}} f\circ T \, d\mu = \int_{\mathcal{M}} f \, d\mu $ for all continuous bounded functions $f$.} and therefore the $L_2(\mu)$-adjoint $\mathcal{K}^\star$ of the Koopman operator (viewed as an operator from $L_2(\mu) $ to $L_2(\mu)$) is given by
\[
\mathcal{K}^\star f = f\circ T^{-1}.
\]
To see this, write
\[
 \langle \mathcal{K} f,g \rangle  = \int_{\mathcal{M}} (f\circ T) \overline{g} \,d\mu = \int_{\mathcal{M}} (f\circ T)\overline{(g\circ T^{-1}\circ T)} \,d\mu = \int_{\mathcal{M}} f \cdot \overline{(g\circ T^{-1} )}\,d\mu = \langle f, \mathcal{K}^\star g\rangle,
\]
which means the operator $g\mapsto g\circ T^{-1}$ is indeed the $L_2(\mu)$-adjoint of $\mathcal{K}$. The relation~(\ref{eq:eigFunctional}) then becomes
\begin{equation}\label{eq:eigFunctional_measurePres_1}
\int_{\mathcal{M}} \mathcal{K}^\star h \, d\nu = \lambda\int_{\mathcal{M}} h\,d\nu
\end{equation}
or
\begin{equation}\label{eq:eigFunctional_measurePres_2}
L(\mathcal{K}^\star h) = \lambda L(h).
\end{equation}

Functionals of the form~(\ref{eq:eigFunctional_measurePres_2}) were called ``generalized eigenfunctions'' by Gelfand and Shilov~ \cite{GelfandandShilov:1964}; here we prefer to call them ``weak eigenfunctions'' or ``eigendistributions'' in order to avoid confusion with generalized eigenfunctions viewed as an extension of the notion of generalized eigenvectors from linear algebra. The measure $\nu$ in~(\ref{eq:eigFunctional_measurePres_1}) is then called ``eigenmeasure''. Here, again, we emphasize the requirement that the limiting functional $L$ (or the measure~$\nu$) be nonzero in order for these objects to be called eigenfunctionals / eigenmeasures.

\subsection{Eigenmeasures of Perron-Frobenius}
We also observe an interesting connection to eigenmeasures of the Perron-Frobenius operator. To see this, set $h:= g\circ T$ in~(\ref{eq:eigFunctional}) to obtain
$
\int_{\mathcal{M}}g\,d\nu = \lambda\int_{\mathcal{M}} g\circ T\,d\nu
$
or, provided that $\lambda\ne 0$,
\begin{equation}
\int_{\mathcal{M}} g\circ T\,d\nu  = \frac{1}{\lambda} \int_{\mathcal{M}}g\,d\nu.
\end{equation}
In other words, if non-zero, the measure $\nu$ is the eigenmeasure of the Perron-Frobenius operator with eiegnvalue~$1/\lambda$. Here, the Perron-Frobenius operator $\mathcal{P}:M(\mathcal{M})\to M(\mathcal{M})$, where $M(\mathcal{M})$ is the space of all complex-valued measures on $\mathcal{M}$, is defined for every $\eta \in M(\mathcal{M})$ and every Borel set $A$ by
\[
(\mathcal{P}\eta)(A) = \eta(T^{-1}(A)).
\]

The results of Section~\ref{KNNconv} are summarized in the following theorem:
\begin{theorem}
Suppose that $\mathcal{M}$ is a compact metric space, $T$ is a homeomorphism, $\mathcal{K}:L_2(\mu)\to L_2(\mu)$ is bounded, the observables $\psi_1,\ldots,\psi_N$ are continuous and the sample points $x_1,\ldots, x_N$ satisfy $x_{i+1} = T(x_i)$ for all $i\in\{1,\ldots,N-1\}$. Let $\lambda_N$ be a bounded sequence of eigenvalues of $\mathcal{K}_{N,N}$,  let $\phi_N$, $\|\phi_N\|_{C(\mathcal{M})}=1$, be the associated normalized eigenfunctions and denote $\nu_N = \phi_N d\hat \mu_N$. Then there exists a subsequence $(N_i)_{i=1}^\infty$ such that $\nu_{N_i}$ and $\hat\mu_{N_i}$ converge weakly to  complex-valued measures $\nu\in M(\mathcal{M})$, $\mu\in M(\mathcal{M})$ and $\lim_{i\to\infty} \lambda_{N_i } = \lambda \in \mathbb{C}$ such that
\[
\int_{\mathcal{M}} h\circ T^{-1} \, d\nu = \lambda\int_{\mathcal{M}} h\,d\nu\quad \forall\,h\in C(\mathcal{M}).
\]
In addition, the measure $\mu$ is invariant under the action of $T$ and
\[
\int_{\mathcal{M}} \mathcal{K}^\star h \, d\nu = \lambda\int_{\mathcal{M}} h\,d\nu \quad \forall\,h\in C(\mathcal{M}),
\]
where $\mathcal{K}^\star$ is the $L_2(\mu)$ adjoint of $\mathcal{K}$, i.e., if nonzero, $\nu$ is a weak eigenfunction (or eigendistribution) of the Koopman operator. Furthermore, if $\lambda \ne 0$, then
\[
\int_{\mathcal{M}} h\circ T\,d\nu  = \frac{1}{\lambda} \int_{\mathcal{M}}h\,d\nu \quad \forall\,h\in C(\mathcal{M}),
\]
i.e., if nonzero, $\nu$ is an eigenmeaure of the Perron-Frobenius operator with eigenvalue $1/\lambda$.
\end{theorem}

\section{Conclusions}
This paper analyzes the convergence of the EDMD operator $\mathcal{K}_{N,M}$, where $M$ is the number of samples and $N$ the number of observables used in EDMD. It was proven in~\cite{klus2015numerical} that as $M\to\infty$,  the operator $\mathcal{K}_{N,M}$ converges to $\mathcal{K}_N$, the orthogonal projection of the action of the Koopman operator on the span of the observables used in EDMD. We analyzed the convergence of $\mathcal{K}_N$ as $N\to\infty$, obtaining convergence in strong operator topology to the Koopman operator and weak convergence of the associated eigenfunctions along a subsequence together with the associated eigenvalues. In particular, any accumulation point of the spectra of $\mathcal{K}_N$ corresponding to a non-zero weak accumulation point of the eigenfunctions lies in the point spectrum of the Koopman operator $\mathcal{K}$. In addition we proved convergence of finite-horizon predictions obtained using $\mathcal{K}_N$ in the $L_2$ norm, a result important for practical applications such as forecasting, estimation and control. Finally we analyzed convergence of $\mathcal{K}_{N,N}$ (i.e., the situation where the number of samples and the number of basis functions is equal) under the assumptions that the sample points lie on the same trajectory. In this case one obtains convergence, along a subsequence, to a weak eigenfunction (or eigendistribution) of the Koopman operator, provided the weak limit is nonzero. This eigendistribution turns out to be also an eigenmeasure of the Perron-Frobenius operator. As a by-product of these results, we proposed an algorithm that, under some assumptions, allows one to construct  $\mathcal{K}_N$ directly, without the need for sampling, thereby eliminating the sampling error.

\new{Future work should focus on non-asymptotic analysis, e.g., on selecting the subspace $\mathcal{F}_N$ such that $\|\mathcal{K}_{N} - \mathcal{K}_{|\mathcal{F}_N} \|$ is minimized and at the same time such that $\mathcal{F}_N$ is rich enough in the sense of containing observables of practical interest (e.g., the state observable). This line of research was already investigated in the context of stochastic systems in~\cite{wu2017variational}, providing an interesting and actionable method for selecting $\mathcal{F}_N$.
}

\label{conc}

\section{Acknowledgments}
The first author would like to thank Mihai Putinar for discussions on a related topic that sparked the work on this paper. We also thank Clancy Rowley for helpful comments on an earlier version of this manuscript as well as to P\' eter Koltai for point out the reference~\cite{klus2015numerical}. This research was supported in part by the ARO-MURI grant W911NF-14-1-0359 and the DARPA grant HR0011-16-C-0116. The research of M. Korda was supported by the Swiss National Science Foundation under grant P2ELP2\_165166.

\bibliographystyle{abbrv}

redpajama_set_name: RedPajamaArXiv


Me, Myself and Technology » Blog Archive » JP6 De-deodexed and zipaligned ROM for GT-I9000 with Voodoo and 2e recovery!
20th Oct 2010: Updated to accommodate Voodoo Pre4 (which contains fixes for some rooting issues, and enables logging by default). For those of you who flashed the previous version, don't worry, the rooting issues were already fixed in the ROM. So there is no need to update. For those who still want to update, you can get Voodoo Pre4 from here, and you will have to flash it via Odin as PDA, with no PIT and re-partition un-checked.
a) Take proper backup of your apps etc with Titanium Backup.
b) Disable Voodoo by creating a folder named "disable lagfix" inside the "Voodoo" folder in your internal sdcard (case matters), and then rebooting. Reboot will take time as Voodoo is disabled.
c) After the phone boots up, shutdown the phone, and flash the CFRoot Kernel by ChainFire for JP6. Download the kernel from XDA or from here. Flash the tar file as PDA with Odin (no PIT needed and don't check re-partition).
1. Disable Voodoo if have it installed (create a folder named "disable lagfix" inside the "Voodoo" folder on your internal sdcard and reboot).
2. Make sure you're running Eclair firmware (if you're on Froyo, make sure you have the 2e recovery).
3. Make sure you have a backup of your /efs (not mandatory, but highly recommended).
1. Download the packaged ROM from here, and put it on your internal sdcard.
2. Download clockworkmod recovery from here. You need Clockworkmod, this won't work with stock recovery.
3. Rename the clockworkmod recovery file to "update.zip" and put it on your internal sdcard.
4. Before shutting down your phone, make sure you remove the "disable lagfix" folder from the "Voodoo" folder.
5. Shutdown phone, wait for it to power down, then boot the phone into recovery mode (press volume up, keep it pressed, then press home button, keep it pressed, then press power button, keep it pressed until you see the Galaxy S screen. Release all three buttons when you see it).
6. Select "apply sdcard:update.zip" (use volume rocker to move up and down, and home button to select). Clockworkmod recovery will boot now (you may have to do apply update.zip twice for Clockworkmod to show up).
7. Once in Clockworkmod, select "install zip from sdcard" (use volume rocker to move up and down and the back button to select).
8. Select "choose zip from sdcard" and then select the ROM zip package. Choose yes then.
9. The update will start (will take some 5 minutes) and during the flashing you will see some strange big characters on the screen, don't worry, that's normal. After the process finishes, you will be dropped back into Clockworkmod recovery. From there, choose go back, and then reboot phone.
10. The first boot will take some time as the ROM will deodex, and also Voodoo will be applied.
1. Voodoo kernel does not have a 2e recovery as of now. Will update the package once it does.
2. You will see a small screen corruption when booting the phone. This is a known issue with Voodoo as of now.
Is this indian ROM.One more thing is that I have xxjpk rom which might have changed my product no.How to check my product no and will it effect upgrading to official froyo.I have the backup of efs folder.
No, this is not an Indian ROM.. It is an European ROM. To check Product Code, install SGS Tools from market.
Is product region is the product code.It is showing GB.
@Gaurav make sure you apply all the manual updates to Google services on the market, you will get FC for most until doing this.
Kalpik you should update to pre6 yo, UNSTABLE ROOT!
1) im from slovenia-europe, so i think yours ROM is ok, right?
and now a read there can be some serious problems, if product code is gone..
-my /efs folder is empty ..
what is this product code? is it IMEI? IMEI on the phone is still the same as is writen on garantie i got.
what is this tool backuping, if /efs folder is empty? how to "refill"-restore it? how that my phone is working if this folder is empty?
how can i restore this backup file, and whay would i? do i even need it? ever? never?
If your IMEI is still intact, don't worry!
why did you said it's not mandatory? did in some part of your procedure, some program, backuped /efs folder somewhere?
from where is phone reading correct IMEI number if there is no files in /efs?
what is sgs toolbox saying to me- from where it makes backups if there is no files in /efs??
Hi Kalpik! First of all let me thank you for this excellent ROM:) 5* reviews all the way.
I do have a few noob questions, I apologise in advance!
You say on XDA forum to use titanium backup in order to backup all apps+data, does the phone have to be rooted already to do this?
You also stated to make a backup of your /efs can you tell me how to do this please?
@undertaker: its not possible that your /efs is empty.. If it were empty, your IMEI would change! What are you using to check the /efs folder?
you are right once again 🙂 my paranioa is over, i think..
it was ES file explorer, that misleaded me..
still, hoping to do manual backup before any more flashes, to be shure..
how come, that your flashing didn't delete /efs folder, since everybody is screaming of deleted folders after flashing with other roms?
What is the quadrant score…????I know it doesn't really matter but still i want to know…Currently i have a lot of apps installed in my i9000..
Hi all, I'm new to this and would rather not end-up with a brick on my hands, so I've a couple of questions.
1) does this (and/or the ROM the superseads it) need me to get my phone SIMunlocked first? I assume so but it's worth checking.
2) will backing-up the /efs folder (and SMS/contacts/apps) allow for a completes restore?
3) If I do brick my phone what and try and get a replacement from my phone co (Orange for the record) what are they likely to say? yes/no/it will cost you/buy a new handset mate.
Thanks, and thanks for the great work.
Thanks kalpik for the prompt reply…it worked as a charm…i am really happy i tried this one out…Nyways a small problem,there still seems to be a wake up lag( if u understand wat i mean)…can u fix dat?? or am i the only one who is facing this issue..thanks in advance….
awesome firmware. keep up the good work.
should i apply that by going into recovery mode for second time.
Yes.. You have to apply the theme via CWM/2e recovery.
All functions seems work ok until know, speed is refreshing :).
I tried to download the clockworkmod recovery but can't be connected to the website. How can i do it?


text:
Hydrogen Is A Bad Car Fuel, But It's The Perfect Boat Fuel
Bradley Brownell
Published 2 years ago: February 5, 2020 at 10:30 am -
Filed to:boatlopnik
hydrogenjalopniktoyota miraizero emissions
There are issues inherent with hydrogen as a fuel for cars. It is incredibly expensive and energy intensive to create, it is difficult to pressurise and transport, and the infrastructure for hydrogen as fuel is far less developed than battery electric charging. A few automotive manufacturers, chiefly Honda and Toyota, have hung their zero emissions program hat on the hydrogen peg, but it's still a very small sliver of the automotive market. It's pretty much only viable in a small area of Southern California near the fuelling stations. As a car fuel, hydrogen straight up sucks.
Toyota and the Energy Observer are proving that hydrogen might be best served as a fuel for traversing the high seas, however. Toyota has adapted what it has learned from the Mirai hydrogen experiment to the Energy Observer, a former racing catamaran which now travels the world preaching the gospel of maritime ZEVs.
The Energy Observer uses a pair of wind turbines and a vast array of solar photo-voltaic cells to both propel the vessel and provide power to its on-board hydrogen-creating electrolysis process. Sea water is essentially zapped into its component parts and the isolated hydrogen is captured to be expended inside the Toyota fuel cell generator. The process emits nothing but oxygen and water out the "tailpipe".
The fuel is the ocean, and the ocean is everywhere. It's free fuel.
In optimum conditions, the boat is propelled entirely by wind and solar. A rack of lithium cells onboard keep the thing running when it's cloudy or calm winds, and Toyota's fuel cell system takes over to produce the boat's propelling energy at night.
This boat is a floating test bed for the future of water transport. I can already imagine a world where this technology is massively scaled to accommodate powering giant cargo ships and fishing vessels.
Shipping and commercial fishing is responsible for nearly one billion tons of CO2 per year, or nearly 3% of global greenhouse gas emissions. I was at a marina the other day and the fuel price said $US5.02 ($7) per gallon of marine grade fuel, or $US4.02 ($6) per gallon if you bought at least 100 gallons. First, this is why I would never own a boat, and second, wouldn't it be great if you boat owners didn't have to buy fuel at all? Especially with the knock on effect being that your boat no longer produces any harmful emissions.
I'm so jazzed that this tech is proving to be viable in the hands of the Energy Observer crew, because this is exactly the kind of big think we need to be committing to right now.
In fact, Japan has committed to powering its fishing industry with large-scale hydrogen-power adaptation. The Japanese government has committed to a 15% reduction in shipping industry emissions by 2030, and this kind of tech could get it there much quicker.
Is Blue Hydrogen Really Better for the Environment than Natural Gas?
Big Oil's New 'Clean' Fuel Is Dirtier Than They Say
Researchers Warn Against Becoming Too Dependent On Hydrogen To Power Cars
This Goopy Paste Could One Day Power Scooters and Motorcycles

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Google Doodle Celebrates Baseball Great Roberto Clemente
0 Posted by Blackthen - October 12, 2018 - IN THE NEWS, LATEST POSTS, SPORTS
Roberto Clemente's legacy as one of baseball's greatest players is matched only by the memory of the selfless sacrifice he made trying to help others in need.
The son of a sugar cane worker in Puerto Rico, Clemente showed athletic promise at an early age, joining the Puerto Rican amateur league in 1952 at the age of 16 and signing a minor league contract with a Brooklyn Dodgers affiliate two years later. In July 1954, Clemente's first home run in North America resulted in an extra-innings walk-off win for the triple-A Montreal Royals.
Later that year, the Pittsburgh Pirates made Clemente its first selection of the rookie draft. As a Pirate, Clemente would go on to win 12 Gold Gloves (tied for most among outfielders), four National League batting titles, two World Series rings, and the World Series MVP for 1971. He had a batting average of over .300 for 13 seasons and is credited with professional baseball's only inside the park, walk-off grand slam.
He recorded his 3,000th and last hit during the final regular-season at-bat of his career in 1972.
Hall of Fame numbers, for certain. But it's also for Clemente's humanitarian efforts that Google, in the spirit of Hispanic Heritage Month, dedicated its Doodle on Friday to the first Latin American player enshrined in Cooperstown.
Clemente spent much of his off-season involved in charity work, delivering baseball equipment and food to those in need in Latin American and Caribbean countries. After a 6.3 earthquake killed thousands and caused widespread damage to Nicaragua on Dec. 23, 1972, Clemente began arranging emergency relief flights to deliver aid to survivors.
Read more https://www.cnet.com/news/google-doodle-honors-baseball-great-and-humanitarian-roberto-clemente/
baseball hall of fameGoogleGoogle DoodleRoberto Clemente


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4 sustainability opportunities in the banking sector
Sandy Eapen
ShutterstockAnton Violin
According to a 2016 study by the Brunswick Group (PDF), only 27 percent of Americans trust banks. It has been almost a decade since the financial crisis of 2008, and while the financial services sector has made progress, it is still in many ways working to earn back stakeholder trust. Sustainability efforts offer a major opportunity for banks to demonstrate their commitment to operating responsibly and making a positive impact.
Banks, in particular, play a major role in our economy, as they provide vast amounts of capital and have the ability to influence other companies and customers across sectors through their products and services. Yet the industry continues to be in the headlines for various environmental, social and governance (ESG) issues, such as ethics violations or potential human rights implications of project finance decisions.
How has sustainability, or corporate responsibility, in banks changed since the crisis, especially in the context of the Paris Agreement, the Sustainable Development Goals (SDGs), and investor-focused and sustainability disclosure initiatives? What more could be done?
To explore these questions, BSR completed a short research study, assessing leading banks across the U.S. and Europe across a few key areas. Here are four key findings.
1. Materiality assessment
Materiality assessments are being conducted to prioritize corporate responsibility issues but could be leveraged to play a greater role in internal engagement and sustainability strategy development. Banks also more closely can align their efforts with the SDGs.
While it is commonplace for banks to spend the time and resources to go through a formal materiality process and publish the results online, in many cases the process seems to be more of a "check-the-box" exercise for reporting than a determinant of strategy and business activities.
The materiality process can be a powerful mechanism to engage and educate senior leaders and get valuable input from external stakeholders. There is a risk that the feedback it provides will be lost if it is not integrated into company strategy.
The SDGs haven't yet played a major role in informing corporate responsibility priorities, although banks understand that the sector will play a critical role in achieving them. While all of the SDGs can be inspirational for organizations, focusing on those that align best with the business strategy and existing corporate responsibility priorities likely will be most impactful for the industry. As a first step, banks are mapping business activities to key SDGs. BNP Paribas has created a formal SDG metric, which measures the share of the bank's loans to companies that contribute exclusively to the SDGs.
2. Communicating commitments
Increasingly, banks are communicating major long-term sustainable financing commitments, which provide an opportunity to link products and services to corporate responsibility; however, they will increasingly need to be transparent about these initiatives.
Bank of America and Citigroup both have 10-year sustainable financing commitments of $100 billion or more. While these big commitments create internal momentum and demonstrate both business and ESG value, what qualifies for this type of funding and how impacts are measured likely varies across banks. Transparency about the methodology and criteria for funding and calculating impacts will help banks add credibility to these initiatives.
Additionally, creating a corporate strategy and mission linked to sustainability, such as ING's purpose-driven Think Forward corporate strategy and Bank of America's Responsible Growth strategy, is key in integrating efforts across the business. BSR's recent report, "Redefining Sustainable Business: Management for a Rapidly Changing World," explores this and many other aspects of how to create a resilient business strategy.
3. ESG metrics
Banks can better establish and communicate focused ESG metrics and targets aligned to their identified material issues.
While banks are providing detailed ESG information in multiple reports and using the GRI standard, the key strategic metric and associated target often integrated with company performance results so far has been sustainable financing performance. Hopefully, we will continue to see more, new ESG metrics integrated in standard company results, internally and externally.
Some banks, such as Barclays, have moved toward publishing an integrated annual report that includes ESG data. While doing this would seem to imply more streamlined reporting, these banks still have been producing supplemental reports to provide the additional detailed ESG disclosures that some stakeholders require.
Governance structures continue to play a key role in engaging employees and driving corporate responsibility and business integration — executive ownership and engagement with the environmental and social risk management teams remain critical.
It is common for banks to have board oversight over ESG issues, such as via a committee. Additionally, cross-functional senior executive committees and other internal councils (for example on sustainable products and human rights) are critical in integrating ESG across the business and engaging subject matter experts. While it requires more internal coordination, core sustainability teams at banks are seeing positive outcomes from these internal working groups and networks of ESG champions dedicated to achieving ESG goals.
One area that needs more consideration is compensation tied to ESG performance. It is encouraging to see BNP Paribas link nine of its 13 corporate responsibility indicators to its variable incentive plan for its 5,000 top managers. More banks should embrace this type of approach.
While banks clearly have made progress in better integrating corporate responsibility, studies show that the industry reputation continues to suffer. There is an opportunity for the sector to do more to engage its leadership, employees, customers and investors on the powerful role it can play in creating a more sustainable world. Doing so could go a long way toward rebuilding trust.
This story first appeared on:
Manager, Advisory Services
@sandyeapen
World Bank renews U.N. Sustainable Development Goals bonds partnership
ByMichael Holder
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ByLibby Bernick
Report Report: Banking, blind spots, carbon capture and climate risk
ByDevon Edwards
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ByCharlotte Bancilhon
Unilever teams with big banks on blockchain for supply chain
Turning Title 24 Regulations into Smart Savings


Permits can be obtained from the Post Office for R85.00 for a year, monthly permits are also available. You can also get the permits at the KZNW office inside the parksboard at Sodwana Bay. The maximum fine for not having a permit is R10, 000.00!
For information on how to obtain a diving permit please contact our office.


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Seattle Photographer Joan Bowers will have her work showcased in the lobby of the Burien Community Center through July 30th.
Bowers specializes in black and white photography, lith prints, Polaroid emulsion transfer and Polaroid image transfer prints.
I grew up in a small Pennsylvania town which was surrounded by the detritus of the anthracite coal mining industry. As much as possible during those early years, my parents took me and my siblings to visit more verdant spaces such as farmlands, forests, riverside banks, and hidden ponds, in order to immerse us in their pleasure with nature's unstinting beauty. The appreciation which they nurtured in each of us continues to influence my relationship with the world around me.
When I was about twelve years old, a Brownie box camera became my introduction to photography; my love for black and white photography goes unabated. In the past several years, I have honed my skills in darkroom processing and printing. Currently, I do lith printing, a multi-step alternative process.
My photographs demonstrate my wish to convey a readiness to see beauty in even the simplest elements: naked trees in winter, water that lies still in ponds or moves through channels cut into the land. These are scenes without people. While the presence of others in the scene is sometimes used to suggest perspective I believe this presence would impinge on the aim of my image – to provide the viewer with a momentary escape into this representation of nature as refuge.


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Original autumn brand. Comes with mesh bag. Free 5pcs storage bags. One of the bottle cap have crack.
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