text:
\section{Introduction}\label{intro}
Let $\mathcal{M}$ be a
 smooth compact manifold of dimension $d$ without boundary, %
 and let $L$ be an elliptic differential
operator of order $2\kappa$, $\kappa\in \mathbb{N}$, on $\mathcal{M}$ with
 infinitely differentiable coefficients
 %
 in local coordinates,  and a trivial null space. %
Then the equation
\begin{equation}\label{pde}
L u=f
\end{equation}
has a unique solution $u$ in the Sobolev space $H^t(\mathcal{M})$, $t\in\mathbb{R}$,  whenever $f\in H^{t-2\kappa}(\mathcal{M})$, and
there are constants $A,B>0$, depending only on $\mathcal{M},L,t$ and the choice of the Sobolev norms, such that
\begin{equation}\label{reg}
A\|f\|_{H^{t-2\kappa}(\mathcal{M})}\le \|u\|_{H^t(\mathcal{M})}\le B\|f\|_{H^{t-2\kappa}(\mathcal{M})},
 \end{equation}
see e.g.\ \cite[Section 6]{Agranovich15}.

Meshless numerical methods are particularly attractive for solving operator equations on manifolds, in particular on the
sphere  \cite{FFprimer15,Freeden98}, because of the difficulties
of creating and maintaining suitable meshes or grids in this setting.
Error bounds for meshless methods for the approximation of functions and specifically
solutions of operator equations on manifolds have been studied for example in
\cite{FuselierWright12,HNRW18,HNW10,LevesleyLuo03,LevesleyRagozin07,MortonNeamtu02,Narcowich95,NRW17,PhamTran14}. However, these
results do not apply to localized finite difference type methods considered below.

Meshless finite difference methods discretize a differential equation  \eqref{pde}
(or similarly a boundary value problem)
on a set of irregular nodes $X=\{x_1,\ldots,x_n\}\subset\mathcal{M}$  with the help of numerical differentiation formulas
\begin{equation}\label{ndf}
Lu(x_i)\approx\sum_{j\in J_i}w_{ij} u(x_j),\quad J_i\subset J:=\{1,\ldots,n\},\qquad i\in I,
\end{equation}
where the coefficients $w_{ij}\in\mathbb{R}$ are obtained by requiring that the formula is exact for certain finite dimensional spaces
of functions, for example polynomials or kernel sums or a combination thereof, and the size of the sets of influence $X_i=\{x_j: j\in J_i\}$ is bounded by a
fixed number $\nu\ll n$. In particular, the
kernel-based formulas are exact for all linear combinations
$\sum_{j\in J_i}c_j K(\cdot,x_j)$, $c_j\in\mathbb{R}$, where $K:\mathcal{M}\times\mathcal{M}\to\mathbb{R}$ is a positive definite kernel. The
discrete approximate solution ${\hat u}\in\mathbb{R}^n$ of \eqref{pde}, such that ${\hat u}_i\approx u(x_i)$, $i=1,\ldots,n$,
 is obtained by solving the sparse linear system
\begin{equation}\label{mfd}
\sum_{j\in J_i}w_{ij} {\hat u}_j=f(x_i), \quad i\in I.
\end{equation}

Numerical performance of methods of this type %
has been studied
e.g.\ in the book \cite{FFprimer15} and papers \cite{BFFB17,FLBWS12,LSH17,PLR18,Shankar17,SWKF15,SDKWT19online,SDT19,SuchdeKuhnert19}, in particular for
differential equations on the sphere and other manifolds.
In contrast to those meshless methods that discretize the weak form of the equations
(see the surveys \cite{Belytschko96,NguRabBorDuf}),
no integration of the trial functions over subdomains is needed,
which is challenging in the meshless setting as the subdomains are not generated
from a few reference shapes controlled by the mesh.
In contrast to the global collocation methods  (see e.g.~\cite{Kansa90b,PowerBarraco02}),
the linear systems are sparse. Moreover, by optimizing the selection of the sets of influence $J_i$, they may be made as
sparse as they are in the mesh based methods, such as the finite element method, with comparable accuracy
\cite{D19arxiv,DavyOanh11,DOT19,ODP17}.

In the classical setting we choose $I=\{1,\ldots,n\}$, and hence \eqref{mfd} is a square linear system.
By using more than $n$ numerical differentiation
formulas \eqref{ndf} we arrive at an overdetermined system, and $\hat u$ may be obtained by the least squares minimization.
In \cite{LSH17} this situation arises from evaluating $Lu$ in the left hand side of \eqref{ndf} on a different set of nodes,
larger than $X$. This is similar to the ``overtesting'' mode that admits, for the collocation and other methods
employing trial functions,  an error analysis under very general conditions
\cite{Schaback10,Schaback16}, which however does not cover %
finite difference type methods \cite{Schaback_error_analysis17}. An extension %
of the classical error analysis that requires that the system matrix of \eqref{mfd} is an $M$-matrix has been proposed
in \cite{DKL84}, and conditions for the $M$-matrix property were investigated in \cite{DKL84,Seibold08}. However, this
approach only applies to special geometric configurations of the sets of influence with low convergence order.

In this paper we derive error bounds for an overdetermined version of the meshless finite difference method
for elliptic differential equations on smooth closed manifolds, solved by least
squares, where the numerical differentiation formulas \eqref{ndf} are generated in blocks corresponding to a family of
overlapping subsets $J_\ell\subset J$, %
\begin{equation}\label{ndfmy}
[Lu(x_j)]_{j\in J_\ell}\approx W_\ell [u(x_j)]_{j\in J_\ell},\quad \ell=1,\ldots,m.
\end{equation}
The local differentiation matrices $W_\ell$ are invertible since they correspond to a reproducing kernel $K$ for a Sobolev
space $H^s(\mathcal{M})$ such that $L$ is positive and self-adjoint with respect to the inner product defined by $K$. This condition
in particular implies that the system matrix has full rank and hence there exists a unique least squares solution
of
\begin{equation}\label{mfdmy}
W_\ell [{\hat u}_j]_{j\in J_\ell}=[f(x_j)]_{j\in J_\ell},\quad \ell=1,\ldots,m.
\end{equation}

Under certain regularity assumptions that involve in particular the quasi-uniformity of $X$ and the existence of an
atlas $\mathcal{A}=\{(U_\ell,\varphi_\ell)\}_{\ell=1}^m$ with sufficiently nice $U_\ell\subset\mathcal{M}$
such that $X\cap U_\ell=X_\ell:=\{x_j:j\in J_\ell\}$ and $X_\ell,X_k$ significantly
overlap whenever $U_\ell\cap U_k\ne\emptyset$, we show in Theorem~\ref{error_bound} that
$$
\max_{j\in J}|u(x_j)-\hat u_j|=\mathcal{O}(h_\mathcal{A}^{s-2\kappa-d-r_0}),
\qquad r_0:=\max\{0,\lfloor \tfrac{d}{2}\rfloor-2\kappa+1\},$$
and there exists $\tilde u\in H^{s+2\kappa}(\mathcal{M})$ such that $\tilde u(x_j)={\hat u}_j$, $j\in J$, and
$$
\|u-\tilde u\|_{H^{2\kappa+r}(\mathcal{M})}=\mathcal{O}(h_\mathcal{A}^{s-2\kappa-d-r}),\qquad 0\le r<s-\kappa,$$
where $h_\mathcal{A}$ is the maximum diameter  of the sets $\varphi_\ell(U_\ell)$. This shows the pointwise
convergence of the method for $h_\mathcal{A}\to 0$ when $s>\max\{2\kappa,\tfrac{d}{2}+1\}+d$, and convergence in $H^{2\kappa}(\mathcal{M})$
when $s>2\kappa+d$.

The results apply in particular to the operators of the form $L=(-\Delta_\mathcal{M}+\alpha I)^\kappa$, $\kappa\in\mathbb{N}$,
$\alpha>0$, where $\Delta_\mathcal{M}$ is the Laplace-Beltrami operator, $I$ is the identity, and $\mathcal{M}=\mathbb{S}^d$ is the $d$-dimensional
sphere. In this case the differentiation matrices $W_\ell$ may be efficiently computed by employing
restrictions to $\mathbb{S}^d$ of the Mat\'ern or Wendland
kernels for the ambient Euclidean space $\mathbb{R}^{d+1}$.

The paper is organized as follows. Section~\ref{prem} is devoted to auxiliary statements on various topics needed in the main parts
of the paper, such as equivalent norms for Sobolev spaces on manifolds, properties of  reproducing kernels and self-adjoint
differential operators, sampling inequalities and local differentiation matrices. The method and error bounds are
presented in Section~\ref{method}, and a proof %
is given in Section~\ref{proof}.

\section{Preliminaries}\label{prem}

We denote the partial derivative of a real function $u$ defined on a subset of $\mathbb{R}^d$ by
$\partial^\alpha u:=\frac{\partial^{|\alpha|}u}{\partial x_1^{\alpha_1}\cdots\partial x_d^{\alpha_d}}$, where
$\alpha\in\mathbb{Z}_+^d$ and $|\alpha|:=\sum_{i=1}^d\alpha_i$. We will extensively use the Leibniz product rule and the estimates that follow from
the multivariate chain rule, see e.g.\ Sections 1.2, 1.63 and 3.41 in \cite{Adams03}.

The cardinality of a finite set $X$ will be denoted $\#X$, and $\partial S$ will stand for the boundary of
a set $S$ in $\mathbb{R}^d$ or $\mathcal{M}$. We will use the usual notations $C(S)$, $C^\infty(S)$, $L^\infty(S)$, $L^2(S)$ for the spaces of continuous, infinitely differentiable,
essentially bounded or square integrable functions on appropriate subsets of $\mathbb{R}^d$ or $\mathcal{M}$.

Apart from the usual restriction  $f|_S$ for a subset $S$ of the domain of definition
of a function $f$, we denote by $f|_{X}$  the vector $[f(x)]_{x\in X}$ when $X$ is a finite set,
and by $w|_I$  the vector $[w_j]_{j\in I}$, for any  $w=[w_j]_{j\in J}$ and a subset $I\subset J$.

In what follows
various ``constants'' denoted $C$, $C_1$, $C_2$, etc.\ will be different at different occurrences and depend on
the manifold $\mathcal{M}$ (and thus on $d$) on default.  We will explicitely list other parameters these constant may
depend on in each case unless stated otherwise.

\subsection{Sobolev spaces on manifolds}\label{Sob}

Recall that in the case of integer $s\ge0$, Sobolev spaces $H^s(\Omega)$ are defined on any open set
$\Omega$ as Hilbert spaces with the norm \cite{Adams03}
\begin{equation}\label{Sobint}
\|u\|_{H^s(\Omega)}=\Big(\sum_{k=0}^s|u|_{H^k(\Omega)}^2\Big)^{1/2},\quad
|u|_{H^k(\Omega)}:=\Big(\sum_{|\alpha|= k}\|\partial^\alpha u\|_{L^2(\Omega)}^2\Big)^{1/2},
\end{equation}
in particular, $H^0(\Omega)=L^2(\Omega)$. When $s\in\mathbb{R}\setminus\mathbb{Z}_+$ and $\Omega$ is a Lipschitz domain in
$\mathbb{R}^d$, we refer to \cite{Agranovich15} for the definition of $H^s(\Omega)$ as the restriction to $\Omega$ of the space
$H^s(\mathbb{R}^d)$ defined as the Bessel potential space. The Bessel norm for $H^s(\mathbb{R}^d)$ is equivalent to \eqref{Sobint}
when $s\in\mathbb{Z}_+$. However, the  definition via restriction, if applied to non-Lipschitz domains and
integer $s$, leads in general to smaller spaces due to the lack of extension of some functions with a finite norm
\eqref{Sobint}, see e.g.~\cite[p.~287]{Burenkov98}. Even for Lipschitz domains, where extension theorems guarantee the equivalence of \eqref{Sobint}
to the restriction norm, the constants of equivalence depend on $\Omega$ and must be taken into account when they influence
estimates that have to be uniform with respect to $\Omega$.

Following the standard definition of Sobolev spaces on smooth manifolds,  we pick a finite smooth atlas
$\hat\mathcal{A}=\{(\hat U_\ell,\hat\varphi_\ell)\}_{\ell=1}^{\hat n}$ and a smooth subordinate
partition of unity $\hat\Gamma=\{\hat\gamma_\ell\}_{\ell=1}^{\hat n}$, see e.g.~\cite{Lee13}, such that $\hat\gamma_\ell\in C^\infty(\mathcal{M})$,
$\hat\gamma_\ell\ge0$, $\supp \hat\gamma_\ell\subset \hat U_\ell$,
$\sum_{\ell=1}^{\hat n}\hat\gamma_\ell=1$, and fix a norm for the Sobolev space $H^s(\mathcal{M})$, $s\in\mathbb{R}$, in  the form
\begin{equation}\label{Snorm}
\|u\|_{H^s(\mathcal{M})}=\Big(\sum_{\ell=1}^{\hat n}\|u\hat\gamma_\ell\circ \hat\varphi_\ell^{-1}\|_{H^s(\mathbb{R}^d)}^2\Big)^{1/2}.
\end{equation}

Below, when we speak of a norm for $H^s(\Omega)$ or $H^s(\mathcal{M})$ we will
always assume that it is generated by an inner product, and the notations $\|\cdot\|_{H^s(\Omega)}$, $\Omega\subset\mathbb{R}^d$,
and $\|\cdot\|_{H^s(\mathcal{M})}$, $\mathcal{M}$ a manifold, are reserved for the specific norms defined in \eqref{Sobint} and \eqref{Snorm},
respectively.

It follows from the properties of $\hat\gamma_\ell$ that the sets $\hat U^{c}_\ell:=\{x\in\mathcal{M}:\hat\gamma_\ell(x)>c\}\subset \hat U_\ell$, $\ell=1,\ldots,\hat n$, form an open cover
of $\mathcal{M}$ when $c=0$, and hence also for all sufficiently small $c>0$.
We say that a finite smooth atlas $\mathcal{A}=\{(U_\ell,\varphi_\ell)\}_{\ell=1}^{n}$ for $\mathcal{M}$ is \emph{$c$-admissible}, where $c>0$, if for each $\ell=1,\ldots,n$ there exists
$k=k(\ell)$ such that $U_\ell\subset \hat U^{c}_k$ and $\varphi_\ell=\hat\varphi_k|_{U_\ell}$. The \emph{covering number}
of $\mathcal{A}$ is defined as
\begin{equation}\label{muA}
\mu_\mathcal{A}:=\max_{1\le \ell\le m}\#\{j: U_j\cap U_\ell\ne\emptyset\}.
\end{equation}

\begin{lemma}\label{admis} Let $\mathcal{A}=\{(U_\ell,\varphi_\ell)\}_{\ell=1}^{n}$ be a $c$-admissible atlas for $\mathcal{M}$ for some $c>0$. Then for all integer $s\ge0$,
\begin{equation}\label{aSob}
C_1\|u\|_{H^s(\mathcal{M})}\le \Big(\sum_{\ell=1}^{n}\|u\circ \varphi_\ell^{-1}\|_{H^s(\varphi_\ell(U_\ell))}^2\Big)^{1/2}
\le C_2\sqrt{\mu_\mathcal{A}}\,\|u\|_{H^s(\mathcal{M})},\quad u\in H^s(\mathcal{M}),
\end{equation}
where $C_1>0$ depends only on $s$,  the choice of the atlas $\hat\mathcal{A}$
and the partition of unity $\hat\Gamma$, and
$C_2$ depends in addition on $c$. %
\end{lemma}

\begin{proof}
Let $L_k:=\{\ell: U_\ell\cap \hat U_k\ne\emptyset\}$. Then $\hat U_k\subset \bigcup_{\ell\in L_k}U_\ell$
and hence
$$%
\|u\hat\gamma_k\circ \hat\varphi_k^{-1}\|_{H^s(\mathbb{R}^d)}^2
=\|u\hat\gamma_k\circ \hat\varphi_k^{-1}\|_{H^s(\hat\varphi_k(\hat U_k))}^2
\le\sum_{\ell\in L_k}\|u\hat\gamma_k\circ \hat\varphi_k^{-1}\|_{H^s(\hat\varphi_k(U_\ell\cap \hat U_k))}^2.
$$%
Since $\hat\varphi_k^{-1}=\varphi_\ell^{-1}\circ\hat\varphi_{k(\ell)}\circ\hat\varphi_k^{-1}$ on
$\hat\varphi_k(U_\ell\cap \hat U_k)$, we have by the Leibniz and the chain rules,
$$
\|u\hat\gamma_k\circ \hat\varphi_k^{-1}\|_{H^s(\hat\varphi_k(U_\ell\cap \hat U_k))}^2
\le C\|u\circ \varphi_\ell^{-1}\|_{H^s(\varphi_\ell(U_\ell\cap \hat U_k))}^2,$$
where $C$ depends only on the size of derivatives of order up to $s$ of the functions $\hat\gamma_k\circ \hat\varphi_k^{-1}$ and the transition maps
$\hat\varphi_{j}\circ\hat\varphi_{k}^{-1}$ of $\hat\mathcal{A}$, and the (nonzero) determinants of their Jacobi matrices.
Finally,
$$
\sum_{\ell=1}^n\sum_{\substack{k=1\\ \ell\in L_k}}^{\hat n}\|u\circ \varphi_\ell^{-1}\|_{H^s(\varphi_\ell(U_\ell\cap \hat U_k))}^2
\le \hat n\sum_{\ell=1}^n\|u\circ \varphi_\ell^{-1}\|_{H^s(\varphi_\ell(U_\ell))}^2,$$
and we deduce the first inequality in \eqref{aSob}.

On the other hand, for each $k=1,\ldots,\hat n$,
$$
\sum_{\substack{\ell=1\\ k(\ell)=k}}^{n}\|u\circ \varphi_\ell^{-1}\|_{H^s(\varphi_\ell(U_\ell))}^2
\le \mu_\mathcal{A} \|u\circ \hat\varphi_k^{-1}\|_{H^s(\hat\varphi_k(\hat U^{c}_k))}^2.$$
By applying the Leibniz product rule to $\frac{1}{\hat\gamma_k}\cdot(u \hat\gamma_k)$ and estimating the $L^2$-norm of the products of (bounded) derivatives of
$\frac{1}{\hat\gamma_k\circ \hat\varphi_k^{-1}}$ with derivatives of $u\circ\hat\varphi_k^{-1}$, we obtain
\begin{equation}\label{loc1}
\|u\circ \hat\varphi_k^{-1}\|_{H^s(\hat\varphi_k(\hat U^{c}_k))}\le C \|u\hat\gamma_k\circ \hat\varphi_k^{-1}\|_{H^s(\mathbb{R}^d)},
\end{equation}
where $C$ depends only on $\hat\mathcal{A}$, $\hat\Gamma$, $c$ and $s$.
This completes the proof of the second inequality in \eqref{aSob}.
\end{proof}

Note that \eqref{aSob} holds without any assumptions about the boundaries of the sets $\varphi_\ell(U_\ell)$. In particular, they do not need to be Lipschitz domains.
If they are, then we can extend the functions $u|_{U_\ell}$ to $\mathcal{M}$, which will be useful in
Section~\ref{pdk} for establishing an equivalent norm for $H^s(\mathcal{M})$ in terms of a reproducing kernel.
For any pair of open sets $\Omega\subset \Omega'\subset\mathbb{R}^d$, let $\mathcal{E}_s(\Omega,\Omega')$ denote the
\emph{extension constant}
$$
\mathcal{E}_s(\Omega,\Omega')
=\sup_{u\in H^s(\Omega)\setminus\{0\}}\inf\{\|\tilde u\|_{H^s(\Omega')}/\|u\|_{H^s(\Omega)}:
\tilde u\in H^s(\Omega'),\, \tilde u|_{\Omega}=u\}.$$
In the case $\Omega'=\mathbb{R}^d$ we write $\mathcal{E}_s(\Omega):=\mathcal{E}_s(\Omega,\mathbb{R}^d)$.
The extension constant $\mathcal{E}_s(\Omega)$ is finite for Lipschitz domains due to extension theorems.

\begin{lemma}\label{ext} Let $U\subset \mathcal{M}$ be a domain such that $U\subset \hat U^c_k$ for
some $c>0$ and $k$, and the extension constant $\mathcal{E}_s(\hat\varphi_k(U))$ is finite
for an integer $s\ge0$.
Then for any $u:U\to\mathbb{R}$ such that $u\circ \hat\varphi_k^{-1}\in H^s(\hat\varphi_k(U))$
there exists a function $\tilde u\in H^s(\mathcal{M})$ such that $\tilde u|_{U}=u$ and
$$
\|\tilde u\|_{H^s(\mathcal{M})}\le C\mathcal{E}_s(\hat\varphi_k(U))\|u\circ \hat\varphi_k^{-1}\|_{H^s(\hat\varphi_k(U))},$$
where $C$ depends only on $\hat\mathcal{A},\hat\Gamma,s,c$.
\end{lemma}

\begin{proof} Let $w\in H^s(\mathbb{R}^d)$ be an extension of $u\circ \hat\varphi_k^{-1}$ such that
$$
\|w\|_{H^s(\mathbb{R}^d)}\le 2 \mathcal{E}_s(\hat\varphi_k(U)) \|u\circ \hat\varphi_k^{-1}\|_{H^s(\hat\varphi_k(U))}.$$
We define $\tilde u$ as follows,
$$
\tilde u(x)=
\begin{cases}
(\chi_{k,c} w\circ \hat\varphi_{k})(x), & \text{if } x\in \hat U_{k},\\
0, &\text{otherwise,}
\end{cases} $$
where $\chi_{k,c}\in C^\infty(\mathbb{R}^d)$ is a cutoff function, see e.g.~\cite{Lee13}, that satisfies
$$
\chi_{k,c}(x)=1,\quad x\in \hat\varphi_k(\hat U^c_k),\quad \text{and}\quad \chi_{k,c}(x)=0,\quad x\notin \hat\varphi_k(\hat U_k).$$
Then $\tilde u|_{U}=u$, and for any $\ell$ by the Leibniz and the chain rules,
\begin{align*}
\|\tilde u\hat\gamma_\ell\circ \hat\varphi_\ell^{-1}\|_{H^s(\mathbb{R}^d)}
&=\|(\hat\gamma_\ell\circ \hat\varphi_\ell^{-1})
(\chi w\circ \hat\varphi_k\circ \hat\varphi_\ell^{-1})\|_{H^s(\hat\varphi_\ell(\hat U_k\cap\hat U_\ell))}\\
&\le C_1\|w\|_{H^s(\hat\varphi_k(\hat U_k\cap\hat U_\ell))},
\end{align*}
where $C_1$ depends only on $c,s$ and the choice of  $\hat\mathcal{A}$, $\hat\Gamma$ and the cutoff function $\chi_{k,c}$.
Clearly, cutoff functions $\chi_{k,c}$, $k=1,\ldots,\hat n$, may be chosen once and for all as soon as other parameters are fixed. Hence
$$
\|\tilde u\|_{H^s(\mathcal{M})}^2=\sum_{\ell=1}^{\hat n}\|\tilde u\hat\gamma_\ell\circ \hat\varphi_\ell^{-1}\|_{H^s(\mathbb{R}^d)}^2
\le 4\hat n C_1^2\mathcal{E}^2_s(\hat\varphi_k(U))\|u\circ \hat\varphi_k^{-1}\|_{H^s(\hat\varphi_k(U))}^2.\qedhere$$
\end{proof}

\subsection{Reproducing kernels}\label{pdk} %

We first recall some basic facts about reproducing kernels and kernel based interpolation, see
\cite{Aronszajn50,Buhmann03,Fasshauer07,FasshauerMcCourt15,PaulsenRaghupathi16,Wendland} for details,
and then discuss reproducing kernels for Sobolev spaces.

Functions $K:\Omega\times \Omega\to\mathbb{R}$ will be called \emph{kernels} on $\Omega$, where $\Omega$ is any set.
For a kernel $K$ and two sets $S,T$, $K|_{S,T}$ stands for the restriction of $K$ to
$S\times T$, and for two finite sets $X,Y$ we also denote by
$K|_{X,Y}$ the matrix $[K(x,y)]_{x\in X,\,y\in Y}$.
To simplify the notation we set $K|_S:=K|_{S,S}$ and $K|_X:=K|_{X,X}$.
For an operator $A$ that can be applied to one or both of the
arguments of the kernel $K$, we write $A_1K$ or $A_2K$  in order to clarify whether $A$ is applied to
the first or the second argument. We will also use the notation $K_A:=A_1K$.

A kernel $K:\Omega\times \Omega\to\mathbb{R}$ is said to be \emph{symmetric} if $K(x,y)=K(y,x)$ for all $x,y\in\Omega$, and \emph{positive (semi-)definite} if
the matrix $K|_X$ is positive (semi-)definite for  any finite subset $X$ of $\Omega$.

If $K$ is a symmetric positive semi-definite kernel on a set $\Omega$, then there is a unique
Hilbert space  $H_K=H_K(\Omega)$ of functions on $\Omega$ called \emph{native space} of $K$, with
the inner product denoted by $(\cdot,\cdot)_K=(\cdot,\cdot)_{K,\Omega}$ and norm by
$\|\cdot\|_{K}=\|\cdot\|_{K,\Omega}$, such that %
\begin{align}
\label{repr1}
K(\cdot,y)\in H_K&\quad\text{for all}\quad y\in\Omega,\\
\label{repr2}
(f,K(\cdot,y))_K=f(y)&\quad\text{for all}\quad f\in H_K,\quad y\in\Omega.
\end{align}
A Hilbert space $H$ of functions on $\Omega$ admits a kernel $K$ with these properties if and only if the linear functional
$\delta_xf:=f(x)$ of point evaluation is bounded on $H$ for all $x\in\Omega$. Then $K$ is a \emph{reproducing kernel} of $H$, and $H$ is said to be a
\emph{reproducing kernel Hilbert space}.

The identity \eqref{repr2} means that $K(\cdot,y)$ is the  Riesz representer of the linear functional $\delta_y\in H_K^*$.
Moreover, the kernel $K$ delivers Riesz representers for all bounded linear functionals.

\begin{lemma}[\cite{Wendland}, Theorem 16.7]\label{gammaK}
For any bounded linear functional $\gamma\in H_K^*$, the Riesz representer of $\gamma$ is the function
$\gamma_1K=\gamma_2K$ that belongs to $H_K$.
\end{lemma}

Assume that $K$ is positive definite. The \emph{kernel sums} of the form
\begin{equation}\label{kint}
\sigma=\sum_{j=1}^n c_j K(\cdot,x_j),\qquad c_j\in\mathbb{R},\quad x_j\in\Omega,
\end{equation}
are dense in $H_K(\Omega)$, and their kernel norm is given explicitely by
\begin{equation}\label{snorm}
\|\sigma\|_{K}^2=\sum_{i,j=1}^nc_i c_j K(x_i,x_j).
\end{equation}

Given $X=\{x_1,\ldots,x_n\}\subset\Omega$, and data $a_1,\ldots,a_n\in\mathbb{R}$, the kernel sum \eqref{kint}
is uniquely determined by the interpolation conditions $\sigma(x_i)=a_i$, $i=1,\ldots,n$. We will call it
the \emph{kernel interpolant} of this data. In the case when $a_i=v(x_i)$, $i=1,\ldots,n$, for some
$v\in H_K$, we say that $\sigma$ is the kernel interpolant  of $v$. It satisfies
\begin{equation}\label{orth}
(v-\sigma,\sigma)_{K}=0
\end{equation}
and hence
\begin{equation}\label{minnorm}
\|v-\sigma\|_{K}^2+\|\sigma\|_{K}^2=\|v\|_{K}^2.
\end{equation}

The restriction $K|_G$ on a subset $G\subset\Omega$ is obviously also a symmetric positive semi-definite kernel.
Thanks to \eqref{snorm}, any kernel sum \eqref{kint} satisfies
\begin{equation}\label{srestr}
\|\sigma\|_{K,G}=\|\sigma\|_{K}\quad\text{whenever}\quad X\subset G.
\end{equation}
The native space of $K|_G$ can be described as follows.

\begin{lemma}[\cite{Aronszajn50}]\label{restrK}
If $K$ is a reproducing kernel on $\Omega$ and $G\subset\Omega$, then $K|_G$ is a reproducing kernel on $G$
with native space $H_K(G)=\{u|_G:u\in H_K(\Omega)\}$, and
$$
\|u\|_{K,G}=\min\{\|\tilde u\|_{K,\Omega}:\tilde u\in H_K(\Omega),\,\tilde u|_G=u\}.$$
\end{lemma}

If $\Omega$ is an open set in $\mathbb{R}^d$, then by Sobolev embedding $H^s(\Omega)$ is a reproducing kernel Hilbert space
as long as $s>d/2$. Every (equivalent) norm for $H^s(\Omega)$ corresponds to a positive definite kernel on $\Omega$.

\begin{lemma}\label{smker}
Let $K:\Omega\times\Omega\to\mathbb{R}$ be a reproducing kernel for $H^s(\Omega)$, $s>d/2$. Then
$\partial^\beta_1 K(x,\cdot)\in H^{s}(\Omega)$ for all $x\in\Omega$ and all $\beta\in\mathbb{Z}_+^d$
satisfying $|\beta|<s-d/2$.
\end{lemma}

\begin{proof} By Sobolev embedding (applied to a ball centered at $x$ and contained in $\Omega$)
the linear functionals $\gamma$ of the form
$\gamma v =\partial^\beta v(x)$, $v\in H^s(\Omega)$, are bounded on $H^s(\Omega)=H_K$ for all $x\in\Omega$ and
$\beta\in\mathbb{Z}^d_+$ with $|\beta|<s-d/2$. Hence the statement follows by Lemma~\ref{gammaK}.
\end{proof}

The next statement follows immediately from Lemma~\ref{restrK}.

\begin{lemma}\label{SobK} Let $\Omega,G$ be two open sets in $\mathbb{R}^d$ with $G\subset\Omega$, and let $K$ be a
reproducing kernel for $H^s(\Omega)$, where $s>d/2$.
Then $H_K(G)\subset H^s(G)$ and
$$
\|u\|_{H^s(G)}\le C_1\|u\|_{K,G},\qquad u \in H_K(G).$$
Moreover, if the extension constant $\mathcal{E}_s(G,\Omega)$ is finite, then
$H_K(G)= H^s(G)$ and
$$
\|u\|_{K,G}\le C_2\mathcal{E}_s(G,\Omega)\|u\|_{H^s(G)},\qquad u \in H_K(G).$$
The constants $C_1,C_2$ depend only on the constants of equivalence between
$\|\cdot\|_{K,\Omega}$ and $\|\cdot\|_{H^s(\Omega)}$.
\end{lemma}

If $\mathcal{M}$ is a smooth closed manifold of dimension $d$, then $H^s(\mathcal{M})$ is a reproducing kernel Hilbert space when $s>d/2$.
It may however be difficult to compute the reproducing kernel such that its native space coincides with a given norm of
$H^s(\mathcal{M})$. %
If an orthonormal basis is known for a Sobolev space, for example for $L^2(\mathcal{M})=H^0(\mathcal{M})$, then reproducing kernels can be constructed via
infinite series, see e.g.~\cite{LevesleyRagozin07,Narcowich95} and \cite[Section 17.4]{Wendland}.
We discuss this in somewhat more detail in Section~\ref{difop}.

If $\mathcal{M}$ is embedded into $\mathbb{R}^m$, $m>d$, then
in view of the trace theorems (see e.g.\ \cite[Theorem 2.3.7]{Agranovich15}) reproducing kernels
$K$ for $H^s(\mathcal{M})$, $s>d/2$, can be obtained by restricting to $\mathcal{M}$ one of explicitely known positive definite kernels defined on $\mathbb{R}^m$ with
the native space norm equivalent to $H^{s+(m-d)/2}(\mathbb{R}^m)$.
The best known examples are Mat\'ern and Wendland kernels.

The  Mat\'ern kernel is given by
$$
M_{s,d}(x,y)=\Phi_{s,d}(x-y),\quad
\Phi_{s,d}(x):=\tfrac{1}{2^{s-1}\Gamma(s)}\mathcal{K}_{s-d/2}(\|x\|_2)\|x\|_2^{s-d/2},\quad x\in\mathbb{R}^m,$$%
where $\mathcal{K}_{\nu}({\bf x})$ denotes the modified Bessel function of second kind.
The native space norm $\|\cdot\|_{\Phi_{s,d}}$ coincides with the Bessel potential
norm of $H^{s+(m-d)/2}(\mathbb{R}^m)$ since the $m$-dimensional Fourier transform
of $\Phi_{s,d}$ is $(1+\|\omega\|_2^2)^{-s-(m-d)/2}$.

The native space of  the Wendland compactly supported kernel
$W_{m,\ell}(x,y)=\phi_{m,\ell}(\|x-y\|_2)$,  $\ell\in\mathbb{Z}_+$, see \cite{Wendland},
 is norm equivalent to  $H^{s+(m-d)/2}(\mathbb{R}^m)$
for $s=(d+1)/2+\ell$  (where $m\ge 3$ if $\ell=0$). Hence, in the case when $s-d/2\in \mathbb{Z}_++\frac12$, the restriction of
$W_{m,r}$, $m>d$, $r=\lfloor s-d/2 \rfloor=\lceil s-d/2\rceil-1$, is a reproducing kernel for $H^s(\mathcal{M})$.

If $U$ is an open set in $\mathcal{M}$ and $\varphi:U\to\mathbb{R}^d$ is a homeomorphism from $U$ to
$\varphi(U)\subset\mathbb{R}^d$,
then %
$K^\varphi(x,y):=K(\varphi^{-1}(x),\varphi^{-1}(y))$, $x,y\in \varphi(U)$ ,
is obviously the reproducing kernel for the Hilbert space
\begin{equation}\label{Hphi}
H_K(\varphi(U))=\{u\circ\varphi^{-1}:u \in H_K(U)\}
\end{equation}
isometric to $H_K(U)$, with the norm given by
$$
\|u\circ\varphi^{-1}\|_{K^\varphi,\varphi(U)}=\|u\|_{K,U}.$$
We show an analogue of Lemma~\ref{SobK}.

\begin{lemma}\label{SobKm} Let $K$ be a reproducing kernel for $H^s(\mathcal{M})$, with an integer $s>d/2$, and let
 $U\subset \mathcal{M}$ be a domain such that $U\subset \hat U^c_k$ for
some $c>0$ and $k$. Then $H_K(\hat\varphi_k(U))\subset H^s(\hat\varphi_k(U))$ and
$$
\|u\circ \hat\varphi_k^{-1}\|_{H^s(\hat\varphi_k(U))}\le C_1\|u\|_{K,U},\qquad u\in H_K(U).$$
Moreover, if the extension constant $\mathcal{E}_s(\hat\varphi_k(U))$ is finite, then
$H_K(\hat\varphi_k(U))= H^s(\hat\varphi_k(U))$ and
$$
\|u\|_{K,U}\le C_2\mathcal{E}_s(\hat\varphi_k(U))\|u\circ \hat\varphi_k^{-1}\|_{H^s(\hat\varphi_k(U))},
\qquad u\in H_K(U).$$
The constants $C_1,C_2$ depend only on $K$, $\hat\mathcal{A}$, $\hat\Gamma$, $c$ and $s$.
\end{lemma}

\begin{proof} Let $v\in H_K(\hat\varphi_k(U))$, that is $v=u\circ \hat\varphi_k^{-1}$ for some $u\in H_K(U)$.
Then $u$ is the restriction to $U$ of a function in
$H_K(\mathcal{M})$, denoted $u$ again. By Lemma~\ref{restrK} we assume without loss of generality that
$\|u\|_{K,U}=\|u\|_{K,\mathcal{M}}$.
As in the proof of Lemma~\ref{admis}, we obtain by \eqref{loc1},
$$
\|u\circ \hat\varphi_k^{-1}\|_{H^s(\hat\varphi_k(U))}\le\|u\circ \hat\varphi_k^{-1}\|_{H^s(\hat\varphi_k(\hat U^{c}_k))}
\le C \|u\hat\gamma_k\circ \hat\varphi_k^{-1}\|_{H^s(\mathbb{R}^d)}\le C \|u\|_{H^s(\mathcal{M})},$$
where $C$ depends only on $\hat\mathcal{A}$, $\hat\Gamma$, $c$ and $s$. Since the norms
$\|\cdot\|_{K,\mathcal{M}}$ and $\|\cdot\|_{H^s(\mathcal{M})}$ are equivalent, the first assertion of the lemma follows.

We now assume that $\mathcal{E}_s(\hat\varphi_k(U))<\infty$. For any $v\in H^s(\hat\varphi_k(U))$, the function
$u=v\circ \hat\varphi_k$ is by Lemma~\ref{ext} the restriction to $U$ of a function $\tilde u\in H^s(\mathcal{M})$
such that
$$
\|\tilde u\|_{H^s(\mathcal{M})}\le C\mathcal{E}_s(\hat\varphi_k(U))\|v\|_{H^s(\hat\varphi_k(U))},$$
where $C$ depends only on $c,s,\hat\mathcal{A},\hat\Gamma$. Since $H^s(\mathcal{M})=H_K(\mathcal{M})$ with equivalent norms,
it follows from Lemma~\ref{restrK} that $\|u\|_{K,U}\le C\|\tilde u\|_{H^s(\mathcal{M})}$, where $C$ depends only on
$K,\hat\mathcal{A},\hat\Gamma$. Hence $v\in H_K(\hat\varphi_k(U))$, and
the second assertion follows.
\end{proof}

\subsection{Differential operators}\label{difop}

Assuming that $\mathcal{M}$ is a smooth compact manifold of dimension $d$ without boundary, we say that
a differential operator $L$ on $\mathcal{M}$ has \emph{smooth coefficients} if its
coefficients $a_{\ell,\alpha}$ in local coordinates,
$$
Lu\circ\hat\varphi_\ell^{-1}
=\sum_{|\alpha|\le m}a_{\ell,\alpha}\partial^\alpha (u\circ\hat\varphi_\ell^{-1}),
\qquad u:\hat U_\ell\to\mathbb{R},\quad a_{\ell,\alpha}:\hat\varphi_\ell(\hat U_\ell)\to\mathbb{R},$$
are infinitely differentiable and uniformly bounded,
such that for some constants $C_{\alpha\beta}$,
\begin{equation}\label{aalpha}
\|\partial^\beta a_{\ell,\alpha}\|_{L^\infty(\hat\varphi_\ell(\hat U_\ell))}\le C_{\alpha\beta},\qquad
\alpha,\beta\in\mathbb{Z}_+^d,\quad |\alpha|\le m,\quad \ell=1,\ldots,\hat n.
\end{equation}

Consider an  elliptic operator $L$ with smooth coefficients and  even order
$m=2\kappa$. In fact, all elliptic operators have even order if $d>2$ \cite{Agranovich15}. For all $t\in\mathbb{R}$,
 $L$ is a bounded linear operator from $H^t(\mathcal{M})$ to $H^{t-2\kappa}(\mathcal{M})$.
Moreover, its inverse is a bounded linear operator from $H^{t-2\kappa}(\mathcal{M})$ to $H^t(\mathcal{M})$ as soon as the null space
$N(L)=\{u: Lu=0\}$
of $L$ is trivial \cite[Theorem 6.2.1]{Agranovich15}, which implies \eqref{reg}.

Since $H^{t_2}(\mathcal{M})$ is compactly embedded in
$H^{t_1}(\mathcal{M})$ for $t_1< t_2$ \cite[Theorem 2.3.1]{Agranovich15}, the operator $L^{-1}:H^t(\mathcal{M})\to H^t(\mathcal{M})$ is compact for
any $t\in R$. Hence the spectrum of $L$ consists of isolated eigenvalues of finite multiplicity with no finite
accumulation points. Assume that $L$, as an unbounded operator on $H^s(\mathcal{M})$ with domain
$H^{s+2\kappa}(\mathcal{M})$, is positive and self-adjoint with respect to an inner product $(\cdot,\cdot)_s$
in $H^s(\mathcal{M})$, that is, $(Lv,v)_s>0$ for all $v\in H^{s+2\kappa}(\mathcal{M})\setminus\{0\}$ and $(Lv,w)_s=(v,Lw)_s$
for all $v,w\in H^{s+2\kappa}(\mathcal{M})$. Then it follows from the spectral theorem
(see e.g.~\cite[Theorem 7.17]{Weidmann80}, where real valued Hilbert spaces are considered)
that there exists an orthonormal basis of $H^s(\mathcal{M})$ consisting of eigenfunctions $e_j$, $j\in\mathbb{N}$, of $L$, such that
$$
Lv=\sum_{j=1}^\infty \lambda_jc_j e_j,\quad L^{-1}v=\sum_{j=1}^\infty \lambda^{-1}_jc_j e_j
\quad\text{for any}\quad v=\sum_{j=1}^\infty c_j e_j\in H^s(\mathcal{M}),$$
where $\lambda_j>0$ are the eigenvalues of $L$ and $c_j\in\mathbb{R}$. This implies for all $k\in\mathbb{N}$,
$$
H^{s-km}(\mathcal{M})=\big\{\sum_{j=1}^\infty\lambda^k_jc_j e_j: \|c\|_2<\infty \big\},\quad
H^{s+km}(\mathcal{M})=\big\{\sum_{j=1}^\infty\lambda^{-k}_jc_j e_j: \|c\|_2<\infty \big\},$$
where $\|c\|_2^2:=\sum_{j=1}^\infty c^2_j$, and by interpolation (see e.g.~\cite[Section 13]{Agranovich15}),
$$
H^t(\mathcal{M})=\big\{\sum_{j=1}^\infty\lambda^{(s-t)/m}_jc_j e_j: \|c\|_2<\infty \big\},\qquad t\in\mathbb{R}.$$
The norm in $H^t(\mathcal{M})$ defined by  $\|\sum_{j=1}^\infty\lambda^{(s-t)/m}_jc_j e_j\|_t:=\|c\|_2$ is equivalent to the standard
Sobolev norms, and $\lambda^{(s-t)/m}_j e_j$, $j=1,2,\ldots$, is an orthonormal basis with respect to this norm.
It is easy to see that $L$ is positive and self-adjoint with respect to the corresponding inner products $(\cdot,\cdot)_t$,
$t\in\mathbb{R}$. Note that eigenfunctions $e_j$ belong to
$H^t(\mathcal{M})$ for all $t\in\mathbb{R}$, and thus to $C^\infty(\mathcal{M})$
since the eigenspaces of $L$ are independent of the choice of $s$ and inner product in $H^s(\mathcal{M})$.
The eigenvalues $\lambda_j$, numbered in nondecreasing order with multiplicities taken into account satisfy
$\lambda_j=j^{2\kappa/d}(C+o(1))$, $j\to\infty$, where $C$ is a positive constant independent of $j$,
see \cite[Theorem 6.1.1]{Agranovich94}.

If $s>d/2$, then $H^s(\mathcal{M})$ is embedded into $C(\mathcal{M})$, and hence $H^s(\mathcal{M})$ is a reproducing kernel Hilbert space.
It is easy to see that its kernel
corresponding to the inner product $(\cdot,\cdot)_s$ is given by
\begin{equation}\label{KfL}
K(x,y)=\sum_{j=1}^\infty e_j(x)e_j(y),
\end{equation}
where the series is a well defined real function on $\mathcal{M}\times\mathcal{M}$ since for each fixed $x\in\mathcal{M}$ the sequence
$e_j(x)=(v_x,e_j)$, $j=1,2,\ldots$, where $v_x\in H^s(\mathcal{M})$ is the Riesz representer of the point evaluation functional at $x$, is
square summable. If $t>d/2$, then the kernel of $H^t(\mathcal{M})$ for the norm $\|\cdot\|_t$ is given by
\begin{equation}\label{Lt}
L^{2(s-t)/m}_1K(x,y)=\sum_{j=1}^\infty \lambda^{2(s-t)/m}_je_j(x)e_j(y),
\end{equation}
where the powers of $L$ are defined by
$$
L^\theta v=\sum_{j=1}^\infty \lambda^\theta_jc_j e_j,\qquad v=\sum_{j=1}^\infty c_j e_j,\qquad \theta\in\mathbb{R},$$
such that
$$
\|L^\theta v\|_t=\|v\|_{t+m\theta},\qquad t,\theta\in\mathbb{R},\quad v\in H^{t+m\theta}(\mathcal{M}).$$
Since $\lambda_j>0$ for all $j$, the kernels $K_{L^\theta}=L^\theta_1K$ are positive definite whenever $s-\theta\kappa > d/2$.

We summarize in the following lemma the main findings of this section.

\begin{lemma}\label{KL}
Let $L$ be an elliptic differential operator  of order $2\kappa$ with smooth coefficients  and  trivial null space,
and let $K$ be a reproducing kernel for $H^s(\mathcal{M})$, where $s>d/2$.
Assume that $L$ is positive and self-adjoint with respect to the inner product of $H^s(\mathcal{M})$ defined by $K$.
Then $K_{L^\theta}=L^\theta_1K$ is a positive definite reproducing kernel for $H^{s-\theta\kappa}(\mathcal{M})$ for any $\theta\in\mathbb{R}$ such that
$s-\theta\kappa > d/2$. In particular, $K_{L^{-1}}$ is a reproducing kernel for $H^{s+\kappa}(\mathcal{M})$.
If $s>\kappa + d/2$, then  $K_L$ is a reproducing kernel for $H^{s-\kappa}(\mathcal{M})$ and
\begin{equation}\label{tK}
\|Lv\|_{K_L}=\|v\|_{K_{L^{-1}}},\qquad v\in H^{s+\kappa}(\mathcal{M}).
\end{equation}
\end{lemma}

Note that Lemma~\ref{KL} also holds for certain classes of pseudodifferential operators on $\mathcal{M}$,
see \cite{Agranovich94,Agranovich15}.

\begin{lemma}\label{smkerL}
Under the hypotheses of Lemma~\ref{KL},
$K(\cdot,x)\in H^{s+2\kappa r}(\Omega)$ for all $x\in\Omega$, where $r$ is any integer satisfying $2\kappa r<s-d/2$.
In particular, $K(\cdot,x)\in H^{s+2\kappa}(\Omega)$ for all $x\in\Omega$ if $s>2\kappa + d/2$.
\end{lemma}

\begin{proof} By Lemma~\ref{smker}, $L^r_1K(x,\cdot)\in H^{s}(\Omega)$ for all $x\in\Omega$ as soon as $2\kappa r<s-d/2$.
Since the kernel  $L^r_1K$ is symmetric, it follows that $L^r_1K(\cdot,x)\in H^{s}(\Omega)$ for all $x\in\Omega$, and the statement
is obtained by applying \eqref{reg} to $u=K(\cdot,x)$ and the operator $L^r$ of order $2\kappa r$.
\end{proof}

In the case of the $d$-dimensional sphere $\mathcal{M}=\mathbb{S}^d$ of particular interest is a class of elliptic pseudodifferential operators
\cite{Freeden98} that can be represented
 in the form
\begin{equation}\label{pseu}
Lv=\sum_{j=1}^\infty \lambda_j c_je_j,\quad v=\sum_{j=1}^\infty  c_je_j,\qquad \lambda_j>0,\quad j\in\mathbb{N},
\end{equation}
where $e_j$ are the spherical harmonics, that is the eigenfunctions of the  Laplace-Beltrami operator $\Delta_{\mathcal{M}}$,
normalized with respect to the standard inner product of $L^2(\mathbb{S}^d)$.  For  $d=2$ these operators have important applications in physical geodesy.

In particular, operators of the form
\begin{equation}\label{LB}
L=(-\Delta_{\mathcal{M}}+\alpha I)^\kappa, \qquad \kappa\in\mathbb{N},\quad \alpha>0,
\end{equation}
where $I$ is the identity operator,
are elliptic differential operators of order $2\kappa$ on $\mathbb{S}^d$ with smooth coefficients and trivial null spaces.
These operators satisfy \eqref{pseu} and are positive and self-adjoint with respect to the inner products
 for Sobolev spaces $H^s(\mathbb{S}^d)$, $s>d/2$, generated by reproducing kernels of the type
\begin{equation}\label{KfLS}
K(x,y)=\sum_{j=1}^\infty \mu_j e_j(x)e_j(y),
\end{equation}
with appropriately decaying positive real sequences $\{\mu_j\}_{j=1}^\infty$. Explicit \emph{zonal} kernels
$K(x,y)=\psi(x^Ty)$ with this property are
obtained as restrictions to $\mathbb{S}^d$ of radially symmetric kernels (radial basis functions)
$K(x,y)=\phi(\|x-y\|_2)$, $x,y\in\mathbb{R}^{d+1}$ since
$\|x-y\|_2=\sqrt{2-2x^Ty}$ if $\|x\|_2=\|y\|_2=1$, see e.g.~\cite[Section 17.2]{Wendland}. In particular, the hypotheses of
Lemma~\ref{KL} are satisfied for the differential operators \eqref{LB} and kernels for $H^s(\mathbb{S}^d)$ obtained by
restricting to $\mathbb{S}^d$ the Mat\'ern and Wendland kernels for $\mathbb{R}^{d+1}$.

If $\mathcal{M}$ is a Cartesian product of multiple spheres, possibly of different dimensions, then appropriate kernels with respect to which operators
\eqref{LB} are positive and self-adjoint are obtained for $H^s(\mathcal{M})$ by taking tensor products of corresponding
spherical restrictions of the Mat\'ern and Wendland kernels. This follows from the fact that
tensor products of kernels may be identified with the kernels of the tensor products of the respective
reproducing kernel Hilbert spaces, see e.g.~\cite{PaulsenRaghupathi16}. This covers in particular the $d$-dimensional torus
$\mathcal{M}=\mathbb{T}^d$ \cite{Narcowich95} and hence the periodic boundary value problems for the operators $L=(-\Delta+\alpha I)^\kappa$ on cuboids, where
$\Delta$ is the $d$-dimensional Laplace operator.

Differential operators \eqref{LB} on arbitrary Riemannian manifolds also satisfy the assumptions of Lemma~\ref{KL}, see \cite{Agranovich15}.
However, closed form formulas for the kernels of the form \eqref{KfLS}, where $e_j$ are the eigenfunctions of $\Delta_\mathcal{M}$, are not available in general.

\subsection{Sampling inequalities}\label{sample}
Sampling inequalities bound a weaker Sobolev norm of a function $f$ in terms of its values on a finite set $X\subset\Omega$ and a stronger Sobolev norm of $f$,
see the survey \cite{RSZ08} and references therein. In Lemma~\ref{sampling} we follow a standard way of proving sampling inequalities, as e.g.\ in  \cite{NWW05,WR05},
but obtain a local version that makes use of the diameter of the domain $\Omega$ as the discretization parameter $h$ instead of the fill distance of $X$.
Therefore our estimates \eqref{sam2}, \eqref{saminf} depend on a polynomial Lebesgue constant that is usually not present explicitly in the estimates thanks to a
requirement that $X$ is sufficiently dense in $\Omega$. We refer to \cite[Section 4]{DavySchaback16} for a demonstration that this requirement is too restrictive for the
setting of local error bounds we will apply the sampling inequalities in Section~\ref{ldm}.

For any finite  $Y\subset\mathbb{R}^d$ we consider the \emph{Lebesgue function}
$$
\lambda_{r}(y,Y)=\sup\{|p(y)|:\|p|_Y\|_\infty\le 1\text{ for all }p\in \Pi^d_r\},
\quad x\in\mathbb{R}^d,\quad r\in\mathbb{N},$$
where $\Pi^d_r$, $r\in\mathbb{N}$,
denotes the space of $d$-variate polynomials of total order at most $r$ (or degree less than $r$).
Note that $\lambda_{r}(y,Y)<\infty$ for all $y\in\mathbb{R}^d$ if and only if $Y$ is a \emph{$r$-determining set}, that is
$p\in \Pi^d_r$ and $p|_Y=0$ imply $p\equiv 0$. For any set $S\subset\mathbb{R}^d$, the \emph{Lebesgue constant} is given by
$$
\lambda_{r}(S,Y)=\sup_{y\in S}\lambda_{r}(y,Y).$$
Note that the \emph{Lebesgue constant} is the reciprocal of the \emph{norming constant} \cite{JSW99}. On the other hand, the Lebesgue function is a special case of a
\emph{growth function} \cite{Davy07,DavySchaback18}, and we rely in  the proof on a general duality theory for growth functions \cite[Theorem 9]{DavySchaback18}
that replaces the ``local polynomial reproduction'' arguments, compare \cite[Chapter 3]{Wendland}.

For any bounded domain $\Omega\subset\mathbb{R}^d$, we  denote by
$\chi(\Omega)$ its \emph{chunkiness parameter} \cite{BrennerScott08}, the quotient of the diameter of $\Omega$ to the maximum radius
of a ball $B\subset\Omega$ such that $\Omega$ is star-shaped with respect to $B$. Clearly, $\chi(\Omega)<\infty$ if and only if
$\Omega$ is bounded and star-shaped with respect to a ball.
It is known that such domains satisfy the Lipschitz
condition, see \cite[Section 4.3]{Burenkov98} %
or \cite[Section 1.3.2]{MazyaPoborchi97}. %

\begin{lemma}\label{sampling} Let $\Omega\subset\mathbb{R}^d$ be a bounded domain with $\chi(\Omega)\le\chi$ for some
constant $\chi<\infty$,
let $X=\{x_1,\ldots,x_n\}\subset\Omega$ be an $r$-determining set for some integer $r>d/2$, and let $v\in H^r(\Omega)$.
We set $h=\diam(\Omega)$.
Then for any integer $k$ with $0\le k<r$,
\begin{equation}\label{sam2}
|v|_{H^k(\Omega)}\le Ch^{-k}\lambda_r(\Omega,X)\big(h^{d/2}\|v|_X\|_\infty + h^{r}|v|_{H^r(\Omega)}\big),
\end{equation}
where $C$ depends only on $r,d$ and $\chi$. Moreover,  for any $\beta\in\mathbb{Z}_+^d$ with $|\beta|<r-d/2$,
\begin{equation}\label{saminf}
|\partial^\beta v(x)|\le Ch^{-|\beta|}\lambda_r(\Omega,X)\big(\|v|_X\|_\infty + h^{r-d/2}|v|_{H^r(\Omega)}\big),
\quad x\in\Omega,
\end{equation}
where $C$ depends only on $r,d$ and $\chi$.
\end{lemma}

\begin{proof}
By Sobolev embedding we identify $v$ with an $m$ times  continuously differentiable function in $\Omega$,
where $m$ is the largest integer such that $m<r-d/2$.
By Proposition 4.3.2,
equation (4.1.18) and Lemma 4.3.8
in \cite{BrennerScott08} there is a polynomial $p\in \Pi^d_r$ such that
\begin{align}
\label{BHinf}
|\partial^\beta v(x)-\partial^\beta p(x)|&\le C_1h^{r-|\beta|-d/2}|v|_{H^r(\Omega)},\quad x\in\Omega,\quad |\beta|< r-d/2,\\
\label{BH}
|v- p|_{H^k(\Omega)}&\le C_2h^{r-k}|v|_{H^r(\Omega)},\quad k\le r,
\end{align}
where both $C_1$ and $C_2$ depend only on $r,d$ and $\chi$.

Let $\alpha\in\mathbb{Z}_+^d$ with $|\alpha|=k$. Since $X$ is $r$-determining, for any $x\in\Omega$ there exist weights
$w_j\in\mathbb{R}$ such that
$$
\partial^\alpha p(x)=\sum_{j=1}^n w_j p(x_j)\quad\text{ and }\quad
\sum_{j=1}^n |w_j|= \sup\big\{\partial^\alpha q(x):q\in \Pi^d_r,\;\|q|_X\|\le 1\big\},$$
see \cite[Theorem 9]{DavySchaback18}.
By Markov inequality (see e.g.~\cite[Proposition 2.2]{NWW05}),
$$
|\partial^\alpha q(x)|\le C_3h^{-k}\|q\|_{L^\infty(\Omega)},\qquad q\in \Pi^d_r,$$ %
where $C_3$ depends only on $r,d$ and $\chi$. Hence
$$
\sum_{j=1}^n |w_j|\le C_3h^{-k}\lambda_r(\Omega,X).$$
By \eqref{BHinf},
$$
|v(x_j)-p(x_j)|\le C_1h^{r-d/2}|v|_{H^r(\Omega)},\quad j=1,\ldots,n.$$
Hence
\begin{align}
\begin{split}\label{pest}
|\partial^\alpha p(x)| &=  \Big|\sum_{j=1}^n w_j v(x_j)+\sum_{j=1}^n w_j \big(p(x_j)-v(x_j)\big)\Big|\\
&\le C_3h^{-k}\lambda_r(\Omega,X)\big(\|v|_X\|_\infty + C_1 h^{r-d/2}|v|_{H^r(\Omega)}\big),
\end{split}
\end{align}
which implies
$$
\|\partial^\alpha p\|_{L^2(\Omega)}
\le C_4C_3h^{-k+d/2}\lambda_r(\Omega,X)\big(\|v|_X\|_\infty + C_1h^{r-d/2}|v|_{H^r(\Omega)}\big),$$
where $C_4$ depends only on $d$.
By \eqref{BH},
$$
\|\partial^\alpha v-\partial^\alpha p\|_{L^2(\Omega)}\le C_2h^{r-k}|v|_{H^r(\Omega)},$$
and \eqref{sam2} follows since
$\|\partial^\alpha v\|_{L^2(\Omega)}
\le \|\partial^\alpha v-\partial^\alpha p\|_{L^2(\Omega)}+\|\partial^\alpha p\|_{L^2(\Omega)}$
and $\lambda_r(\Omega,X)\ge1$. Similarly, \eqref{saminf} follows directly from \eqref{BHinf} and \eqref{pest}.
\end{proof}

\subsection{Extension constants}\label{extc}

Since we will need to bound the extension constants $\mathcal{E}_s(\hat\varphi_k(U))$ uniformly for families of local subdomains
$U\subset\mathcal{M}$, we consider the question of estimating $\mathcal{E}_s(\Omega)$ for Lipschitz domains $\Omega\subset\mathbb{R}^d$.

A bounded domain $\Omega$ satisfies the Lipschitz condition if it has a locally Lipschitz boundary, that is, each point on the
boundary of $\Omega$ has a neighborhood whose intersection with $\partial\Omega$ is the graph of a function satisfying the
Lipschitz condition.

Since $\partial\Omega$ is compact, there is a finite system $\mathcal{L}$ consisting of $N_\mathcal{L}$ open sets in $\mathbb{R}^d$ that cover
$\partial\Omega$ such
that their intersections with $\partial\Omega$ are the graphs of continuous functions with Lipschitz constants not exceeding
some number $M_\mathcal{L}$. Such a system is not unique, and  a smaller $M_\mathcal{L}$ may be obtained at the expense of increasing $N_\mathcal{L}$.
We will call any such system $\mathcal{L}$ a \emph{Lipschitz cover} of $\partial\Omega$. Apart from $N_\mathcal{L}$ and $M_\mathcal{L}$,
another parameter of $\mathcal{L}$ important for the extension constant is a number $r_\mathcal{L}>0$ such that
for each $x\in\partial\Omega$ the  ball with radius $r_\mathcal{L}$ centered at $x$  is contained in one of the sets of $\mathcal{L}$.
It is easy to see that a positive $r_\mathcal{L}$ exists for any Lipschitz cover.

The Stein Extension Theorem \cite{Stein} shows that $\mathcal{E}_s(\Omega)$ is bounded from the above by a constant depending only on
$d,s,N_\mathcal{L},M_\mathcal{L}$ and $r_\mathcal{L}$.

Since we will need bounds for $\mathcal{E}_s(\Omega)$ for domains with diameter tending to zero, we will  use
the parameter $\tau_\mathcal{L}:=r_\mathcal{L}/\diam(\Omega)$ in place of $r_\mathcal{L}$.
The following estimate follows immediately from the results in \cite[Section 3.1.5]{MazyaPoborchi97}.

\begin{lemma}\label{extce} Let $\Omega\subset\mathbb{R}^d$ be a bounded domain star-shaped with respect to a ball, and let $\mathcal{L}$ be a Lipschitz cover
of its boundary. If $\chi(\Omega)\le\chi$, $N_\mathcal{L}\le N$, $M_\mathcal{L}\le M$ and $\tau_\mathcal{L}\ge \tau>0$, then for any integer $s\ge0$,
$$
\mathcal{E}_s(\Omega)\le Ch^{-d/2},\qquad h=\diam(\Omega),$$
where $C$ depends only on $d,s,\chi,N,M$ and $\tau$.
\end{lemma}

\subsection{Local differentiation matrices}\label{ldm}
Let $L$ be a differential operator of order $m$ on $\mathcal{M}$, $K$ the reproducing kernel of a Hilbert space $H_K(\mathcal{M})$
embedded into $C^m(\mathcal{M})$, and $X=\{x_1,\ldots,x_n\}\subset\mathcal{M}$. For any $u\in C^m(\mathcal{M})$, an approximation of $Lu$ can be
obtained by applying $L$ to the kernel interpolant \eqref{kint},
\begin{align*}
\sigma&=\sum_{j=1}^n c_j K(\cdot,x_j),\quad \sigma(x_j)=u(x_j),\quad j=1,\ldots,n,\\
Lu&\approx L\sigma=\sum_{j=1}^n c_j K_L(\cdot,x_j).
\end{align*}
In particular, the approximation  $L\sigma|_X$ of $Lu|_X$ can be computed with the help of the \emph{differentiation matrix}
$W_X$,
\begin{equation}\label{difm}
L\sigma|_X=W_X u|_X ,\quad\text{where}\quad W_X:=K_L|_{X}\,(K|_{X})^{-1}.
\end{equation}

If $U$ is an open set in $\mathcal{M}$ and $\varphi:U\to\mathbb{R}^d$ is a homeomorphism from $U$ to
$U^\varphi:=\varphi(U)\subset\mathbb{R}^d$, then we denote by $L^\varphi$ the differential operator $L$ in the local coordinates defined
by $\varphi$, that is
$$
L^\varphi v:=L(v\circ\varphi)\circ\varphi^{-1}$$
 for any sufficiently smooth function $v$ on
$U^\varphi$. Then the kernel $K^\varphi(x,y)=K(\varphi^{-1}(x),\varphi^{-1}(y))$ of the space
$H_K(U^\varphi)$ defined in \eqref{Hphi} obviously satisfies
\begin{equation}\label{L1Kphi}
L^\varphi_iK^\varphi=(L_iK)^\varphi,\quad i=1,2.
\end{equation}
In particular,
$$
W_X=K^\varphi_L|_{X^\varphi}\,(K^\varphi|_{X^\varphi})^{-1},$$
where $K^\varphi_L:=L^\varphi_1K^\varphi$ and $X^\varphi:=\varphi(X)$.

\begin{lemma}\label{lde} Let $L$ be a differential operator on $\mathcal{M}$ of order $m$ with smooth coefficients, and
let $K$ be a reproducing kernel for $H^s(\mathcal{M})$, with an integer $s>m+d/2$.
Furthermore, let $X=\{x_1,\ldots,x_n\}\subset U$ for an open set $U\subset \hat U^c_k\subset\mathcal{M}$
for some $k\in\{1,\ldots,\hat n\}$ and $c>0$. For $\varphi=\hat\varphi_k$, assume that   $U^\varphi$
is star-shaped with respect to a ball, with $\chi(U^\varphi)\le\chi<\infty$, and
$X^\varphi$ is an $r$-determining set for some $r>m+d/2$ such that $K(\cdot,x)\in H^r(\mathcal{M})$ for all $x\in\mathcal{M}$.
Then for any $u\in H^r(\mathcal{M})$,
\begin{equation}\label{lde1}
\|Lu|_X-W_X u|_X\|_\infty\le
C_1\lambda_r(U^\varphi,X^\varphi)h^{r-m-d/2}|u^\varphi-\sigma^\varphi|_{H^r(U^\varphi)},
\end{equation}
where $h=\diam(U^\varphi)$, $u^\varphi:=u\circ\varphi^{-1}$, $\sigma^\varphi:=\sigma\circ\varphi^{-1}$ and
$C_1$ depends only on $r,d,\chi$ and $L$.
Moreover, if  $X^\varphi$ is an $s$-determining set and $u\in H^s(\mathcal{M})$, then
\begin{align}\label{lde2}
\|Lu|_X-W_X u|_X\|_\infty&\le
C_2\lambda_s(U^\varphi,X^\varphi)h^{s-m-d/2}\|u\|_{K,U}\\
\label{lde3}
&\le
C_3\lambda_s(U^\varphi,X^\varphi)\mathcal{E}_s(U^\varphi)h^{s-m-d/2}\|u^\varphi\|_{H^s(U^\varphi)},
\end{align}
where $C_2,C_3$ depend in addition on $K$, $\hat\mathcal{A}$, $\hat\Gamma$ and $c$.
\end{lemma}

\begin{proof} By Sobolev embedding, the condition $r>m+d/2$ ensures that $u\in C^m(\mathcal{M})$, hence $Lu$ is well defined as a continuous function on $\mathcal{M}$
as soon as $u\in H^r(\mathcal{M})$.
Since $U^\varphi$ is a Lipschitz domain, it follows that $\mathcal{E}_s(U^\varphi)<\infty$, and Lemma~\ref{SobKm} shows that
$K^\varphi$ is a reproducing kernel for $H^s(U^\varphi)$. Since $s>m+d/2$, $H^s(U^\varphi)$ is embedded
into $C^m(\mathcal{M})$. Moreover, $\sigma^\varphi\in H^{r}(U^\varphi)$ as a linear combination of the functions $K^\varphi(\cdot,x_j)$, $j=1,\ldots,n$.

In view of \eqref{difm} and \eqref{L1Kphi}, the $i$-th component of the vector $Lu|_X-W_X u|_X$
has the form
$$
Lu(x_i)-L\sigma(x_i)=L^\varphi u^\varphi(x^\varphi_i)-L^\varphi \sigma^\varphi(x^\varphi_i),$$
where  $x^\varphi_i=\varphi(x_i)$. Since $(u^\varphi-\sigma^\varphi)|_{X^\varphi}=0$,
the sampling inequality \eqref{saminf} implies
$$
|L^\varphi u^\varphi(x^\varphi_i)-L^\varphi \sigma^\varphi(x^\varphi_i)|
\le C\lambda_r(U^\varphi,X^\varphi)h^{r-m-d/2}|u^\varphi-\sigma^\varphi|_{H^r(U^\varphi)},$$
where $C$ depends only on $r,d,\chi$ and the constants in \eqref{aalpha}, which implies \eqref{lde1}.

In the case $r=s$ the condition that $K(\cdot,x)\in H^s(\mathcal{M})$ for all $x\in\mathcal{M}$ is satisfied since $K$ is  a reproducing kernel for $H^s(\mathcal{M})$, and
we have by Lemma~\ref{SobKm},
$$
|u^\varphi-\sigma^\varphi|_{H^s(U^\varphi)}\le \|u^\varphi-\sigma^\varphi\|_{H^s(U^\varphi)}\le
C\|u-\sigma\|_{K,U},$$
where $C$ depends only on $K$, $\hat\mathcal{A}$, $\hat\Gamma$, $c$ and $s$.
Then the minimum norm property
\eqref{minnorm} of the kernel interpolant implies
$\|u-\sigma\|_{K,U}\le \|u\|_{K,U}$, and \eqref{lde2} follows.

Finally, by Lemma~\ref{SobKm}, $\|u\|_{K,U}\le C\mathcal{E}_s(U^\varphi)\|u^\varphi\|_{H^s(U^\varphi)}$,
 where $C$ depends only on $K$, $\hat\mathcal{A}$, $\hat\Gamma$, $c$ and $s$, which implies \eqref{lde3}.
\end{proof}

Different types of local error bounds for kernel based numerical differentiation can be found in
\cite{DavySchaback16,DavySchaback19}.

Note that for an operator $L$ and kernel $K$ satisfying the hypotheses of Lemma~\ref{KL}, the estimate \eqref{lde1} holds,
thanks to Lemma~\ref{smkerL}, for all $r$ satisfying $2\kappa + d/2 < r< 2s-d/2$ such that $r-s$ is a multiple of $2\kappa$.
Hence, it holds in this case for all integer $r$ with $2\kappa + d/2 < r\le s+2\kappa$.

\section{A meshless finite difference method}\label{method}

Assume that $L$ is an elliptic differential operator on $\mathcal{M}$ of order $2\kappa$ with smooth coefficients
and trivial null space, and  $K$ is a reproducing kernel for $H^s(\mathcal{M})$, where $s>\kappa+d/2$, such that
$L$ is positive and self-adjoint with respect to the inner product $(\cdot,\cdot)_K$ of $H^s(\mathcal{M})$ defined by $K$.
As in Section~\ref{Sob}, we denote by $\|u\|_{H^s(\mathcal{M})}$ the norm of $H^s(\mathcal{M})$ associated with a fixed finite smooth atlas
$\hat\mathcal{A}=\{(\hat U_\ell,\hat\varphi_k)\}_{k=1}^{\hat n}$ and a smooth
partition of unity $\hat\Gamma=\{\hat\gamma_k\}_{k=1}^{\hat n}$ subordinate to it.

We consider the equation
$Lu=f$, with continuous $f$, and look for a discrete solution ${\hat u}\in\mathbb{R}^n$  to approximate $u$
on a given set of \emph{nodes} $X=\{x_1,\ldots,x_n\}\subset\mathcal{M}$.

Let
$X_\ell =\{x_j:j\in J_\ell \}\ne\emptyset$, $\ell =1,\ldots,m$, with $J_\ell \subset J:=\{1,\ldots,n\}$, be such
that  $X_\ell=X\cap U_\ell$, $\ell =1,\ldots,m$,
for an atlas  $\mathcal{A}=\{(U_\ell,\varphi_\ell)\}_{\ell=1}^m$ of $\mathcal{M}$.
Then $\bigcup_{\ell =1}^m X_\ell =X$. We set
$X_\ell^\varphi:= \varphi_\ell(X_\ell)$, $U_\ell^\varphi:= \varphi_\ell(U_\ell)$,
$n_\ell:=\# X_\ell$.
Consider the local differentiation matrices
$W_{\ell}:=W_{X_\ell}\in\mathbb{R}^{n_\ell\times n_\ell}$ of \eqref{difm}, that is
\begin{equation}\label{Wl}
W_{\ell}=K_L|_{X_\ell}(K|_{X_\ell})^{-1}
=K^{\varphi_\ell}_L|_{X_\ell^\varphi}(K^{\varphi_\ell}|_{X_\ell^\varphi})^{-1},
\qquad \ell=1,\ldots,m.
\end{equation}
The kernel $K_L$ is well defined and positive definite by Lemma~\ref{KL}, and hence the matrices $W_{\ell}$ are symmetric and positive definite.

We now set up an overdetermined linear system
\begin{equation}\label{ods}
W\!Mv=F
\end{equation}
with unknown vector $v\in\mathbb{R}^N$, the block diagonal matrix
$$
W=\diag(W_1,\ldots,W_m):=
\matr{W_1 & \cdots & 0\\
\vdots & \ddots &\vdots\\
0 &\cdots & W_m}\in \mathbb{R}^{N\times N},\quad N:=n_1+\cdots+n_m,$$
the incidence matrix
$$
M=\matr{M_1\\ \vdots \\ M_m}\in\mathbb{R}^{N\times n},\quad
M_\ell=[\delta_{ij}]_{i\in J_\ell,j\in J}\in\mathbb{R}^{n_\ell\times n},\quad
\delta_{ij}=\begin{cases} 1 & \text{if }i=j,\\
0 & \text{otherwise,}
\end{cases}$$
and the right hand side vector
$$
F=\matr{F_1\\ \vdots \\ F_m }\in\mathbb{R}^N,\quad
F_\ell:=f|_{X_\ell}.$$
The matrix $W\!M$ has the full rank $n$ since $W$ is non-singular and $\rank M=n$.

We determine the discrete solution $\hat u\in\mathbb{R}^n$ as the unique least squares solution of \eqref{ods}, such that
\begin{equation}\label{leastsq}
\|W\!M\hat u-F\|_2=\min_{v\in\mathbb{R}^n}\|W\!Mv-F\|_2.
\end{equation}

Note that \eqref{ods} can be more explicitely written as
\begin{equation}\label{ods1}
W_\ell \,v|_{J_\ell} = f|_{X_\ell},\quad \ell=1,\ldots,m,
\end{equation}
and hence, in line with the main idea of meshless generalized finite differences,
 each linear equation in \eqref{ods} is derived from a numerical differentiation formula
$$
Lu(x_i)\approx \sum_{j\in J_\ell} w^{(\ell)}_{i,j}u(x_j),\quad i\in J_\ell,$$
where the weight vector $[w^{(\ell)}_{i,j}]_{j\in J_\ell}$ is the row of $W_\ell$ corresponding to the node $x_i$.
Since
\begin{equation}\label{ods2}
\|W\!Mv-F\|_2^2=\sum_{\ell=1}^m \|W_\ell \,v|_{J_\ell} - f|_{X_\ell}\|_2^2,
\end{equation}
we in fact minimize in \eqref{leastsq} the sum of squared residuals of the numerical differentiation on the subsets
$X_\ell$.

Thus, ${\hat u}$ is well-defined for any node set $X$ and any atlas $\mathcal{A}$ as soon as $L$
is positive and self-adjoint with respect to the inner product defined by a reproducing kernel
$K$ for $H^s(\mathcal{M})$ with $s>\kappa+d/2$.
However, in order to estimate the error of the discrete solution ${\hat u}$,
 \begin{equation}\label{der}
\|\hat u-u|_{X}\|_\infty,
\end{equation}
we will assume that
\begin{equation}\label{As}
\text{$s$ is an integer satisfying $s>2\kappa+d/2$,}
\end{equation}
and make the following  assumptions about  $\mathcal{A}$ and $X$:

\renewcommand{\labelenumi}{(A\arabic{enumi})}
\newcommand{\Aref}[1]{(A\ref{#1})}

\begin{enumerate}

\item\label{A1} $\mathcal{A}$ is $c$-admissible with respect to $\hat\mathcal{A}=\{(\hat U_\ell,\hat\varphi_k)\}_{k=1}^{\hat n}$ and
$\hat\Gamma$, for some $c=c_\mathcal{A}>0$.

\item\label{A2} Each $U_\ell^\varphi$ is bounded and star-shaped with respect to a ball. Thus,
$$
\chi_\mathcal{A}:=\max_{\ell=1,\ldots,n}\chi(U_\ell^\varphi)$$
is finite.

\item\label{A3}
For all $\ell$ and $p$ such that %
$U_p\cap U_\ell\ne\emptyset$, the set
$\varphi_\ell(X_p\cap X_\ell)$ is an $(s+\kappa)$-determining set.

\end{enumerate}
\renewcommand{\labelenumi}{\arabic{enumi}.}

For $r\in\mathbb{Z}_+$, let
\begin{align*}
\lambda_{r}(\mathcal{A},X):=
\max_{1\le\ell\le m}\max_{i\in I_\ell}
\lambda_{r}(U_{\ell}^\varphi,\varphi_\ell(X_i\cap X_\ell)),
\end{align*}
where
$$
I_\ell:=\{i: U_i\cap U_\ell\ne\emptyset\}.$$
By \Aref{A3}, $\lambda_{s+\kappa}(\mathcal{A},X)<\infty$.

We will also need some further parameters of $X$ and $\mathcal{A}$. The main parameter to measure the error against will be
$$
h_\mathcal{A}=\max_{1\le \ell\le m}h_\ell,\quad\text{where}\quad h_\ell:=\diam(U_\ell^\varphi).$$
Furthermore, let
$$
\nu_{X,\mathcal{A}}:=\max_{1\le \ell\le m}n_\ell.$$

We denote by $\delta_{X,\mathcal{A}}$
the \emph{separation distance} of the set $X$ with respect to the atlas $\mathcal{A}$,
that is, the largest $\delta>0$ such that for each $\ell =1,\ldots,m$,
the open balls of radius $\delta$ centered at $\varphi_\ell(x_j)$ for all $j\in J_\ell$
are pairwise disjoint and contained in $U_\ell^\varphi$,
\begin{equation}\label{sep}
\delta_{X,\mathcal{A}}:=\min_{1\le \ell\le m}\min\big\{\dist(X_\ell^\varphi,\partial U_\ell^\varphi),\,\tfrac12\min_{j\in J_\ell}
\dist\big(\varphi_\ell(x_j),X_\ell^\varphi\setminus\{\varphi_\ell(x_j)\}\big)\big\}.
\end{equation}
To measure the quasi-uniformity of $X$, we define
$$
q_{X,\mathcal{A}}:=h_\mathcal{A}/\delta_{X,\mathcal{A}}.$$
Note that
$$
\max_{1\le i,j\le m}\frac{h_i}{h_j}\le q_{X,\mathcal{A}}$$
since $\delta_{X,\mathcal{A}}<h_\ell\le h_\mathcal{A}$ for all $\ell$.

It follows from \Aref{A2} that the domains $U_\ell^\varphi$ satisfy the Lipschitz condition. For each $\ell$, we
choose a Lipschitz cover $\mathcal{L}$ of $\partial U_\ell^\varphi$ and consider $N_\ell=N_\mathcal{L}$, $M_\ell=M_\mathcal{L}$,
$\tau_\ell=\tau_\mathcal{L}$, as defined in Section~\ref{extc}. Let
$$
N_\mathcal{A}:=\max_{\ell=1,\ldots,n}N_\ell,\qquad
M_\mathcal{A}:=\max_{\ell=1,\ldots,n}M_\ell,\qquad
\tau_\mathcal{A}:=\min_{\ell=1,\ldots,n}\tau_\ell,$$
and
$$
\mathcal{E}_r(\mathcal{A}):=\max_{\ell=1,\ldots,n}\mathcal{E}_r(U_\ell^\varphi),\quad r\in\mathbb{Z}_+.$$

Finally, we choose a smooth partition of unity $\Gamma=\{\gamma_\ell\}_{\ell=1}^n$ subordinate to $\mathcal{A}$  such that
\begin{equation}\label{pum}
\gamma_\ell\in C^\infty(\mathcal{M}),\quad\gamma_\ell\ge0,\quad %
\supp \gamma_\ell\subset U_\ell,\quad
\sum_{\ell=1}^n\gamma_\ell=1,
\end{equation}
and set
\begin{equation}\label{eta}
\eta_r(\mathcal{A}):=\max_{1\le\ell\le m}\max_{|\alpha|\le r}\|\partial^\alpha g_\ell\|_{L^\infty(\mathbb{R}^d)},\qquad r\in\mathbb{Z}_+,
\end{equation}
where
$$
g_\ell(x):=
\begin{cases}
(\gamma_\ell\circ\varphi_\ell^{-1})(h_\ell x), & \text{if }h_\ell x\in U_{\ell}^\varphi,\\
0, &\text{otherwise.}
\end{cases}$$

\begin{example}\label{pume}\rm
Assume that  each $U_{\ell}^\varphi$ contains a ball $B_\ell$ of radius
$\rho_\ell$ centered at the origin of $\mathbb{R}^d$ such that $h_\ell\le \chi_\mathcal{A} \rho_\ell$ for  each $x\in\mathcal{M}$ there is an $\ell$ with
$\|\varphi_\ell(x)\|_2\le \alpha\rho_\ell$ for some fixed positive $\alpha<1$. (In particular, we may choose
$U_\ell=\hat\varphi_k^{-1}(y_\ell+B_\ell)$ for suitable centers $y_\ell\in \varphi_k(\hat U_k^c)$ and radii $\rho_\ell$,
and define the subsets $X_\ell$ of $X$ as $X_\ell=X\cap U_\ell$.)
Let $\Psi:\mathbb{R}^d\to\mathbb{R}$ be an infinitely differentiable bump function supported on the unit ball of $\mathbb{R}^d$,
for example $\Psi(x)=\psi(\|x\|_2^2)$, with $\psi\in C^\infty[0,\infty)$, $\psi(0)=1$, $\psi^{(k)}(0)=0$,
$k\in\mathbb{N}$, $\psi(t)>0$ and $\psi'(t)\le0$ for $0\le t<1$, and $\psi(t)=0$ for $t\ge1$.
We set
$$
\bar\gamma_\ell(x):=
\begin{cases}
\Psi(\varphi_\ell(x)/\rho_\ell ), & x\in U_{\ell},\\
0, & x\in\mathcal{M}\setminus U_\ell,
\end{cases}$$
and
$$
\gamma_\ell(x)=\bar\gamma_\ell(x)/\bar\gamma(x),\quad \ell=1,\ldots,m,\qquad
\bar\gamma(x):=\sum_{i=1}^m \bar\gamma_i(x).$$
Then
$$
\bar\gamma(x)\ge \psi(\alpha^2)>0,\quad x\in\mathcal{M},$$
each function $g_\ell$ is supported in the ball of radius $\rho_\ell/h_\ell$ centered at the origin,
and for $x$ in this ball,
$$
g_\ell(x)=\frac{\Psi(h_\ell x/\rho_\ell)}{\bar\gamma(\varphi_\ell^{-1}(h_\ell x))},$$
with
\begin{align*}
\bar\gamma(\varphi_\ell^{-1}(h_\ell x))
&=\sum_{i\in I_\ell(x)}\Psi\big((\varphi_i\circ \varphi_\ell^{-1})(h_\ell x)/\rho_i\big),\\
I_\ell(x)&:=\{i: \varphi_\ell^{-1}(h_\ell x)\in U_i\}\subset\{i: U_i\cap U_\ell\ne\emptyset\}.
\end{align*}
Then by the Leibniz and the chain rules it is easy to see that $\eta_r(\mathcal{A})$ can be bounded above in terms of
(upper bounds for)
$$
\chi_\mathcal{A},\quad  \mu_\mathcal{A},\quad q_\mathcal{A},\quad
\max_{|\alpha|\le r}\|\partial^\alpha (\hat\varphi_i\circ \hat\varphi_j^{-1})\|_{L^\infty(U_j^\varphi)}\quad
\text{and}\quad 1/\psi(\alpha^2).
$$
\end{example}

Note that more general domains  e.g.\ unions of ellipsoids
may be used instead of balls as supports of the partition of unity functions $\gamma_\ell$
if we want to estimate the smallest possible constant $\eta_r(\mathcal{A})$
 for a given choice of  $U_\ell$ and $X_\ell=X\cap U_\ell$.

We stress that in contrast to the partition of unity methods \cite{BabuskaMelenk96,GriebelSchweitzer02,LSH17},
$\Gamma$ is not used in the computation of the discrete solution ${\hat u}\approx u|_X$, and we only need
the existence of partitions of unity with bounded constants $\eta_r(\mathcal{A})$ in order to prove the error bounds.

\begin{theorem}\label{error_bound}
Let $L$ be an elliptic differential operator  of order $2\kappa$ with smooth coefficients
and trivial null space, and let $K$ be a reproducing kernel for a Sobolev space $H^s(\mathcal{M})$ with an integer
$s$ satisfying $s>2\kappa+d/2$,
such that $L$  is positive and self-adjoint with respect to the inner product defined by $K$.
Assume that a set of nodes $X$ and an atlas $\mathcal{A}=\{(U_\ell,\varphi_\ell)\}_{\ell=1}^m$ satisfy
\Aref{A1}--\Aref{A3}. For any $f\in H^{s-\kappa}(\mathcal{M})$, let $u\in H^{s+\kappa}(\mathcal{M})$ be the
solution of $Lu=f$, and let $\hat u$ be the discrete approximate solution  determined by \eqref{leastsq}.
Then there exists a function $\tilde u\in H^{s+2\kappa}(\mathcal{M})$ such that $\tilde u|_X={\hat u}$ and for all
$0\le r<s-\kappa$,
\begin{equation}\label{Hb}
\|u-\tilde u\|_{H^{2\kappa+r}(\mathcal{M})}\le
C_1h_\mathcal{A}^{s-2\kappa-d-r}\|u\|_{H^s(\mathcal{M})}
+C_2h_\mathcal{A}^{s-\kappa-d/2-r}\|u\|_{H^{s+\kappa}(\mathcal{M})},
\end{equation}
where $C_1$ and $C_2$ are independent of $f,u$ and $h_\mathcal{A}$.
If $\mu_\mathcal{A}\le\mu$, $\lambda_{s+\kappa}(\mathcal{A},X)\le\lambda$, $c_\mathcal{A}\le c$, $\chi_\mathcal{A}\le\chi$, $\nu_{X,\mathcal{A}}\le\nu$,
$q_{X,\mathcal{A}}\le q$, $\eta_{s+\kappa}(\mathcal{A})\le\eta$, $N_\mathcal{A}\le N$, $M_\mathcal{A}\le M$, $\tau_\mathcal{A}\ge\tau$,
for some positive real $\mu,\lambda,c,\chi,\nu,q,\eta,N,M,\tau$,
then there are constants $C_1,C_2$ in \eqref{Hb}
depending only on $\mu,\lambda,c,\chi,\nu,q,\eta,N,M,\tau$.
Moreover, %
\begin{equation}\label{infb}
\|u|_{X}-\hat u\|_\infty\le C_1h_\mathcal{A}^{s-2\kappa-d-r_0}\|u\|_{H^s(\mathcal{M})}
+C_2h_\mathcal{A}^{s-\kappa-d/2-r_0}\|u\|_{H^{s+\kappa}(\mathcal{M})},
\end{equation}
where $r_0=\max\{0,\lfloor \frac{d}{2}\rfloor-2\kappa+1\}$, with the same properties of the constants $C_1,C_2$.
\end{theorem}

The proof of Theorem~\ref{error_bound} will be given in Section~\ref{proof}.

Note that convergence in $H^{2\kappa+r}(\mathcal{M})$ as $h_\mathcal{A}\to0$ follows from \eqref{Hb} only when $0\le r<s-2\kappa-d$, thus under
a stricter assumption than \eqref{As},
$$
s>2\kappa+d,$$
and the discrete convergence follows from \eqref{infb} under the assumption that
$$
s>\max\{2\kappa,\tfrac{d}{2}+1\}+d.$$
In particular, in the discrete case in order for the power of $h_\mathcal{A}$ to be positive, we need
$$
s>2\kappa+d+r_0=\max\{2\kappa+d,d+\lfloor \tfrac{d}{2}\rfloor+1\},$$
which is equivalent to $s>\max\{2\kappa,\tfrac{d}{2}+1\}+d$ since $s$ is integer.

For example, for $\kappa=1$ and $d\le 3$ we obtain the Sobolev norm convergence with the order
$$
\|u-\tilde u\|_{H^{2+r}(\mathcal{M})}=\mathcal{O}(h_\mathcal{A}^{s-d-2-r})\quad\text{if}\quad s\ge d+3,\quad 0\le r\le s-d-3,$$
and the discrete norm convergence
$$
\|u|_X-{\hat u}\|_{\infty}=\mathcal{O}(h_\mathcal{A}^{s-d-2})\quad\text{if}\quad s\ge d+3.$$

In fact, local interpolation functions $\tilde u_\ell$ of the proof in Section~\ref{proof} may be a good suggestion for the evaluation
of the approximate solution and derived quantities such as gradient.  Alternatively, for the same purpose
we may use the function $\tilde u$ of the proof (see Section~\ref{proof}), which however requires an implementation of the partition of unity $\Gamma$,
or the kernel interpolant to $\hat u$, which requires solving a non-sparse linear system with the matrix $K|_X$ and evaluation of
a kernel sum with $n$ terms.

\section{Proof of Theorem~\ref{error_bound}}\label{proof}
Recall that we assume that $\mu_\mathcal{A}\le \mu$,  $\lambda_{s+\kappa}(\mathcal{A},X)\le\lambda$,
$c_\mathcal{A}\le c$, $\chi_\mathcal{A}\le\chi$, $\nu_{X,\mathcal{A}}\le\nu$,
$q_{X,\mathcal{A}}\le q$, $\eta_{s+\kappa}(\mathcal{A})\le\eta$. Then also $\lambda_{r}(\mathcal{A},X)\le\lambda$ for all $r\le s+\kappa$, in particular for $r=s$ and $r=s-\kappa$.
The constants denoted by $C,C_1,C_2,\ldots$ in
this section will be independent of $f,u,h_\mathcal{A},\delta_{X,\mathcal{A}}$ and the Lipschitz cover parameters  $N_\mathcal{A},M_\mathcal{A},\tau_\mathcal{A}$.
They in general depend on $\mathcal{M},L,K$, the atlas
$\hat\mathcal{A}$ and the partition of unity $\hat\Gamma$ that we use to fix the Sobolev norms, as well as on
$\mu,\lambda,%
c,\chi,\nu,q,\eta$.
We assume without loss of generality that $h_\mathcal{A}\le1$. Then also $\delta_{X,\mathcal{A}}\le1$.

By \eqref{lde3} and \Aref{A1}--\Aref{A3},
$$
\|W_{\ell} u|_{X_\ell} - Lu|_{X_\ell}\|_\infty\le C\mathcal{E}_{s}(U^\varphi_\ell)h_\mathcal{A}^{s-2\kappa-d/2}\|u\circ\varphi_\ell^{-1}\|_{H^s(U^\varphi_\ell)},
\quad \ell=1,\ldots,m.$$
Since, in view of \eqref{ods2},
$$
\|W\!M\hat u-F\|_2^2\le\|W\!Mu|_X-F\|_2^2%
=\sum_{\ell=1}^{m}\|W_{\ell} u|_{X_\ell} - Lu|_{X_\ell}\|_2^2,$$
we obtain by Lemma~\ref{admis},
\begin{equation}\label{eps}
\varepsilon_1:=\|W\!M\hat u-F\|_2\le C\mathcal{E}_{s}(\mathcal{A})h_\mathcal{A}^{s-2\kappa-d/2}\|u\|_{H^s(\mathcal{M})}.
\end{equation}

We will construct a function $\tilde u\in H^{s+\kappa}(\mathcal{M})$  such that
$\tilde u|_{X}=\hat u$, and estimate its distance to $u$ in Sobolev spaces of lower order as in \eqref{Hb}. We define local interpolants $\tilde u_\ell$
on the sets $X_\ell$ and then combine them together with the help of a partition of unity.

For each $\ell=1,\ldots,m$, let
$$
\tilde u_\ell(x)=\sum_{j\in J_\ell}c_j K(x,x_j),$$
where the coefficients $c_j$ are uniquely determined by the interpolation conditions
\begin{equation}\label{uellint}
\tilde u_\ell|_{X_\ell}=\hat u|_{J_\ell}.
\end{equation}
By \eqref{Wl},
$L\tilde u_\ell|_{X_\ell}=W_\ell \tilde u_\ell|_{X_\ell}$. Hence
\begin{equation}\label{giell}
L \tilde u_\ell|_{X_\ell}= W_\ell \hat u|_{J_\ell}.
\end{equation}
Moreover,  by Lemma~\ref{smkerL}, $\tilde u_\ell\in H^{s+2\kappa}(\mathcal{M})$.
By \eqref{reg} and Lemma~\ref{KL},
\begin{equation}\label{uelln}
\|\tilde u_\ell\|_{H^{s+\kappa}(\mathcal{M})}\le C\|L\tilde u_\ell\|_{K_L}.
\end{equation}
It follows from  \eqref{giell} and \eqref{ods2}
 that
\begin{equation}\label{eps4}
\sum_{\ell=1}^{m}\|L \tilde u_\ell|_{X_\ell}-Lu|_{X_\ell}\|_2^2=
\sum_{\ell=1}^{m}\|W_\ell \hat u|_{J_\ell}-Lu|_{X_\ell}\|_2^2=\|W\!M\hat u-F\|_2^2=\varepsilon_1^2.
\end{equation}

Our next goal is to estimate the  $K_L$-norm of $L\tilde u_\ell$. Note that due to \eqref{srestr},
$\|L\tilde u_\ell\|_{K_L}=\|L\tilde u_\ell\|_{K_L,U_\ell}$ since $X_\ell\subset U_\ell$. Let
$$
u_\ell\in\spann\{K(\cdot,x_j):\,j\in J_\ell\}$$
be the kernel sum determined by the interpolation conditions
$$
L u_\ell|_{X_\ell}=L u|_{X_\ell}.$$ %
Then $L u_\ell$ is the $K_L$-kernel interpolant of $f=Lu$.
By Lemma~\ref{KL}, $K_L$ is a reproducing kernel for $H^{s-\kappa}(\mathcal{M})$,
and by Lemma~\ref{SobKm} and \Aref{A2}, $H_{K_L}(U_\ell)=H^{s-\kappa}(U_\ell^\varphi)$. Hence,  by \eqref{srestr} and the minimum norm property
\eqref{minnorm}, since $X_\ell\subset U_\ell$,
$$
\|L u_\ell\|_{K_L}=\|L u_\ell\|_{K_L,{U_\ell}}\le \|f\|_{K_L,{U_\ell}},$$
and by Lemma~\ref{SobKm},
$$
\|f\|_{K_L,{U_\ell}}\le C\mathcal{E}_{s-\kappa}(U^\varphi_\ell)\|f \circ\varphi_\ell^{-1}\|_{H^{s-\kappa}(U_\ell^\varphi)}.$$
It follows by Lemma~\ref{admis} that
\begin{equation}\label{b1}
\sum_{\ell=1}^m\|L u_\ell\|_{K_L}^2\le C\mathcal{E}^2_{s-\kappa}(\mathcal{A})\|f\|_{H^{s-\kappa}(\mathcal{M})}^2.
\end{equation}
Consider the functions $v_\ell=L\tilde u_\ell - Lu_\ell$, $\ell=1,\ldots,m$, that satisfy
$$
\sum_{\ell=1}^{m}\|v_\ell|_{X_\ell}\|_2^2\le\varepsilon_1^2,$$
thanks to \eqref{eps4}. Let
$$
\psi_\ell=\sum_{j\in J_\ell}v_\ell(x_j)\,\Psi_{j,\delta}^\ell,
\qquad \Psi_{j,\delta}^\ell(x):=
\begin{cases}\Psi\big(\tfrac{\varphi_\ell(x)-\varphi_\ell(x_j)}{\delta}\big),& x\in U_\ell,\\
0, & x\in\mathcal{M}\setminus U_\ell,
\end{cases}$$
where $\Psi(x)=\psi(\|x\|_2^2)$ is a radially symmetric bump function in $C^\infty(\mathbb{R}^d)$ supported on the unit ball,
as in Example~\ref{pume},
and $\delta=\delta_{X,\mathcal{A}}\le1$. %
Then the supports of the functions
$\Psi_{j,\delta}^\ell$, $j\in  J_\ell$, are pairwise disjoint and hence
$$
\psi_\ell|_{X_\ell}=v_\ell|_{X_\ell}$$
and
$$
|\psi_\ell\circ \varphi_\ell^{-1}|_{H^r(U_\ell^\varphi)}=\delta^{-r+d/2}|\Psi|_{H^r(\mathbb{R}^d)}
\|v_\ell|_{X_\ell}\|_2,\qquad r\in\mathbb{Z}_+,$$
which implies (since in particular $\delta\le1$),
\begin{align*}
\sum_{\ell=1}^{m}\|\psi_\ell\circ \varphi_\ell^{-1}\|_{H^{s-\kappa}(U_\ell^\varphi)}^2
\le \varepsilon_1^2\,\sum_{r=0}^{s-\kappa}|\Psi|_{H^{r}(\mathbb{R}^d)}^2\,\delta^{-2r+d}
\le \varepsilon_1^2\,\|\Psi\|_{H^{s-\kappa}(\mathbb{R}^d)}^2\,\delta^{-2(s-\kappa)+d}.
\end{align*}
By \eqref{srestr}, the minimum norm property \eqref{minnorm} of $v_\ell$ as the $K_L$-kernel interpolant of
$\psi_\ell\in H_{K_L}(U_\ell)$,  and by Lemma~\ref{SobKm},
$$
\|v_\ell\|_{K_L}=\|v_\ell\|_{K_L,U_\ell}\le \|\psi_\ell\|_{K_L,U_\ell}
\le C\mathcal{E}_{s-\kappa}(U^\varphi_\ell)\|\psi_\ell\circ \varphi_\ell^{-1}\|_{H^{s-\kappa}(U_\ell^\varphi)}.$$
Since $\delta^{-1}\le qh_\mathcal{A}^{-1}$,
$$ %
\sum_{\ell=1}^{m}\|v_\ell\|_{K_L}^2
\le C\mathcal{E}^2_{s-\kappa}(\mathcal{A})\varepsilon_1^2h_\mathcal{A}^{-2(s-\kappa)+d}.
$$ %
Combining this with \eqref{b1}, we obtain for $L\tilde u_\ell=Lu_\ell+v_\ell,$
\begin{equation}\label{b3}
\Big(\sum_{\ell=1}^{m}\|L \tilde u_\ell\|_{K_L}^2\Big)^{1/2}
\le \mathcal{E}_{s-\kappa}(\mathcal{A})\big(C_1\|f\|_{H^{s-\kappa}(\mathcal{M})}+C_2\varepsilon_1h_\mathcal{A}^{-s+\kappa+d/2} \big).
\end{equation}

We set
$$
\tilde u :=\sum_{\ell=1}^m \gamma_\ell \tilde u_\ell,$$
where $\{\gamma_\ell\}_{\ell=1}^m$ is the smooth partition of unity \eqref{pum} subordinate to $\mathcal{A}$.
Note that $\tilde u\in H^{s+2\kappa}(\mathcal{M})$, which follows from the same property of $\tilde u_\ell$.
It follows from \eqref{uellint} and \eqref{pum} that
\begin{equation}\label{uint}
\tilde u|_{X} = \hat u.
\end{equation}

We now use \eqref{b3} and the properties of the partition of unity
in order to obtain an estimate for $\|L\tilde u\|_{H^{s-\kappa}(\mathcal{M})}$.
By Lemma~\ref{admis},
$$
\|L\tilde u\|_{H^{s-\kappa}(\mathcal{M})}^2
\le C\sum_{\ell=1}^m\|L\tilde u\circ \varphi_\ell^{-1}\|_{H^{s-\kappa}(U_\ell^\varphi)}^2.$$
For each $\ell=1,\ldots,m$,
$$
\|L\tilde u\circ \varphi_\ell^{-1}\|_{H^{s-\kappa}(U_\ell^\varphi)}
\le \|L\tilde u_\ell\circ \varphi_\ell^{-1}\|_{H^{s-\kappa}(U_\ell^\varphi)}+
\|L(\tilde u-\tilde u_\ell)\circ \varphi_\ell^{-1}\|_{H^{s-\kappa}(U_\ell^\varphi)}.$$
By Lemma~\ref{SobKm} and \eqref{srestr},
$$
\|L\tilde u_\ell\circ \varphi_\ell^{-1}\|_{H^{s-\kappa}(U_\ell^\varphi)}\le C \|L\tilde u_\ell\|_{K_L},$$
and we will be able to estimate the terms of this type by applying \eqref{b3}.
It follows from \eqref{aalpha} and the Leibniz rule that
$$
\|L(\tilde u-\tilde u_\ell)\circ \varphi_\ell^{-1}\|_{H^{s-\kappa}(U_\ell^\varphi)}
\le C \|(\tilde u-\tilde u_\ell)\circ \varphi_\ell^{-1}\|_{H^{s+\kappa}(U_\ell^\varphi)}.$$
By the definition of $\tilde u$,
$$
\tilde u-\tilde u_\ell
=\sum_{p\in I_\ell\setminus\{\ell\}}
\gamma_p(\tilde u_p -\tilde u_{\ell})
\quad\text{in $U_\ell$},\qquad I_\ell=\{i: U_i\cap U_\ell\ne\emptyset\}.$$
Applying the Leibniz rule again, we obtain
for  $|\alpha|\le s+\kappa$,
$$
\partial^\alpha\big( (\tilde u - \tilde u_{\ell})\circ\varphi_\ell^{-1}\big)
=\sum_{p\in I_\ell\setminus\{\ell\}}
\sum_{\beta\le\alpha}{\alpha\choose\beta}
\partial^{\alpha-\beta}(\gamma_p\circ \varphi_\ell^{-1})
\,\partial^{\beta}\big((\tilde u_p -\tilde u_{\ell})\circ \varphi_\ell^{-1}\big)\quad\text{in $U_\ell^\varphi$}.$$
By \eqref{eta},
$$
\|\partial^{\alpha-\beta}(\gamma_p\circ \varphi_\ell^{-1})\|_{L^\infty(U_\ell^\varphi)}
=h_\ell^{|\beta-\alpha|}\|\partial^{\alpha-\beta}g_\ell\|_{L^\infty(\mathbb{R}^d)}
\le h_\ell^{|\beta-\alpha|}\eta_{s+\kappa}(\mathcal{A})\le\eta h_\ell^{|\beta-\alpha|}.$$
Hence
$$
\|\partial^\alpha\big( (\tilde u - \tilde u_{\ell})\circ\varphi_\ell^{-1}\big)\|_{L^2(U_\ell^\varphi)}
\le C\sum_{p\in I_\ell\setminus\{\ell\}}
\sum_{\beta\le\alpha}{\alpha\choose\beta} h_\ell^{|\beta-\alpha|}
\|\partial^{\beta}\big((\tilde u_p-\tilde u_\ell)\circ \varphi_\ell^{-1}\big)\|_{L^2(U_\ell^\varphi)}.$$
Let $p\in I_\ell\setminus\{\ell\}$. By \eqref{uellint},
$$
\tilde u_p|_{X_p\cap X_\ell}={\hat u}|_{J_p\cap J_\ell}= \tilde u_\ell|_{X_p\cap X_\ell}.$$
Hence, it follows from \eqref{sam2} that for $|\beta|<s+\kappa$,
$$
\|\partial^{\beta}\big((\tilde u_p-\tilde u_\ell)\circ \varphi_\ell^{-1}\big)\|_{L^2(U_\ell^\varphi)}
\le C h_\ell^{s+\kappa-|\beta|}
|(\tilde u_p-\tilde u_\ell)\circ\varphi_\ell^{-1}|_{H^{s+\kappa}(U_\ell^\varphi)}.$$
By Lemma~\ref{admis},
$$
|(\tilde u_p-\tilde u_\ell)\circ\varphi_\ell^{-1}|_{H^{s+\kappa}(U_\ell^\varphi)}
\le C\|\tilde u_p-\tilde u_\ell\|_{H^{s+\kappa}(\mathcal{M})}.$$
By \eqref{uelln},
$$
\|\tilde u_p-\tilde u_\ell\|_{H^{s+\kappa}(\mathcal{M})}\le
C(\|L\tilde u_p\|_{K_L}+\|L\tilde u_\ell\|_{K_L}).$$
By combining these estimates, since $h_\mathcal{A}\le 1$
and hence $h_\ell^{s+\kappa-|\alpha|}\le1$, we obtain
\begin{equation}\label{utmutl}
\|(\tilde u-\tilde u_\ell)\circ \varphi_\ell^{-1}\|_{H^{s+\kappa}(U_\ell^\varphi)}
\le C\sum_{p\in I_\ell\setminus\{\ell\}}
(\|L\tilde u_p\|_{K_L}+\|L\tilde u_\ell\|_{K_L}).
\end{equation}
This implies
$$
\sum_{\ell=1}^m
\|L(\tilde u-\tilde u_\ell)\circ \varphi_\ell^{-1}\|_{H^{s-\kappa}(U_\ell^\varphi)}^2
\le C
\sum_{\ell=1}^m\|L\tilde u_\ell\|_{K_L}^2.$$
Summarizing the estimates, we get
$$
\|L\tilde u\|_{H^{s-\kappa}(\mathcal{M})}
\le C
\Big(\sum_{\ell=1}^m \|L\tilde u_\ell\|_{K_L}^2\Big)^{1/2},$$
and hence by \eqref{b3},
\begin{equation}\label{b4}
\|L\tilde u\|_{H^{s-\kappa}(\mathcal{M})}
\le C_1\|f\|_{H^{s-\kappa}(\mathcal{M})}+C_2\varepsilon_1h_\mathcal{A}^{-s+\kappa+d/2}.
\end{equation}

Finally, we will estimate Sobolev norms of $L u-L \tilde u$ by applying Lemma~\ref{sampling} locally, which in turn will give us estimates for $u-\tilde u$ thanks to
\eqref{reg}.
Since $X=\cup_{\ell=1}^mX_\ell$, it follows from \eqref{eps4} that
\begin{equation}\label{eps12}
\|L \tilde u|_{X}-L u|_{X}\|_2\le\Big(\sum_{\ell=1}^m\|L \tilde u|_{X_\ell}-L u|_{X_\ell}\|_2^2\Big)^{1/2}
\le \varepsilon_1 + \varepsilon_2,
\end{equation}
where
$$
\varepsilon_2:=\Big(\sum_{\ell=1}^m\|L \tilde u|_{X_\ell}-L \tilde u_\ell|_{X_\ell}\|_2^2\Big)^{1/2}.$$
By \eqref{giell} and \eqref{uint},
$$
L \tilde u_\ell|_{X_\ell}= W_{\ell} \hat u|_{J_\ell}=W_{\ell}\tilde u|_{X_\ell}.$$
By construction, $\tilde u_\ell$ is the kernel interpolant to $\tilde u$ on $X_\ell$, compare \eqref{uellint} and
\eqref{uint}. Hence by \eqref{lde1},
$$
\|L \tilde u|_{X_\ell}- L \tilde u_\ell|_{X_\ell}\|_\infty
 =\|L \tilde u|_{X_\ell}- W_{\ell}\tilde u|_{X_\ell}\|_\infty\le
Ch_\ell^{s-\kappa-d/2}
|(\tilde u-\tilde u_\ell)\circ\varphi_\ell^{-1}|_{H^{s+\kappa}(U_\ell^\varphi)}.$$
In view of \eqref{utmutl},
\begin{align*}
\varepsilon_2^2
\le \nu \sum_{\ell=1}^m\|L \tilde u|_{X_\ell}-L \tilde u_\ell|_{X_\ell}\|_\infty^2
\le C h_\mathcal{A}^{2s-2\kappa-d}
\sum_{\ell=1}^m\|L\tilde u_\ell\|_{K_L}^2.
\end{align*}
and by \eqref{b3} we get
\begin{equation}\label{eps2}
\varepsilon_2\le \mathcal{E}_{s-\kappa}(\mathcal{A})\big(
C_1h_\mathcal{A}^{s-\kappa-d/2}\|Lu\|_{H^{s-\kappa}(\mathcal{M})}+C_2\varepsilon_1\big).
\end{equation}
Let $v=L \tilde u-Lu$. Then $v\in H^{s-\kappa}(\mathcal{M})$ and
$$
\varepsilon_3:=\|v\|_{H^{s-\kappa}(\mathcal{M})}\le \|L\tilde u\|_{H^{s-\kappa}(\mathcal{M})}+\|Lu\|_{H^{s-\kappa}(\mathcal{M})}.$$
Hence by \eqref{b4}, %
\begin{equation}\label{eps3}
\varepsilon_3\le \mathcal{E}_{s-\kappa}(\mathcal{A})\big(C_1\|Lu\|_{H^{s-\kappa}(\mathcal{M})}+C_2\varepsilon_1h_\mathcal{A}^{-s+\kappa+d/2}\big),
\end{equation}
and by \eqref{eps12},
$$
\|v|_{X}\|_2\le \varepsilon_1+\varepsilon_2.$$
By Lemma~\ref{sampling}, for any integer $r$ with $0\le r< s-\kappa$, and each $\ell=1,\ldots,m$,
$$
|v\circ \varphi_\ell^{-1}|_{H^r(U_\ell^\varphi)}
\le Ch_\ell^{-r}
\big(h_\ell^{d/2}\|v|_{X_\ell}\|_\infty
+h_\ell^{s-\kappa}|v\circ \varphi_\ell^{-1}|_{H^{s-\kappa}(U_\ell^\varphi)}\big).$$
Hence by Lemma~\ref{admis},
 \begin{align*}
\|v\|_{H^r(\mathcal{M})}^2&\le C\sum_{\ell=1}^m\|v\circ \varphi_\ell^{-1}\|_{H^r(U_\ell^\varphi)}^2\\
&\le C
\Big(\sum_{\ell=1}^m h_\ell^{d-2r}\|v|_{X_\ell}\|_\infty^2
+h_\mathcal{A}^{2(s-\kappa-r)}\sum_{\ell=1}^m |v\circ \varphi_\ell^{-1}|_{H^{s-\kappa}(U_\ell^\varphi)}^2\Big)\\
&\le C_1h_\mathcal{A}^{d-2r} \|v|_{X}\|_2^2
+C_2h_\mathcal{A}^{2(s-\kappa-r)}\|v\|_{H^{s-\kappa}(\mathcal{M})}^2.
\end{align*}

It follows in view of \eqref{reg} that for any $0\le r< s-\kappa$,
\begin{equation}\label{eps123}
\|u-\tilde u\|_{H^{r+2\kappa}(\mathcal{M})}\le
C_1h_\mathcal{A}^{d/2-r} (\varepsilon_1+\varepsilon_2)
+C_2h_\mathcal{A}^{s-\kappa-r}\varepsilon_3.
\end{equation}
By  \eqref{eps}, \eqref{eps2} and \eqref{eps3},
\begin{align*}
\varepsilon_1+\varepsilon_2&\le
C_1\mathcal{E}_{s}(\mathcal{A})h_\mathcal{A}^{s-2\kappa-d/2}\|u\|_{H^s(\mathcal{M})}
+C_2\mathcal{E}_{s-\kappa}(\mathcal{A})h_\mathcal{A}^{s-\kappa-d/2}\|Lu\|_{H^{s-\kappa}(\mathcal{M})},\\
\varepsilon_3&\le \mathcal{E}_{s-\kappa}(\mathcal{A})\big(C_1\|Lu\|_{H^{s-\kappa}(\mathcal{M})}+C_2\mathcal{E}_{s}(\mathcal{A})h_\mathcal{A}^{-\kappa}\|u\|_{H^s(\mathcal{M})}\big).
\end{align*}
Hence by \eqref{eps123},
$$
\|u-\tilde u\|_{H^{r+2\kappa}(\mathcal{M})}\le
\mathcal{E}_{s-\kappa}(\mathcal{A})\big(C_1\mathcal{E}_s(\mathcal{A})
h_\mathcal{A}^{s-2\kappa-r}\|u\|_{H^s(\mathcal{M})}
+C_2h_\mathcal{A}^{s-\kappa-r}\|Lu\|_{H^{s-\kappa}(\mathcal{M})}\big).$$
By using \eqref{reg} again, we arrive at
\begin{equation}\label{errorE}
\|u-\tilde u\|_{H^{r+2\kappa}(\mathcal{M})}\le
\mathcal{E}_{s-\kappa}(\mathcal{A})
\big(C_1\mathcal{E}_s(\mathcal{A})
h_\mathcal{A}^{s-2\kappa-r}\|u\|_{H^s(\mathcal{M})}
+C_2h_\mathcal{A}^{s-\kappa-r}\|u\|_{H^{s+\kappa}(\mathcal{M})}\big),
\end{equation}
which implies  \eqref{Hb} in view of Lemma~\ref{extce}.

To show  \eqref{infb} we use Sobolev embedding of $H^{2\kappa+r}(\mathcal{M})$ into $C(\mathcal{M})$ as soon as $2\kappa+r>d/2$.
It is easy to see that $r_0$ in \eqref{infb} is the smallest integer $r$ satisfying this inequality.

\bibliographystyle{abbrv} %
                                  %

redpajama_set_name: RedPajamaArXiv


Do you love reading, meeting new people and hanging out with friends? You can combine all three of these things when you join our Bookworms Book Club! This book club welcomes all children between 3rd and 6th grade or up to age 12.
We meet every 4th Thursday of the month at 4:30pm in the Harnett County Public Library's meeting room. Join us for snacks, friends and fun activities.
Read the series that's sold more than 2 million copies--if you dare! Warning: this description has not been authorized by Pseudonymous Bosch. As much as he'd love to sing the praises of his book (he is very vain), he wouldn't want you to hear about his brave 11-year old heroes, Cass and Max-Ernest. Or about how a mysterious box of vials, the Symphony of Smells, sends them on the trail of a magician who has vanished under strange (and stinky) circumstances. And he certainly wouldn't want you to know about the hair-raising adventures that follow and the nefarious villains they face. You see, not only is the name of this book secret, the story inside is, too. For it concerns a secret. A Big Secret.


Per grew up in South Minneapolis attending St. Peder's Lutheran Church. Every Summer he would head to Sugar Creek Bible Camp to participate in a new adventure every year. Later, after finishing his college education at Augustana College (now University) in Sioux Falls, SD, he returned to work at Sugar Creek Bible Camp as a High School counselor and then later as one of the Assistant Program Directors. His first youth ministry job was in Delano, MN at Light of Christ Lutheran Church before he found his way to Farmington Lutheran in February of 2015. During his time at FLC, he has enjoyed walking alongside the families of the Confirmation students and growing the High School ministry into a program where our youth feel cared for.
Shannon grew up in Plymouth and attended three different churches of various faiths. Shannon received her BA in Theatre Performance from Augsburg College (now University) in 2011. After graduating, she worked with a handful of organizations with youth of various ages in different theatrical, teaching, and administrative roles. Most recently, she was the Community Engagement Program Assistant and Teaching Artist for Children's Theatre Company's Neighborhood Bridges Program. Shannon has called FLC her church home since 2013 when her and her husband, Nick, decided to get married here. They have since added two little boys to their family, Henrik (3 years) and Jameson (3 months). She loves being a part of the youth program here at FLC and supporting the youth and their families as they grow and learn together.
Farmington Lutheran Church offers scholarship opportunities to those who are planning to attend a Lutheran college. Applications will be excepted until April 30, 2019.
Scholarships are granted to students attending freshman through senior year.
FLC's Endowment committee offers scholarships to those who attend a Lutheran college, while working on an undergrad degree.
Question? Fill out the form below and we will get in touch!


Message Keeper: Birdie All The Best!
Made this mini card for a colleague who is leaving tomorrow. I made a 3"x3" mini card because I am going to insert the card into a transparent gift pouch together with the farewell gift.
I adopt the sketch from Tuesday Morning Sketches #240 with twist 3 B's as shown above.
A bird (that is one B's) is stamped on a white card stock and colored using Copic markers. It is distressed with ink to tone down the whiteness as well to mimic the sky. Then a speech bubble is stamped on a white card stock with the sentiment. It is trimmed and distressed as well. The bird is attached onto a butterfly (That is another B's but afraid it cannot be seen well. So I have some extra.) pattern paper follow by the speech bubble using pop dots.
A black ribbon (that is another B's) is tied to the botton of the butterfly pattern paper. An button (that is another B's) with yellow twine as bow (another B's) is attached onto the black ribbon as shown in the sketch. It is later attached onto a yellow card base using pop dots.
Very cute card. Nice use of our sketch. Thank you for joining us this week at TMS.
What a cute card! Love the tweet little bird! I am sure your colleague will love it! Thanks for joining us at Tuesday Morning Sketches!
How cute.i like the large caption cloud! Thank you for using our sketch at Tuesday Morning Sketches.
Thanks for joining us at Crafty Ribbons Challenges!


In a recent post we looked at an introduction to the study of covenants, or what is commonly called Covenant Theology. In that post we mentioned that views and opinions on Covenant Theology have been widespread and numerous throughout church history, most significantly in more recent times. In this post, we'll look at a little more background to its historical development through its key proponents to establish the historical validity of covenant theology as a biblical expression of redemptive history to help better wrap our minds around the concept of covenants.
In providing a historical survey, my hope is that it will be shown how much of the views that dominate present evangelicalism are in fact a departure from the more historical views held by the church throughout her history and are the development of modern inventions, influenced by either liberalism or dispensationalism, that have become embraced as the excepted norm rather than questioned as to their validity.
In beginning, perhaps it would be an unfair angle to take by saying the doctrinal development of covenant theology begins in Scripture as God unfolds His plan of redemption to man via covenants, as I'm sure all alternate views would claim a similar origin. Avoiding that argument, as well as any incipient thought on covenants that may have been present in the early Church Fathers (though certainly Augustine would be an early champion), our focus brings us squarely to the time of the Protestant Reformation of the 16th Century. Additionally, as we'll see in a future post on the definition of covenants, the concept of covenants was alive and well in the Ancient Near East and was employed between numerous secular states and kings. However, as it relates to God's plan of redemption unfolded throughout Scripture, covenant theology, as with many other mature doctrinal positions, owes its development to the Reformation.
While re-hashing this monumental event is certainly outside the scope of a brief history on covenants, it should be noted that through the protest from Catholicism, the early Protestant Church began the seeds of developing a robust understanding of the covenants to help define their views on Church membership, baptism, the Lord's Supper, separation between Church and State, etc. Principally, our overview of covenant theology begins in this period with the Father of the Reformation himself, Martin Luther.
In Martin Luther's objections against the Catholic Church, we begin to see the formulation of clear distinctions between law and gospel, which really lay at the heart of covenant theology. In formulating the early Reformers understanding of justification by faith alone, as opposed to the Catholic notion of justification by works, Luther realized that undue attention had been given to the law/gospel discussion. R. Scott Clark summarizes Luther's thought, "The law demands perfect obedience, and the gospel announces Christ's perfect obedience to that law, his death and his resurrection for his people." This statement can only be further defined and defended by understanding the biblical covenants, specifically the relationship between the Old and the New or the Covenant of Works and Covenant of Grace (to be further defined later).
While there were other lesser known figures that were advancing some of these early thoughts on covenants, such as Swiss Reformed theologian Johannes Oecolampadius (1482–1531) and Heinrich Bullinger (1504–75), it was through John Calvin (1509–1564) that covenant theology found its fullest treatment. Through the publication of his Institutes of Christian Religion and to a lesser extent his commentaries on the Bible, Calvin laid the groundwork for the development of covenant theology, albeit from a paedobaptist perspective; more on that in a moment.
These covenant terms may be unfamiliar for now, but as we progress on our journey of unfolding the biblical covenants, we'll take the time to define and defend each.
As covenant thought advanced into the 17th Century, with the height of Puritanism underway, we find expressions of covenant theology in many of their writings. This is especially true in such prominent Puritans as William Perkins and William Ames. Perhaps the more well-known of the two is Ames through his work, The Marrow of Divinity. As an aside, this publication by Ames was the "basic instruction manual at Harvard" during the colonial period and was "long considered to be the best summary of Calvinist theology." Other notable Puritan writers who published expressions on the covenants were John Ball (1585-1640), Peter Bulkeley (1583-1659), Thomas Blake (1597-1657), Samuel Petto (1642-1711), Francis Turretin (1623-1687), Johannes Cocceius (1609–1669), and perhaps the author of the most significant work on paedobaptist covenant theology, Herman Witsius (1636-1708).
Prominent among the Puritans was John Owen, whose commentary on Hebrews contains a developed, mature thinking on the covenants. While his exposition on Hebrews 8 has garnered much attention recently, his efforts to address the covenants throughout his commentary should not go unnoticed, particularly when he comments on Hebrews 6 and 7 as well as several of the articles in the commentary's "Preliminary Exercitations". As we'll see later, Owen's thoughts on the covenants were more closely aligned with the Baptist understanding of continuity/discontinuity between Old and New Covenants, though he remained a paedobaptist; perhaps lending itself to the notion that had the Puritans more time to develop their thoughts and distance themselves from Rome, the more likely their departure from paedobaptism. With Owen, we arrive at those who were instrumental in the formulation of the Westminster Confession of Faith, published in 1646 which served to solidify the paedobaptist understanding of covenants.
To this point in our historical survey, much of the focus has been on the development of covenant theology from the paedobaptist or Presbyterian perspective. This is understandable given the climate of Roman Catholicism from which the Reformers had split, though clearly some influence remained. Unfortunately, until some more recent publications, much of the Baptist influence during this same time period has gone unnoticed and this has unfairly served to create a migration away from covenant theology in more modern times, especially among Baptists. While not as well-known because of the lack of preserved historical documents and publications, these early Baptists formulated well their own understanding of covenant theology and were by no means considered inferior theologians. Some of the more prominent Baptists who wrote concerning covenants were: John Spilsbury (1598-1668), Henry Lawrence (1600-1664), Thomas Patient (?-1666), John Bunyan (1628-1688), Edward Hutchinson (?), Nehemiah Coxe (?-1688), Thomas Grantham (?-16920, and Benjamin Keach (1640-1704).
Among these early Baptist theologians who were expressing their own version of Covenant Theology specifically written to counter the notion of infant baptism were Spilsbury and most notably Nehemiah Coxe. Not only did Coxe formulate an understanding of the Abrahamic Covenant that was defining for the historical Baptist expression of covenants and defense of believer's baptism, but he was likely the chief architect of the 1689 Second London Baptist Confession. The parallels of this confession with the Westminster Confession have been well documented, but it is in their differences among the chapters that articulate the views of covenants where the 1689 shines brightest for Baptist history.
With this brief overview of the Baptists, this brings our timeline up through the 17th and into the 18th century where much of the diversity in covenant thought begins to rear its head. For now, it should be clearly stated that from the early Reformation into the late 18th century, covenant theology was the dominant view and expression among Christian thought. Though distinctions remained between paedobaptists and Baptists, the development of additional views was largely non-existent and covenant theology remained the dominant explanation for God's unfolding plan of redemption. However, this would not remain the case going into the subsequent century. It's on this point that we will pick up our overview of covenant history next time.
Understanding that the roots of covenant theology spring from the mountains of the Reformation may not appear significant to everyone. To reiterate, this is not to say that covenant theology had no place whatsoever in the minds of the early church fathers through the medieval period, only that through the Reformation these thoughts became more developed and organized. Like a daisy (or tulip) from a dung pile, covenant theology's mature development from medieval Catholicism cannot be understated. The issues over which the Reformers were protesting had at their foundation a proper understanding and application of covenant theology. From there, the Baptists "protested" further in defining their own understanding of Church membership and who could/could not be baptized. For many, this portion of the discussion that includes Baptists has gone unnoticed and consequently many have come to view covenant theology as equivalent to paedobaptism. Perhaps this explains the devolution in thought from early Reformed Baptist Covenant Theology to many of the variations of views that have migrated over and over throughout the last two centuries. Lord willing, we will examine some of these next time.
 It should be noted that Turretin was neither British, nor Puritan in the classic sense of the word.
 It should be noted that Owen wrote a paper on the defense of infant baptism but never published it during his lifetime.
In keeping with our discussion of the new covenant, as we take a parenthetical break from our study of end times, I included the following chapter from the 1689 Baptist Confession of Faith on the mediation of Christ over the New Covenant, which He inaugurated with His blood (Matt. 26:28). I hope to follow up with a post discussing this further, particularly several passages from Hebrews.


Hello fellow tarot lovers and mystics. It's Thursday, so we're back with another round of #tarotthursdaythree. I'm going to keep this intro. short because I think most people who visit my page are getting pretty familiar with this Thursday ritual, and speaking of ritual, our question set this week is all about spells, ritual and the tools we use for crafting or card reading.
If you are unfamiliar with this process, you can read more about it here. And don't forget, not only is this exercise open to all mystical bloggers across all platforms, but I also invite anyone interested to submit their own questions for use on an upcoming Thursday.