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=1$ with the optimal constant $d=3\cdot 2^{n-1}$, see \cite{BS1}. In the early 2000s the Finiteness Principle was proved by C.~Fefferman for all $k$ and $n$ for regular moduli of continuity $\omega$ (i.e., $\omega(1)=1$), see \cite{F1}. The upper bound for | the constant $d$ in the Fefferman proof was reduced later to $d=2^{ k+n \choose k}$ by Bierstone and P. Milman \cite{BM} and independently and by a different method by Shvartsman \cite{S}. The obtained results (and the Finiteness Principle in general) ad |
mit the following reformulation in terms of geometric characteristics of closed unit balls $B_b^{k,\omega}(S)$ of $G_b^{k,\omega}(S)$. Specifically,
let $B_b^{k,\omega}(S;m)\subset B_b^{k,\omega}(S)$, $m\in{\mbf N}$, be the balanced closed convex hull of t | he union of all finite-dimensional balls $B_b^{k,\omega}(S')\subset G_b^{k,\omega}(S')$, $S'\subset S$, ${\rm card}\, S'\le m$.
\begin{Th}\label{teo1.5}
There exist constants $d\in{\mbf N}$ and $c\in (1,\infty)$ such that
\[
B_b^{k,\omega}(S;d)\subset B_ |
b^{k,\omega}(S)\subset c\cdot B_b^{k,\omega}(S;d).
\]
Here for $k=0$, $d=2$ (-\,optimal) and $c=1$, for $n=1$, $d=k+2$ (-\,optimal) and $c$ depends on $k$ only, for $k=1$, $d=3\cdot 2^{n-1}$ (-\,optimal) and $c$ depends on $k$ and $n$ only, and for $k\ge 2 | $, $d=2^{ k+n \choose k}$ and $c=\frac{\tilde c}{\omega(1)}$, where $\tilde c$ depends on $k$ and $n$ only.
\end{Th}
\subsection{Complementability of Spaces ${\mathbf G_b^{k,\omega}(S)}$} We begin with a result describing bounded linear operators on $G_b^ |
{k,\omega}({\mbf R}^n)$. To this end,
for a Banach space $X$ by $C_b^{k,\omega}({\mbf R}^n;X)$ we denote the Banach space of $X$-valued $C^k$ functions on ${\mbf R}^n$ with norm defined similarly to that of Definition \ref{def1} with absolute values replac | ed by norms $\|\cdot\|_X$ in $X$.
Let $\mathcal L\bigl(X_1;X_2\bigr)$ stand for the Banach space of bounded linear operators between Banach spaces $X_1$ and $X_2$ equipped with the operator norm.
\begin{Th}\label{te1.6}
The restriction map to the set $\{\d |
elta_x^0\, :\, x\in {\mbf R}^n\}\subset G_b^{k,\omega}({\mbf R}^n)$ determines an isometric isomorphism between $\mathcal L\bigl(G_b^{k,\omega}({\mbf R}^n);X\bigr)$ and $C_b^{k,\omega}({\mbf R}^n;X)$.
\end{Th}
Let $q_S: C_b^{k,\omega}({\mbf R}^n)\rightar | row C_b^{k,\omega}(S)$ be the quotient map induced by the restriction of functions on ${\mbf R}^n$ to $S$. A right inverse $T\in {\mathcal L}(C_b^{k,\omega}(S); C_b^{k,\omega}({\mbf R}^n))$ for $q_S$ (i.e., $q_S\circ T={\rm id}$) is called a {\em linear ex |
tension operator}. The set of such operators is denoted by $Ext(C_b^{k,\omega}(S); C_b^{k,\omega}({\mbf R}^n))$.
\begin{D}\label{def1.5}
An operator $T\in Ext(C_b^{k,\omega}(S); C_b^{k,\omega}({\mbf R}^n))$ has depth $d\in{\mbf N}$ if for all $x\in{\mbf R} | ^n$ and $f\in C_b^{k,\omega}(S)$,
\begin{equation}\label{equ1.6}
(Tf)(x)=\sum_{i=1}^d \lambda_i^x\cdot f(y_i^x),
\end{equation}
where $y_i^x\in S$ and $\lambda_i^x$ depend only on $x$.
\end{D}
Linear extension operators of finite depth exist. For $k=0$ ( |
the Lipschitz case) the Whitney-Glaeser linear extension operators $C_{b}^{0,\omega}(S)\rightarrow C_{b}^{0,\omega}({\mbf R}^n)$, see \cite{Gl}, have depth $d$ depending on $n$ only and norms bounded by a constant depending on $n$ only.
In the 1990s bound | ed linear extension operators $C_{b}^{1,\omega}(S)\rightarrow C_{b}^{1,\omega}({\mbf R}^n)$ of depth $d$ depending on $n$ only with norms bounded by a constant depending on $n$ only were constructed by Yu.~Brudnyi and Shvartsman \cite{BS2}. Recently bound |
ed linear extensions operators of depth $d$ depending on $k$ and $n$ only
were constructed by Luli \cite{Lu} for all spaces $C_b^{k,\omega}(S)$; their norms are bounded by $\frac{C}{\omega(1)}$, where $C\in (1,\infty)$ is a constant depending on $k$ and $ | n$ only.
(Earlier such extension operators were constructed for finite sets $S$ by C.~Fefferman \cite[Th.\,8]{F2}.)
In the following result we identify $(G_b^{k,\omega}(S))^*$ with $C_b^{k,\omega}(S)$ by means of the isometric isomorphism of Theorem \ref |
{te1.2}.
\begin{Th}\label{teo1.6}
For each $T\in Ext(C_b^{k,\omega}(S); C_b^{k,\omega}({\mbf R}^n))$ of finite depth there exists a bounded linear projection $P:G_b^{k,\omega}({\mbf R}^n)\rightarrow G_b^{k,\omega}(S)$ whose adjoint $P^*=T$.
\end{Th}
\begin | {R}\label{rem1.7}
{\rm
It is easily seen that if $T$ has depth $d$ and is defined by \eqref{equ1.6}, then
\[
p(x):=P(\delta_x^0)=\sum_{i=1}^d \lambda_i^x\cdot \delta_{y_i^x}^0
\quad {\rm for\ all}\quad x\in{\mbf R}^n.
\]
Moreover, $p\in C_b^{k,\omega}({\m |
bf R}^n; G_b^{k,\omega}(S))$ and has norm equal to $\|T\|$ by Theorem \ref{te1.6}.
}
\end{R}
\subsection{Approximation Property}
Recall that a Banach space $X$ is said to have the {\em approximation property}, if,
for every compact set $K\subset X$ and e | very $\varepsilon > 0$, there exists an operator $T : X\to X$ of
finite rank so that $\|Tx-x\|\le\varepsilon$ for every $x\in K$.
Although it is strongly believed that the class of spaces with the approximation property
includes practically all spaces wh |
ich appear naturally in analysis, it is not known yet even
for the space $H^\infty$ of bounded holomorphic functions on the open unit disk. The first example of a space which fails to have the approximation property was
constructed by Enflo \cite{E}. Sinc | e Enflo's work several other examples of such spaces were
constructed, for the references see, e.g., \cite{L}.
A Banach space has the $\lambda$-{\em approximation property}, $1\le\lambda<\infty$, if it has the approximation property with the approximatin |
g
finite rank operators of norm $\le\lambda$. A Banach space is said to have the {\em bounded approximation property}, if it has the $\lambda$-approximation property for some $\lambda$. If $\lambda=1$, then the space is said to have the {\em metric approxi | mation property}.
Every Banach spaces with a basis has the bounded approximation property. Also, it is known that the approximation property does not imply the bounded approximation property, see \cite{FJ}. It was established by Pe\l czy\'nski \cite{P} |
that a separable Banach space has the bounded approximation property if and only if it is isomorphic to a complemented subspace of a separable Banach space with a basis.
Next, for Banach spaces $X,Y$ by ${\mathcal F}(X,Y)\subset {\mathcal L}(X,Y)$ we den | ote the subspace of linear bounded operators of finite rank $X\to Y$.
Let us consider the trace mapping $V$ from the projective tensor product $Y^*\hat{\otimes}_\pi X\to {\mathcal F}(X,Y)^*$ defined by
\[
(Vu)(T)={\rm trace}(Tu),\quad\text{where}\quad u\in |
Y^*\hat{\otimes}_\pi X,\ T\in {\mathcal F}(X,Y),
\]
that is, if $u=\sum_{n=1}^\infty y_n^*\otimes x_n$, then $(Vu)(T)=\sum_{n=1}^\infty y_n^*(Tx_n)$.
It is easy to see that $\|Vu\|\le \|u\|_\pi$. The $\lambda$-bounded approximation property of $X$ is eq | uivalent to the fact that
$\|u\|_\pi\le\lambda\|Vu\|$ for all Banach spaces $Y$. This well-known result (see, e.g., \cite[page 193]{DF}) is essentially due to Grothendieck \cite{G}.
Our result concerning spaces $G_b^{k,\omega}(S)$ reads as follows.
\begi |
n{Th}\label{te1.3}
\begin{enumerate}
\item Spaces $G_b^{k,\omega}({\mbf R}^n)$ have the $\lambda$-approximation property with \penalty-10000 $\displaystyle \lambda=\lambda(k,n,\omega):=1+C\cdot\lim_{t\rightarrow\infty}\,\mbox{$\frac{1}{\omega(t)}$}$,
wher | e $C$ depends on $k$ and $n$ only.
\item All the other spaces $G_b^{k,\omega}(S)$ have the $\lambda$-approximation property with \penalty-10000 $\lambda= C'\cdot\lambda(1,n,\omega)$, where $C'$ is a constant depending on $n$ only, if $k=0,1$, and with $\la |
mbda=\frac{C''\cdot \lambda(k,n,\omega)}{\omega(1)}$, where $C''$ is a constant depending on $k$ and $n$ only, if $k\ge 2$.
\end{enumerate}
\end{Th}
If $\lim_{t\rightarrow\infty}\omega(t)=\infty$, then (1) implies that the corresponding space $G_b^{k,\omeg | a}({\mbf R}^n)$ has the metric approximation property. In case $\lim_{t\rightarrow\infty}\omega(t)<\infty$, one can define the new modulus of continuity $\widetilde\omega$ (cf. properties (i) and (ii) in its definition) by the formula
\[
\widetilde\omega(t |
)=\max\{\omega(t),t\},\quad t\in (0,\infty).
\]
It is easily seen that spaces $C_b^{k,\omega}({\mbf R}^n)$ and $C_b^{k,\widetilde\omega}({\mbf R}^n)$ are isomorphic. Thus $G_b^{k,\omega}({\mbf R}^n)$ is isomorphic to space $G_b^{k,\widetilde\omega}({\mbf | R}^n)$ having the metric approximation property. However, the distortion of this isomorphism depends on $\omega$. So, in general, it is not clear whether $G_b^{k,\omega}({\mbf R}^n)$ itself has the metric approximation property.
In fact, in some cases sp |
aces $G_b^{k,\omega}(S)$ still have the metric approximation property. E.g., by the classical result of Grothendieck \cite[Ch.\,I]{G},
separable dual spaces with the approximation property have the metric approximation property.
The class of such spaces $G | _b^{k,\omega}(S)$ is studied in the next section.
\begin{R}\label{k}
{\rm It is not known, even for the case $k=0$, whether all spaces $C_b^{k,\omega}({\mbf R}^n)$ have the approximation property (for some results in this direction for $k=0$ see, e.g., \ci |
te{K}).
}
\end{R}
At the end of this section we formulate a result describing the structure of operators in ${\mathcal L}(G_b^{k,\omega}({\mbf R}^n);X)$, where $X$ is a separable Banach space with the $\lambda$-approximation property. In particular, it can | be applied to $X=G_b^{k,\omega}(S)$ and $\lambda:=\lambda(S,k,n,\omega)$ the constant of the approximation property for $G_b^{k,\omega}(S)$ of Theorem \ref{te1.3}\,(2).
\begin{Th}\label{teor1.10}
There exists the family of norm one vectors $\{v_j\}_{j\in{ |
\mbf N}}\subset X$ and given $H\in {\mathcal L}(G_b^{k,\omega}({\mbf R}^n);X)$ the family of functions $\{h_j\}_{j\in{\mbf N}}\subset C_b^{k,\omega}({\mbf R}^n)$ of norms $\le 32\cdot\lambda^2\cdot\|H\|$ such that for all $x\in{\mbf R}^n$, $\alpha\in{\mb | f Z}_+^n$, $|\alpha|\le k$,
\begin{equation}\label{equa1.7}
H(\delta_x^\alpha)=\sum_{j=1}^\infty D^\alpha h_j(x)\cdot v_j
\end{equation}
(convergence in $X$).
\end{Th}
\begin{R}
{\rm If $X=G_b^{k,\omega}(S)$ and $H\in\mathcal L(G_b^{k,\omega}({\mbf R}^n);G |
_b^{k,\omega}(S))$ is a projection onto $G_b^{k,\omega}(S)$, then in addition to \eqref{equa1.7} we have
\begin{equation}\label{equa1.8}
\delta_x^0=\sum_{j=1}^\infty h_j(x)\cdot v_j\quad {\rm for\ all}\quad x\in S.
\end{equation}
In this case, the adjoint | $H^*$ of $H$ belongs to $Ext(C_b^{k,\omega}(S);C_b^{k,\omega}({\mbf R}^n))$ and for all $x\in{\mbf R}^n$, $\alpha\in{\mbf Z}_+^n$, $|\alpha|\le k$, the extension $H^*f$ of $f\in C_b^{k,\omega}(S)$ satisfies
\begin{equation}\label{equa1.9}
D^\alpha (H^*f) |
(x):=\sum_{j=1}^\infty D^\alpha h_j(x)\cdot f(v_j).
\end{equation}
}
\end{R}
\subsection{Preduals of ${\mathbf G_b^{k,\omega}(S)}$ Spaces}
Let $C_{0}^{k,\omega}({\mbf R}^n)$ be the subspace of functions $f\in C_b^{k,\omega}({\mbf R}^n)$ such that
\begin{ | itemize}
\item[(i)] for all $\alpha\in{\mbf Z}_+^n$, $0\le |\alpha|\le k$,
\[
\lim_{\|x\|\rightarrow\infty}D^\alpha f(x)=0;
\]
\item[(ii)] for all $\alpha\in{\mbf Z}_+^n$, $|\alpha|=k$,
\[
\lim_{\|x-y\|\rightarrow 0}\frac{D^\alpha f(x)-D^\alpha f(y)}{\omeg |
a(\|x-y\|)}=0.
\]
\end{itemize}
It is easily seen that $C^{k,\omega}_0({\mbf R}^n)$ equipped with the norm induced from $C^{k,\omega}_b({\mbf R}^n)$ is a Banach space. By $C^{k,\omega}_0(S)$ we denote the trace of $C^{k,\omega}_0({\mbf R}^n)$ to a closed | subset $S\subset{\mbf R}^n$ equipped with the trace norm.
If $\lim_{t\rightarrow 0^+}\,\frac{t}{\omega(t)}> 0$ (see condition (i) for $\omega$ in section~1.1), then clearly, the corresponding space $C^{k,\omega}_0({\mbf R}^n)$ is trivial. Thus we may nat |
urally assume that $\omega$ satisfies the condition
\begin{equation}\label{omega2}
\lim_{t\rightarrow 0^+}\,\frac{t}{\omega(t)}=0.
\end{equation}
In the sequel, the weak$^*$ topology of $C_b^{k,\omega}(S)$ is defined by means of functionals in $G_b^{k,\om | ega}(S)\subset
\bigl(C_b^{k,\omega}({\mbf R}^n)\bigr)^*$. Convergence in the weak$^*$ topology is described in section~4.2.
\begin{Th}\label{te1.4}
Suppose $\omega$ satisfies condition \eqref{omega2}.
\begin{enumerate}
\item
Space
$\bigl(C^{k,\omega}_0( |
{\mbf R}^n)\bigr)^*$ is isomorphic to $G_b^{k,\omega}({\mbf R}^n)$, isometrically if $\displaystyle \lim_{t\rightarrow\infty}\omega(t)=\infty$.
\item If there exists a weak$^*$ continuous operator
$T\in Ext(C_b^{k,\omega}(S);C_b^{k,\omega}({\mbf R}^n))$ s | uch that $T\bigl(C_0^{k,\omega}(S)\bigr)\subset C_0^{k,\omega}({\mbf R}^n)$, then $\bigl(C^{k,\omega}_0(S)\bigr)^*$ is isomorphic to $G_b^{k,\omega}(S)$.
\end{enumerate}
\end{Th}
From the first part of the theorem we obtain (for $\omega$ satisfying \eqref{ |
omega2}):
\begin{C}\label{cor1.10}
The space of $C^\infty$ functions with compact supports on ${\mbf R}^n$ is dense in $C^{k,\omega}_0({\mbf R}^n)$. In particular, all spaces $C^{k,\omega}_0(S)$ are separable.
\end{C}
It is not clear whether the condition | of the second part of the theorem is valid for all spaces $C_b^{k,\omega}(S)$ with $\omega$ subject to \eqref{omega2}. Here we
describe a class of sets $S$ satisfying this condition. As before, by $\mathcal P_{k,n}$ we denote the space of real polynomials |
on ${\mbf R}^n$ of degree $k$, and by $Q_r(x)\subset {\mbf R}^n$ the closed cube centered at $x$ of sidelength $2r$.
\begin{D}\label{wm}
A point $x$ of a subset $S\subset{\mbf R}^n$ is said to be weak $k$-Markov if
\[
\varliminf_{r\rightarrow 0}\left\{\s | up_{p\in\mathcal P_{k,n}\setminus 0}\left(\frac{\sup_{Q_r(x)}|p|}{\sup_{Q_r(x)\cap S}|p|} \right) \right\}<\infty .
\]
A closed set $S\subset{\mbf R}^n$ is said to be weak $k$-Markov if it contains a dense subset
of weak $k$-Markov points.
\end{D}
The |
class of weak $k$-Markov sets, denoted by ${\rm Mar}^*_k({\mbf R}^n)$, was introduced and studied by Yu.~Brudnyi and the author, see
\cite{BB1, B}. It contains, in particular, the closure of any open set, the Ahlfors $p$-regular compact subsets of ${\mbf | R}^n$ with $p > n-1$, a wide class of fractals of arbitrary positive Hausdorff measure, direct products $\prod_{j=1}^l S_j$, where $S_j\in {\rm Mar}^*_k({\mbf R}^{n_j})$, $1\le j\le l$, $n=\sum_{j=1}^l n_j$, and closures of unions of any combination of su |
ch sets. Solutions of the Whitney problems (see sections 1.2 and 1.3 above) for sets in ${\rm Mar}^*_k({\mbf R}^n)$ are relatively simple, see \cite{BB1}.
We prove the following result.
\begin{Th}\label{te1.11}
Let $S'\in {\rm Mar}^*_k({\mbf R}^n)$ and | $\omega$ satisfy \eqref{omega2}. Suppose $H:{\mbf R}^n\rightarrow{\mbf R}^n$ is a differentiable map such that
\begin{itemize}
\item[(a)]
the entries of its Jacobian matrix belong to $C_b^{k-1,\omega_o}({\mbf R}^n)$, where $\omega_o$ satisfies
\begin{equ |
ation}\label{equ1.8}
\lim_{t\rightarrow 0^+}\frac{\omega_o(t)}{\omega(t)}=0;
\end{equation}
\item[(b)]
the map $H|_{S'}:S'\rightarrow S=:H(S')$ is a proper retraction.\footnote{I.e., $S\subset S'$ and $H|_{S'}(x)=x$ for all $x\in S$, and for each compact | $K\subset S$ its preimage $(H|_{S'})^{-1}(K)$ is compact.}
\end{itemize}
Then the condition of Theorem \ref{te1.4} holds for $C_b^{k,\omega}(S)$. Thus $G_b^{k,\omega}(S)$ is isomorphic to $\bigl(C^{k,\omega}_0(S)\bigr)^*$ and so $G_b^{k,\omega}(S)$ and $ |
C^{k,\omega}_0(S)$ have the metric approximation property.
\end{Th}
\begin{R}
{\rm (1)
In addition to weak $k$-Markov sets $S\subset{\mbf R}^n$, Theorem \ref{te1.11} is valid, e.g., for a compact subset $S$ of a $C^{k+1}$-manifold
$M\subset{\mbf R}^n | $ such that the base of the topology of $S$ consists of relatively open subsets of Hausdorff dimension $> {\rm dim}\,M - 1$. Indeed, in this case there exist tubular open neighbourhoods $U_M\subset V_M\subset{\mbf R}^n$ of $M$ such that ${\rm cl}(U_M)\sub |
set V_M$ together with a $C^{k+1}$ retraction $r: U_M\rightarrow M$. Then, due to the hypothesis for $S$, the base of topology of $S':=r^{-1}(S)\cap {\rm cl}(U_M) $ consists of relatively open subsets
of Hausdorff dimension $>n-1$ and so $S'\in {\rm Mar}_ | p^*({\mbf R}^n)$ for all $p\in{\mbf N}$, see, e.g., \cite[page\,536]{B}. Moreover, it is easily seen that $r|_{S'}$ is the restriction to $S'$ of a map $H\in C_b^{k+1}({\mbf R}^n; {\mbf R}^n)$. Decreasing $V_M$, if necessary, we may assume that $S'$ is co |
mpact, and so the triple $(H, S', S)$ satisfies the hypothesis of the theorem.
\noindent (2) Under conditions of Theorem \ref{te1.11}, $C_b^{k,\omega}(S)$ is isomorphic to the second dual of $C^{k,\omega}_0(S)$.}
\end{R}
\section{Proof of Theorem \ref | {te1.2}}
By $\delta_x^\alpha$, $x\in{\mbf R}^n$, $\alpha\in\mathbb Z^n_+$, we denote the evaluation functional $D^\alpha|_{\{x\}}$. By definition each $\delta_x^\alpha$, $|\alpha|\le k$, belongs to $\bigl(C^{k,\omega}_b({\mbf R}^n)\bigr)^*$ and has norm $\ |
le 1$. Similarly, functionals $\frac{\delta_x^\alpha-\delta_y^\alpha}{\omega(\|x-y\|)}$, $|\alpha|=k$, $x,y\in{\mbf R}^n$, $x\ne y$, belong to $\bigl(C^{k,\omega}_b({\mbf R}^n)\bigr)^*$ and have norm $\le 1$.
\begin{Proposition}\label{p2.1}
The closed unit | ball $B$ of $\bigl(C^{k,\omega}_b({\mbf R}^n)\bigr)^*$ is the balanced weak$\,^*$ closed convex hull of the set $V$ of all functionals $\delta_x^\alpha$, $|\alpha|\le k$, and $\frac{\delta_x^\alpha-\delta_y^\alpha}{\omega(\|x-y\|)}$, $|\alpha|=k$, $x,y\in |
{\mbf R}^n$, $x\ne y$.
\end{Proposition}
\begin{proof}
Clearly, $V\subset B$ and therefore the required hull $\widehat V\subset B$ as well. Assume, on the contrary, that $\widehat V\ne B$. Then due to the Hahn-Banach theorem there exists an element $f\in C | ^{k,\omega}_b({\mbf R}^n)$ of norm one such that $\sup_{v\in\widehat V}|v(f)|\le c<1$. Since $V\subset \widehat V$, this implies
\[
\|f\|_{C^{k,\omega}_b({\mbf R}^n)}\le c<1,
\]
a contradiction proving the result.
\end{proof}
Let $X$ be the minimal closed |
subspace of $\bigl(C^{k,\omega}_b({\mbf R}^n)\bigr)^*$ containing $V$.
\begin{Proposition}\label{p2.2}
$X^*$ is isometrically isomorphic to $C^{k,\omega}_b({\mbf R}^n)$.
\end{Proposition}
\begin{proof}
For $h\in X^*$ we set $H(x):=h(\delta^0_x)$, $x\in\mat | hbb R^n$.
Let $e_1,\dots, e_n$ be the standard orthonormal basis in ${\mbf R}^n$. By the mean-value theorem for functions in $C^{k,\omega}_b({\mbf R}^n)$ we obtain, for all $\alpha\in\mathbb Z^n_+$, $|\alpha|<k$,
$x\in{\mbf R}^n$,
\begin{equation}\label{eq |
2.6}
\lim_{t\rightarrow 0}\frac{\delta^{\alpha}_{x+t\cdot e_i}-\delta^\alpha_x}{t}=\delta^{\alpha+e_i}_x
\end{equation}
(convergence in $\bigl(C^{k,\omega}_b({\mbf R}^n)\bigr)^*)$. From here by induction we deduce easily that $H\in C^k({\mbf R}^n)$ and fo | r all $\alpha\in\mathbb Z^n_+$, $|\alpha|\le k$,
$x\in{\mbf R}^n$,
\[
h(\delta^\alpha_x)=D^\alpha H(x).
\]
This shows that $H\in C^{k,\omega}_b({\mbf R}^n)$ and $\|H\|_{C^{k,\omega}_b({\mbf R}^n)}\le \|h\|_{X^*}$. Considering $H$ as the bounded linear func |
tional on $\bigl(C^{k,\omega}_b({\mbf R}^n)\bigr)^*$ we obtain that
$H|_V=h|_V$. Thus, by the definition of $X$,
\[
H|_{X}=h.
\]
Since the unit ball of $X$ is $B\cap X$,
\[
\|h\|_{X^*}\le \|H\|_{C^{k,\omega}_b({\mbf R}^n)}\, \bigl(\le \|h\|_{X^*}\bigr).
\ | ]
Hence, the correspondence $h\mapsto H$ determines an isometry $I :X^*\rightarrow C_b^{k,\omega}({\mbf R}^n)$. Since the restriction of each $H\in C_b^{k,\omega}({\mbf R}^n)$, regarded as the bounded linear functional on $\bigl(C^{k,\omega}_b({\mbf R}^n) |
\bigr)^*$, to $X$ determines some $h\in X^*$, map $I$ is surjective.
This completes the proof of the proposition.
\end{proof}
Note that equation \eqref{eq2.6} shows that the minimal closed subspace $G_b^{k,\omega}({\mbf R}^n)\subset\bigl(C_b^{k,\omega}({\ | mbf R}^n)\bigr)^*$ containing all $\delta_x^0$, $x\in{\mbf R}^n$, coincides with $X$. Thus,
$\bigl(G_b^{k,\omega}({\mbf R}^n)\bigr)^*$ is isometrically isomorphic to $C^{k,\omega}_b({\mbf R}^n)$; this proves Theorem \ref{te1.2} for $S={\mbf R}^n$.
\begin{ |
C}\label{cor2.3}
The closed unit ball of $G^{k,\omega}_b({\mbf R}^n)$ is the balanced closed convex hull of the set $V$ of all functionals $\delta_x^\alpha$, $|\alpha|\le k$, and $\frac{\delta_x^\alpha-\delta_y^\alpha}{\omega(\|x-y\|)}$, $|\alpha|=k$, $x,y | \in{\mbf R}^n$, $x\ne y$.
\end{C}
\begin{proof}
The closed unit ball of $G^{k,\omega}_b({\mbf R}^n)$ is
$B\cap G^{k,\omega}_b({\mbf R}^n)$. Since the weak$^*$ topology of $\bigl(C^{k,\omega}_b({\mbf R}^n)\bigr)^*$ induces the weak topology of $G^{k,\omega} |
_b({\mbf R}^n)$ and the weak closure of the balanced convex hull of $V$ coincides with the norm closure of this set, the result follows from Proposition \ref{p2.1}.
\end{proof}
Now, let us consider the case of general $S\subset\mathbb R^n$. Let $h\in \bigl | (G_b^{k,\omega}(S)\bigr)^*$. We set $H(x):=h(\delta_x^0)$, $x\in S$. Due to the Hahn-Banach theorem, there exists $\tilde h\in \bigl(G_b^{k,\omega}({\mbf R}^n)\bigr)^*$ such that $\tilde h|_{G_b^{k,\omega}(S)}=h$ and $\|\tilde h\|_{(G_b^{k,\omega}({\mbf R} |
^n))^*}=\|h\|_{(G_b^{k,\omega}(S))^*}$. Let us define $\widetilde H(x)=\tilde h(\delta_x^0)$, $x\in {\mbf R}^n$. According to Proposition \ref{p2.2},
$\widetilde H\in C^{k,\omega}_b({\mbf R}^n)$ and $\|\widetilde H\|_{C_b^{k,\omega}({\mbf R}^n)}=\|\tilde | h\|_{(G_b^{k,\omega}({\mbf R}^n))^*}$. Moreover, $\widetilde H|_S=H$. This implies that $H\in C_b^{k,\omega}(S)$ and has norm $\le \|h\|_{(G_b^{k,\omega}(S))^*}$. Hence, the correspondence $h\mapsto H$ determines a bounded nonincreasing norm linear injecti |
on $I_S:\bigl(G_b^{k,\omega}(S)\bigr)^*\rightarrow C_b^{k,\omega}(S)$.
Let us show that $I_S$ is a surjective isometry. Indeed, for $H\in C_b^{k,\omega}(S)$ there exists $\widetilde H\in C_b^{k,\omega}({\mbf R}^n)$ such that $\widetilde H|_S=H$ and $\|\wid | etilde H\|_{C_b^{k,\omega}({\mbf R}^n)}=\|H\|_{C_b^{k,\omega}(S)}$. In turn, due to Proposition \ref{p2.2} there exists $\tilde h\in \bigl(G_b^{k,\omega}({\mbf R}^n)\bigr)^*$ such that $\widetilde H(x)=\tilde h(\delta_x^0)$, $x\in{\mbf R}^n$, and $\|\wide |
tilde H\|_{C_b^{k,\omega}({\mbf R}^n)}=\|\tilde h\|_{(G_b^{k,\omega}({\mbf R}^n))^*}$. We set $h:=\tilde h|_{G_b^{k,\omega}(S)}$. Then $h\in \bigl(G_b^{k,\omega}(S)\bigr)^*$ and $H(x)=h(\delta_x^0)$, $x\in S$, i.e.,
$I_S(h)=H$ and
\[
\bigl(\|h\|_{(G_b^{k, | \omega}(S))^*}\ge\bigr)\, \|I_S(h)\|_{C_b^{k,\omega}(S)}\ge \|h\|_{(G_b^{k,\omega}(S))^*}.
\]
The proof of Theorem \ref{te1.2} is complete.
\section{Proofs of Theorems \ref{teo1.5}, \ref{te1.6}, \ref{teo1.6}}
\subsection{Proof of Theorem \ref{teo1.5}}
\ |
begin{proof}
According to the Finiteness Principle
there exist constants $d\in{\mbf N}$ and $c\in (1,\infty)$ such that for all $f\in C_b^{k,\omega}(S)$,
\begin{equation}\label{e3.13}
\sup_{S'\subset S\,;\, {\rm card}\,S'\le d}\|f\|_{C_b^{k,\omega}(S')}\ | le\|f\|_{C_b^{k,\omega}(S)}\le c\cdot\left(\sup_{S'\subset S\,;\, {\rm card}\,S'\le d}\|f\|_{C_b^{k,\omega}(S')}\right).
\end{equation}
Here for $k=0$, $d=2$ (-\,optimal) and $c=1$, see \cite{McS}, for $n=1$, $d=k+2$ (-\,optimal) and $c$ depends on $k$ onl |
y, see \cite{M}, for $k=1$, $d=3\cdot 2^{n-1}$ (-\,optimal) and $c$ depends on $k$ and $n$ only, see \cite{BS1}, and for $k\ge 2$, $d=2^{ k+n \choose k}$ and $c=\frac{\tilde c}{\omega(1)}$, where $\tilde c$ depends on $k$ and $n$ only, see \cite{F1} and \c | ite{BM}, \cite{S}.
Considering $f$ as the bounded linear functional on $G_b^{k,\omega}(S)$, we get from \eqref{e3.13} the required implications
\[
B_b^{k,\omega}(S;d)\subset B_b^{k,\omega}(S)\subset c\cdot B_b^{k,\omega}(S;d).
\]
Indeed, suppose, on the c |
ontrary, that there exists $v\in B_b^{k,\omega}(S)\setminus c\cdot B_b^{k,\omega}(S;d)$. Let $f\in C_b^{k,\omega}(S)$ be such that
\[
\sup_{c\cdot B_b^{k,\omega}(S;d)}|f|<|f(v)|.
\]
By the definition of $B_b^{k,\omega}(S;d)$ the left-hand side of the pr | evious inequality coincides with $c\cdot\bigl(\sup_{S'\subset S\,;\, {\rm card}\,S'\le d}\,\|f\|_{C_b^{k,\omega}(S')}\bigr)$. Hence,
\[
c\cdot\left(\sup_{S'\subset S\,;\, {\rm card}\,S'\le d}\|f\|_{C_b^{k,\omega}(S')}\right)<|f(v)|\le \|f\|_{C_b^{k,\omega} |
(S)},
\]
a contradiction with \eqref{e3.13}.
\end{proof}
\subsection{Proof of Theorem \ref{te1.6}}
\begin{proof}
We set
\begin{equation}\label{e4.14}
r_{X}(F)(s):=F(\delta_s^0),\quad F\in \mathcal L(G_b^{k,\omega}({\mbf R}^n); X),\quad s\in {\mbf R}^n.
\en | d{equation}
Applying the arguments similar to those of Proposition \ref{p2.2} we obtain
\[
r_{X}(F)\in C_b^{k,\omega}({\mbf R}^n;X)\quad {\rm and}\quad \|r_X(F)\|_{C_b^{k,\omega}({\mbf R}^n;X)}\le \|F\|_{\mathcal L(G_b^{k,\omega}({\mbf R}^n);X)}.
\]
On th |
e other hand, for each $\varphi\in X^*$, $\|\varphi\|_{X^*}=1$, function $r_{{\mbf R}}(\varphi\circ F)\in C_b^{k,\omega}({\mbf R}^n)$. So, since $r_{{\mbf R}}(\varphi\circ F)=\varphi (r_{X}(F))$,
\[
\|\varphi\circ F\|_{(G_b^{k,\omega}({\mbf R}^n))^*}=\|r_ | {{\mbf R}}(\varphi\circ F)\|_{C_b^{k,\omega}({\mbf R}^n)}=\|\varphi (r_{{\mbf R}^n;X}(F))\|_{C_b^{k,\omega}({\mbf R}^n)}\le \|r_{X}(F)\|_{C_b^{k,\omega}({\mbf R}^n;X)}.
\]
Taking supremum over all such $\varphi$ we get
\[
\| F\|_{\mathcal L(G_b^{k,\omega}( |
{\mbf R}^n);X)}\le \|r_{X}(F)\|_{C_b^{k,\omega}({\mbf R}^n;X)}\, \bigl(\le \| F\|_{\mathcal L(G_b^{k,\omega}({\mbf R}^n);X)}\bigr).
\]
This shows that $r_{X}:\mathcal L(G_b^{k,\omega}({\mbf R}^n);X)\rightarrow C_b^{k,\omega}({\mbf R}^n;X)$ is an isometry. | Let us prove that it is surjective.
Since every finite subset of ${\mbf R}^n$ is interpolating for $C_b^{k,\omega}({\mbf R}^n)$,
the set of vectors $\delta_s^0\in G_b^{k,\omega}({\mbf R}^n)$, $s\in{\mbf R}^n$, is linearly independent. Hence, each $f\in C |
_b^{k,\omega}({\mbf R}^n;X)$ determines a linear map $\hat f:{\rm span}\{\delta_s^0\, :\, s\in {\mbf R}^n\}\rightarrow X$,
\[
\hat f\left(\sum_{j}c_j\delta_{s_j}^0\right):=\sum_j c_j f(s_j),\quad \sum_{j}c_j\delta_{s_j}^0\in {\rm span}\{\delta_s^0\, :\, s | \in {\mbf R}^n\}\, (\subset G_b^{k,\omega}({\mbf R}^n)).
\]
Next, for each $\varphi\in X^*$, $\|\varphi\|_{X^*}=1$, function $\varphi\circ f\in C_b^{k,\omega}({\mbf R}^n)$ and
\[
\|\varphi\circ f\|_{C_b^{k,\omega}({\mbf R}^n)}\le\|f\|_{C_b^{k,\omega}({\m |
bf R}^n;X)}.
\]
Since $r_{{\mbf R}}:\bigl(G_b^{k,\omega}({\mbf R}^n)\bigr)^*\rightarrow C_b^{k,\omega}({\mbf R}^n)$ is an isometric isomorphism, there exists $\ell_{\varphi\circ f}\in \bigl(G_b^{k,\omega}({\mbf R}^n)\bigr)^*$ such that $r_{{\mbf R}}(\ell_{ | \varphi\circ f})=\varphi\circ f$. Clearly, $\ell_{\varphi\circ f}$ coincides with $\varphi\circ\hat f$ on ${\rm span}\{\delta_s^0\, :\, s\in {\mbf R}^n\}$ and for all $v\in G_b^{k,\omega}({\mbf R}^n)$,
\[
\begin{array}{r}
\displaystyle
|\ell_{\varphi\cir |
c f}(v)|\le \| \ell_{\varphi\circ f}\|_{(G_b^{k,\omega}({\mbf R}^n))^*}\cdot \|v\|_{G_b^{k,\omega}({\mbf R}^n)}
=\|\varphi\circ f\|_{C_b^{k,\omega}({\mbf R}^n)}\cdot \|v\|_{G_b^{k,\omega}({\mbf R}^n)}\medskip\\
\displaystyle \le \|f\|_{C_b^{k,\omega}({\ | mbf R}^n;X)}\cdot \|v\|_{G_b^{k,\omega}({\mbf R}^n)}.
\end{array}
\]
These imply that $\hat f:{\rm span}\{\delta_s^0\, :\, s\in {\mbf R}^n\}\rightarrow X$ is a linear continuous operator of norm $\le \|f\|_{C_b^{k,\omega}({\mbf R}^n;X)}$. Hence, it extend |
s
to a bounded linear operator $F:{\rm cl}({\rm span}\{\delta_s^0\, :\, s\in {\mbf R}^n\})=:G_b^{k,\omega}({\mbf R}^n)\rightarrow X$ such that $r_{X}(F)=f$. Thus,
$r_{X}(F):\mathcal L(G_b^{k,\omega}({\mbf R}^n);X)\rightarrow C_b^{k,\omega}({\mbf R}^n;X)$ | is an isometric isomorphism.
The proof of the theorem is complete.
\end{proof}
\subsection{Proof of Theorem \ref{teo1.6}}
\begin{proof}
Without loss of generality we may assume that $T$ has depth $d$ and is defined by \eqref{equ1.6}.
Let $T:\bigl(C_b^{k |
,\omega}({\mbf R}^n)\bigr)^*\rightarrow \bigl(C_b^{k,\omega}(S)\bigr)^*$ be the adjoint of $T$ and $q_S^*:\bigl(C_b^{k,\omega}(S)\bigr)^*\rightarrow \bigl(C_b^{k,\omega}({\mbf R}^n)\bigr)^*$ the adjoint of the quotient map $q_S: C_b^{k,\omega}({\mbf R}^n)\ | rightarrow C_b^{k,\omega}(S)$.
Clearly, $q_S^*$ is an isometric embedding which maps the closed subspace of $\bigl(C_b^{k,\omega}(S)\bigr)^*$ generated by $\delta$-functionals of points in $S$ isometrically onto $G_b^{k,\omega}(S)\subset \bigl(C_b^{k,\omeg |
a}({\mbf R}^n)\bigr)^*$. We define
\begin{equation}\label{proj}
P:=q_S^*\circ T^*.
\end{equation}
By the definition of $T_S$, for each $\delta_x^0\in G_b^{k,\omega}({\mbf R}^n)$, $x\in{\mbf R}^n\setminus S$, and $f\in C_b^{k,\omega}(S)$ we have, for some $ | y_i^x\in S$,
\[
(P\delta_x^0)(f)=\delta_x^0(Tq_S f)=\sum_{i=1}^d \lambda_i^x\cdot f(y_i^x)=\left(\sum_{i=1}^d \lambda_i^x\cdot\delta_{y_i^x}^0\right)(f).
\]
Hence,
\begin{equation}\label{equ3.11}
P\delta_x^0=\sum_{i=1}^d \lambda_i^x\cdot\delta_{y_i^x}^0\qu |
ad {\rm for\ all}\quad x\in{\mbf R}^n\setminus S.
\end{equation}
Since $T\in Ext(C_b^{k,\omega}(S);C_b^{k,\omega}({\mbf R}^n))$,
\begin{equation}\label{equ3.12}
P\delta_x^0=\delta_x^0\quad {\rm for\ all}\quad x\in S.
\end{equation}
Thus $P$ maps $G_b^{k,\o | mega}({\mbf R}^n)$ into $G_b^{k,\omega}(S)$ and is identity on $G_b^{k,\omega}(S)$. Hence, $P$ is a bounded linear projection of norm $\|P\|\le \|T q_S\|\le \|T\|$.
Next, under the identification $\bigl(G_b^{k,\omega}(S)\bigr)^*=C_b^{k,\omega}(S)$ for all |
closed $S\subset{\mbf R}^n$ (see Theorem \ref{te1.2}), for all $x\in{\mbf R}^n\setminus S$, and $f\in C_b^{k,\omega}(S)$ we have by \eqref{equ3.11}
\[
(P^*f)(\delta_x^0)=f(P\delta_x^0)=f\left(\sum_{i=1}^d \lambda_i^x\cdot\delta_{y_i^x}^0\right)=\sum_{i=1} | ^d \lambda_i^x\cdot f(y_i^x)=(Tf)(\delta_x^0).
\]
The same identity is valid for $x\in S$, cf. \eqref{equ3.12}.
This implies that $P^*=T$ and completes the proof of the theorem.
\end{proof}
\section{Proofs of Theorems \ref{te1.3} and \ref{teor1.10}}
S |
ections 4.1 and 4.2 contain auxiliary results used in the proof of Theorem \ref{te1.3}.
\subsection{Jackson Theorem}
Recall that the {\em Jackson kernel} $J_N$ is the trigonometric polynomial of degree $2\widetilde N$, where $\widetilde N:= \left\lfloor\fr | ac N2\right\rfloor$, given by the formula
\[
J_N(t)=\gamma_N\left(\frac{\sin\frac{\widetilde N t}{2}}{\sin \frac t2}\right)^4,
\]
where $\gamma_N$ is chosen so that $\displaystyle \int_{-\pi}^\pi J_N(t)\, dt =1$.
For a $2\pi$-periodic real function $f\in |
C({\mbf R})$
we set
\begin{equation}\label{jack1}
(L_N f)(x):=\int_{-\pi}^\pi f(x-t)J_N(t)\, dt,\quad x\in\mathbb R.
\end{equation}
Then the classical {\em Jackson theorem} asserts (see, e.g., \cite[Ch.\,V]{T}): $L_N f$ is a real trigonometric polynomial | of degree at most $N$ such that
\begin{equation}\label{jack2}
\sup_{x\in (-\pi,\pi)}|f(x)-(L_N f)(x)|\le c\,\omega\bigl(f, \mbox{$\frac 1N$}\bigr),
\end{equation}
for a numerical constant $c>0$;
here $\omega(f,\cdot)$ is the modulus of continuity of $f$.\ |
subsection{Convergence in the Weak$^*$ Topology of ${\mathbf C_b^{k,\omega}({\mbf R}^n)}$} In the proof of Theorem \ref{te1.3} we use the following result. As before, we equip $C_b^{k,\omega}({\mbf R}^n)$ with the weak$^*$ topology induced by means of fun | ctionals in $G_b^{k,\omega}({\mbf R}^n)\subset \bigl(C_b^{k,\omega}({\mbf R}^n)\bigr)^*$.
\begin{Proposition}\label{prop3.1}
A sequence $\{f_i\}_{i\in{\mbf N}}\subset C_b^{k,\omega}({\mbf R}^n)$ weak$\,^*$ converges to $f\in C_b^{k,\omega}({\mbf R}^n)$ if |
and only if
\begin{itemize}
\item[(a)]
\[
\sup_{i\in{\mbf N}}\|f_i\|_{C_b^{k,\omega}({\mbf R}^n)}<\infty;
\]
\item[(b)]
For all $\alpha\in\mathbb Z_+^n$, $0\le |\alpha|\le k$, $x\in{\mbf R}^n$
\[
\lim_{i\rightarrow\infty}D^\alpha f_i(x)=D^\alpha f(x).
\]
\ | end{itemize}
\end{Proposition}
\begin{proof}
Without loss of generality we may assume that $f=0$.
If $\{f_i\}_{i\in{\mbf N}}$ weak$^*$ converges to $0$, then (a) follows from the Banach-Steinhaus theorem and (b) from the fact that each $\delta_x^\alpha\in |
G_b^{k,\omega}({\mbf R}^n)$.
Conversely, suppose that $\{f_i\}_{i\in{\mbf N}}\subset C_b^{k,\omega}({\mbf R}^n)$ satisfies (a) and (b) with $f=0$. Let $g\in G_b^{k,\omega}({\mbf R}^n)$. According to Corollary \ref{cor2.3}, given $\varepsilon>0$ there exis | t $J\in{\mbf N}$ and families $c_{j\alpha}\in {\mbf R}$, $x_{j\alpha}\in{\mbf R}^n$, $1\le j\le J$, $\alpha\in {\mbf Z}_+^n$, $0\le |\alpha|<k$, and
$d_{j\alpha}\in {\mbf R}$, $x_{j\alpha}, y_{j\alpha}\in{\mbf R}^n$, $x_{j\alpha}\ne y_{j\alpha}$, $1\le j |
\le J$, $\alpha\in {\mbf Z}_+^n$, $|\alpha|=k$, such that
\[
g=\sum_{j,\alpha}c_{j\alpha}\delta_{x_{j\alpha}}^\alpha+\sum_{j,\alpha}d_{j\alpha}\frac{\delta_{x_{j\alpha}}^\alpha-\delta_{y_{j\alpha}}^\alpha}{\omega(\|x_{j\alpha}-y_{j\alpha} \|)}+g',
\]
where |
\[
\sum_{j,\alpha}|c_{j\alpha}|+\sum_{j,\alpha}|d_{j\alpha}|\le \|g\|_{G_b^{k,\omega}({\mbf R}^n)}\quad {\rm and}\quad \|g'\|_{G_b^{k,\omega}({\mbf R}^n)}<\frac{\varepsilon}{2M},\quad M:=\sup_{i\in{\mbf N}}\|f_i\|_{C_b^{k,\omega}({\mbf R}^n)}.
\]
Further, |
due to condition (b), there exists $I\in{\mbf N}$ such that for all $i\ge I$,
\[
\left|f_i\left(\sum_{j,\alpha}c_{j\alpha}\delta_{x_{j\alpha}}^\alpha+\sum_{j,\alpha}d_{j\alpha}\frac{\delta_{x_{j\alpha}}^\alpha-\delta_{y_{j\alpha}}^\alpha}{\omega(\|x_{j\al | pha}-y_{j\alpha} \|)} \right) \right|<\frac{\varepsilon}{2}.
\]
Also, for such $i$,
\[
|f_i(g')|\le \|f_i\|_{C_b^{k,\omega}({\mbf R}^n)}\cdot \|g'\|_{G_b^{k,\omega}({\mbf R}^n)}< M\cdot \frac{\varepsilon}{2M}=\frac{\varepsilon}{2}.
\]
Combining these in |
equalities we obtain for all such $i$:
\[
|f_i(g)|<\varepsilon.
\]
This shows that $\lim_{i\rightarrow\infty}f_i(g)=0$. Thus $\{f_i\}_{i\in{\mbf N}}$ weak$^*$ converges to $0$, as required.
\end{proof}
\subsection{Proof of Theorem \ref{te1.3}\,(1)}
We se | t
\[
\mathbb K_N^n:=\bigl\{x=(x_1,\dots, x_n)\in{\mbf R}^n\, :\, \max_{1\le i\le n}|x_i|\le N\bigr\}.
\]
Let $\rho:\mathbb R^n\rightarrow [0,1]$ be a fixed $C^\infty$ function with support in the cube $\mathbb K_2^n$, equals one on the unit cube $\mathbb K |
_1^n$. For a natural number $\ell$ we set $\rho_\ell(x):=\rho(x/\ell)$, $x\in\mathbb R^n$. Then there exist constants $c_{k,n}$ (depending on $k$ and $n$) such that
\begin{equation}\label{rhol}
\sup_{x\in\mathbb R^n}|D^\alpha\rho_\ell(x)|\le \frac{c_{k,n}} | {\ell^{|\alpha|}}\quad {\rm for\ all}\quad \alpha\in\mathbb Z_+^n\quad {\rm with }\quad |\alpha|\le k+1.
\end{equation}
Let $f\in C_b^{k,\omega}(\mathbb R^n)$. We define a
$8\ell\sqrt n\,$-periodic in each variable function $f_\ell$ on $\mathbb R^n$ by |
\begin{equation}\label{eq3.10}
f_\ell(v+x)=\rho_\ell(x)\cdot f(x),\qquad v+x\in 8\ell\sqrt n\cdot\mathbb Z^n+\mathbb K_{4\ell\sqrt n}^n.
\end{equation}
Note that $f_\ell$ coincides with $f$ on the cube $\mathbb K_\ell^n$.
\begin{Lm}\label{norm}
There exi | sts a constant $C_\ell=C(\ell,k,n,\omega)$ (i.e., depending on $\ell, k,n$ and $\omega$) such that
\[
\lim_{\ell\rightarrow\infty}C_{\ell}=1+c_{k,n}\cdot 4\sqrt n\cdot (k+1)\cdot\lim_{t\rightarrow\infty}\,\frac{1}{\omega(t)},
\]
and
\[
\|f_\ell\|_{C_b^{ |
k,\omega}(\mathbb R^n)}\le C_\ell\cdot\|f\|_{C_b^{k,\omega}(\mathbb R^n)}.
\]
\end{Lm}
\begin{proof}
We use the standard multi-index notation.
According to the general Leibniz rule, for $\alpha\in\mathbb Z_+^n$, $|\alpha|\le k$,
\[
D^\alpha f_\ell=\sum_{\n | u\, :\, \nu\le\alpha}\binom{\alpha}{\nu}(D^\nu\rho_\ell)\cdot (D^{\alpha-\nu}f)\quad {\rm on}\quad \mathbb K_{4\ell\sqrt n}^n .
\]
From here and \eqref{rhol} we get for $\alpha\in \mathbb Z_+^n$, $|\alpha|\ge 1$,
\[
\begin{array}{l}
\displaystyle
\sup_{x\i |
n\mathbb R^n}|D^\alpha f_\ell(x)|\le \|f\|_{C_b^k(\mathbb R^n)}\cdot\left(\sum_{\nu\, :\, 0< \nu\le\alpha}\binom{\alpha}{\nu}\cdot\frac{c_{k,n}}{\ell^{|\nu|}}+1\right)\medskip\\
\displaystyle \le \|f\|_{C_b^k(\mathbb R^n)}\cdot\left(c_{k,n}\cdot\left(\left | (1+\frac 1\ell \right)^{|\alpha|}-1\right)+1\right)\le \|f\|_{C_b^k(\mathbb R^n)}\cdot\left( \left(1+\frac 1\ell\right)^{|\alpha|-1}\cdot \frac{c_{k,n}\cdot |\alpha|}{\ell}+1 \right).
\end{array}
\]
Hence,
\[
\|f_\ell\|_{C_b^k(\mathbb R^n)}\le\left( \lef |
t(1+\frac 1\ell\right)^{\max\{k-1,0\}}\cdot \frac{c_{k,n}\cdot k}{\ell}+1 \right)\cdot \|f\|_{C_b^k(\mathbb R^n)}=:C_1(\ell,k,n)\cdot \|f\|_{C_b^k(\mathbb R^n)}.
\]
Similarly, for $\alpha\in\mathbb Z_+^n$, $|\alpha|=k$, and $x,y \in \mathbb K_{2\ell}^n$ | using properties of $\omega$ we obtain\smallskip
\begin{equation}\label{eq2.9}
\begin{array}{lr}
\qquad\\
\displaystyle
|D^\alpha f_\ell (x)-D^\alpha f_\ell (y)|\medskip\\
\displaystyle \le \sum_{\nu\, :\, \nu\le\alpha}\binom{\alpha}{\nu}\cdot \bigl(| D^\ |
nu \rho_\ell(x)-D^\nu\rho_\ell(y)|\cdot |D^{\alpha-\nu}f(x) |+| D^\nu\rho_\ell(y)|\cdot |D^{\alpha-\nu}f(x)-D^{\alpha-\nu}f(y)|\bigr)\medskip\\
\displaystyle \le \|f\|_{C_b^{k,\omega}(\mathbb R^n)}
\cdot\left(\sum_{\nu\, :\, 0<\nu\le\alpha}\binom{\alpha}{ | \nu}\cdot c_{k,n}\cdot\left(\frac{\|x-y\|}{\ell^{|\nu|+1}}+\frac{\|x-y\|}{\ell^{|\nu|}}\right)+\frac{c_{k,n}\cdot\|x-y\|}{\ell}+\omega(\|x-y\|)\right)\medskip\\
\displaystyle = \|f\|_{C_b^{k,\omega}(\mathbb R^n)}\cdot\left(c_{k,n}\cdot\left(\left(1+\frac |
1\ell\right)^{k+1}-1\right)\cdot \|x-y\|+\omega(\|x-y\|\right)\medskip\\
\displaystyle \le \|f\|_{C_b^{k,\omega}(\mathbb R^n)}\cdot \omega(\|x-y\|)\cdot\left( c_{k,n}\cdot \left(1+\frac 1\ell\right)^{k}\cdot\frac{4\ell\sqrt n \cdot (k+1)}{\ell\cdot\omega | \bigl(4\ell \sqrt n\bigr)}+1\right)\medskip\\
\displaystyle =:\|f\|_{C_b^{k,\omega}(\mathbb R^n)}\cdot \omega(\|x-y\|)\cdot C_2(\ell,k,n,\omega).
\end{array}
\end{equation}
Observe that
\begin{equation}\label{equ4.18}
\lim_{\ell\rightarrow\infty}C_{2}(\ell |
,k,n,\omega)=1+c_{k,n}\cdot 4\sqrt n\cdot (k+1)\cdot\lim_{t\rightarrow\infty}\,\frac{1}{\omega(t)}.
\end{equation}
Next, assume that $x,y\in {\rm supp}\, f_\ell$. Since the case $x,y\in \mathbb K_{2\ell}^n$ was considered above, without loss of generality | we may assume that $x\in \mathbb K_{2\ell}^n$ and
$y\in v+\mathbb K_{2\ell}^n$ for some $v\in (8\ell\sqrt n\cdot\mathbb Z^n)\setminus\{0\}$. Then for $y':=y-v\in \mathbb K_{2\ell}^n$ we have $D^\alpha f_\ell(y')=D^\alpha f_\ell(y)$. Also,
\[
\|x-y\ |
|=\|x-y'-v\|\ge \|v\|-\|x-y'\|\ge 8\ell\sqrt n- 4\ell\sqrt n=4\ell\sqrt n.
\]
Therefore for $\alpha\in\mathbb Z_+^n$, $|\alpha|=k$,
\[
\begin{array}{l}
\displaystyle
|D^\alpha f_\ell (x)-D^\alpha f_\ell (y)|= |D^\alpha f_\ell (x)-D^\alpha f_\ell (y') | |\le C_2(\ell,k,n,\omega)\cdot\|f\|_{C_b^{k,\omega}(\mathbb R^n)}\cdot \omega(\|x-y'\|)\medskip\\
\displaystyle \le C_2(\ell,k,n,\omega)\cdot\|f\|_{C_b^{k,\omega}(\mathbb R^n)}\cdot \omega(4\ell\sqrt n)\le C_2(\ell,k,n,\omega)\cdot\|f\|_{C_b^{k,\omega}(\m |
athbb R^n)}\cdot \omega(\|x-y\|).
\end{array}
\]
Finally, in the case $x\in {\rm supp}\, f_\ell \cap \mathbb K_{2\ell}^n$, $y\not\in {\rm supp}\, f_\ell$, there exists a point $y'\in \mathbb K_{2\ell}^n$ lying on the interval joining $x$ and $y$ such t | hat $f_\ell(y')=0$. Since $\|x-y'\|\le \|x-y\|$ and inequality \eqref{eq2.9} is valid for $x$ and $y'$, a similar inequality is valid for $x$ and $y$. Hence, combining the considered cases we conclude that (cf. \eqref{eq5})
\[
|f_\ell|_{C_b^{k,\omega}(\mat |
hbb R^n)}\le C_2(\ell,k,n,\omega)\cdot\|f\|_{C_b^{k,\omega}(\mathbb R^n)}.
\]
Therefore the inequality of the lemma is valid with
\[
C(\ell,k,n,\omega):=\max\{C_1(\ell,k,n), C_2(\ell,k,n,\omega)\},
\]
see \eqref{equ4.18}.
\end{proof}
We set
\begin{equa | tion}\label{eq2.10}
\lambda:=\frac{4\ell\sqrt n}{\pi}.
\end{equation}
For a natural number $N$ and $x=(x_1,\dots, x_n)\in\mathbb R^n$ we define
\begin{equation}\label{eq2.11}
(E_Nf_\ell)(x)=\int_{-\pi}^\pi\cdots\int_{-\pi}^\pi f_\ell(x_1-\lambda t_1,\dots, |
x_n-\lambda t_n)J_N(t_1)\cdots J_N(t_n)\, dt_1\cdots dt_n.
\end{equation}
\begin{Lm}\label{lem2.2}
Function $E_Nf_\ell$ satisfies the following properties:
\begin{itemize}
\item[(a)] $(E_Nf_\ell)(\lambda x)$, $x\in\mathbb R^n$, is the trigonometric polyn | omial of degree at most $N$ in each coordinate;
\item[(b)]
\[
\|E_Nf_\ell\|_{C_b^{k,\omega}(\mathbb R^n)}\le C_\ell\cdot\|f\|_{C_b^{k,\omega}(\mathbb R^n)};
\]
\item[(c)] There is a constant $c_N$, $\lim_{N\rightarrow\infty} c_N=0$, such that
\[
\|f_\ell |
-E_Nf_\ell\|_{C_b^{k}(\mathbb R^n)}\le c_N\cdot\|f\|_{C_b^{k,\omega}(\mathbb R^n)}.
\]
\end{itemize}
\end{Lm}
\begin{proof}
(a) If we set $f_\ell^\lambda (x):=f_\ell(\lambda x)$, $x=(x_1,\dots, x_n)\in \mathbb R^n$, then
\[
E_Nf_\ell(\lambda x)=(L_N^1\cdo | ts L_N^nf_\ell^\lambda)(x),
\]
where $L_N^i$ is the Jackson operator \eqref{jack1} acting on univariate functions in variable $x_i$, $1\le i\le n$. This and the properties of $L_N^i$ give the required statement.\smallskip
(b) According to definition \eqre |
f{eq2.11}, for each $\alpha\in\mathbb Z_+^n$, $|\alpha|\le k$,
\begin{equation}\label{eq2.12}
\sup_{x\in\mathbb R^N}|D^\alpha(E_Nf_\ell)(x)|=\sup_{x\in\mathbb R^n}|(E_N D^\alpha f_\ell)(x)|\le \|f_\ell\|_{C_b^k(\mathbb R^n)}\le C_\ell\cdot \|f\|_{C_b^k(\ma | thbb R^n)}.
\end{equation}
In turn, if $|\alpha|=k$ and $x,y\in\mathbb R^n$, then
\begin{equation}\label{eq2.13}
\begin{array}{l}
\displaystyle
|D^\alpha (E_Nf_\ell)(x)-D^\alpha(E_Nf_\ell)(y)|\le
|(E_N D^\alpha f_\ell)(x)- (E_ND^\alpha f_\ell)(y)|\ |
medskip\\
\displaystyle \le |f_\ell|_{C_b^{k,\omega}(\mathbb R^n)}\cdot \omega(\|x-y\|)\le
C_\ell\cdot |f|_{C_b^{k,\omega}(\mathbb R^n)}\cdot \omega(\|x-y\|).
\end{array}
\end{equation}
Thus \eqref{eq2.12} and \eqref{eq2.13} give the required statemen | t.\smallskip
(c) For each $\alpha\in\mathbb Z_+^n$, $|\alpha|\le k-1$, using \eqref{eq2.11} and due to \eqref{jack2} one obtains\smallskip
\begin{equation}\label{eq2.14}
\\
\begin{array}{l}
\displaystyle
\sup_{x\in\mathbb R^n}|D^\alpha\bigl( f_\ell |
-E_Nf_\ell\bigr)(x)|=\sup_{x\in\mathbb R^n}|D^\alpha f_\ell(x)-(E_ND^\alpha f_\ell)(x)|\medskip\\
\displaystyle \le \sup_{x\in\mathbb R^n}\bigl\{|D^\alpha f_\ell(\lambda x)-(L_N^1D^\alpha f_\ell)(\lambda x)|+|\bigl(L_N^1\bigl(D^\alpha f_\ell-L_N^2D^\alph | a f_\ell\bigr)\bigr)(\lambda x)|+\,\cdots \medskip\\
\displaystyle + |\bigl(L_N^1\cdots L_N^{n-1}\bigl(D^\alpha f_\ell-L_N^nD^\alpha f_\ell\bigr)\bigr)(\lambda x)| \bigr\}\le
\frac{c\cdot n}{N}\cdot\|f_\ell\|_{C_b^k(\mathbb R^n)}\le \frac{c\cdot n\cdot |
C_\ell}{N}\cdot\|f\|_{C_b^k(\mathbb R^n)}.
\end{array}
\end{equation}
Similarly, for $|\alpha|=k$, we have\smallskip
\begin{equation}\label{eq2.15}
\sup_{x\in\mathbb R^n}|D^\alpha(f_\ell -E_Nf_\ell)(x)|\le c\cdot n\cdot C_\ell\cdot\omega(\mbox{$\frac | 1N$})\cdot |f|_{C_b^{k,\omega}(\mathbb R^n)}.
\end{equation}
Equations \eqref{eq2.14} and \eqref{eq2.15} imply that the required statement is valid with
\[
c_N:=c\cdot n\cdot C_\ell\cdot \max\left\{\mbox{$\frac 1N ,\, \omega(\frac 1N)$}\right\}.
\]
|
\end{proof}
\begin{proof}[Proof of Theorem \ref{te1.3}\,(1)] For $f\in C_b^{k,\omega}({\mbf R}^n)$ we set
\begin{equation}\label{eq3.18}
L_{N,\ell}f:=E_Nf_\ell.
\end{equation}
According to Lemma \ref{lem2.2}, $L_{N,\ell}: C_b^{k,\omega}({\mbf R}^n)\right | arrow C_b^{k,\omega}({\mbf R}^n)$ is a finite rank bounded linear operator of norm $\le C_\ell$.
\begin{Lm}\label{lem3.3}
Operators $L_{N,\ell}$ are weak$\,^*$ continuous.
\end{Lm}
\begin{proof} Since $C_b^{k,\omega}({\mbf R}^n)=\bigl(G_b^{k,\omega}( |
{\mbf R}^n)\bigr)^*$ and $G_b^{k,\omega}({\mbf R}^n)$ is separable, $C_b^{k,\omega}({\mbf R}^n)$ equipped with the weak$^*$ topology is a Frechet space. Then $L_{N,\ell}$ is weak$^*$ continuous if and only if for each sequence $\{f_i\}_{i\in{\mbf N}}\subs | et C_b^{k,\omega}({\mbf R}^n)$ weak$^*$ converging to $0\in C_b^{k,\omega}({\mbf R}^n)$ the sequence $\{L_{N,\ell}f_i\}_{i\in{\mbf N}}$ weak$\,^*$ converges to $0$ as well. Note that such a sequence $\{f_i\}_{i\in{\mbf N}}$ is bounded in $C_b^{k,\omega}({ |
\mbf R}^n)$ due to the Banach-Steinhaus theorem.
Then $\{L_{N,\ell}f_i\}_{i\in{\mbf N}}$ is bounded as well and according to Proposition \ref{prop3.1} we must prove only that
\begin{equation}\label{eq3.19}
\lim_{i\rightarrow\infty}D^\alpha (L_{N,\ell}f_ | i)(x)=0\quad {\rm for\ all}\quad \alpha\in{\mbf Z}_+^n,\ 0\le |\alpha|\le k,\ x\in{\mbf R}^n.
\end{equation}
Further, since $D^\alpha (L_{N,\ell}f_i)(x)=(E_ND^\alpha (f_i)_\ell)(x)$ for such $\alpha$ and $x$, $\{D^\alpha (f_i)_\ell\}_{i\in{\mbf N}}$ is a |
bounded sequence of continuous functions and $E_N$ is the convolution operator with the absolutely integrable kernel, to establish \eqref{eq3.19} it suffices to prove (due to the Lebesgue dominated convergence theorem) that
\[
\lim_{i\rightarrow\infty}D | ^\alpha (f_i)_\ell(x)=0 \quad {\rm for\ all}\quad \alpha\in{\mbf Z}_+^n,\ 0\le |\alpha|\le k,\ x\in{\mbf R}^n.
\]
The latter follows directly from the definition of $(f_i)_\ell$, see \eqref{eq3.10}, the general Leibniz rule and the fact that $D^\alpha f_ |
i(x)\rightarrow 0$ as $i\rightarrow\infty$ for all the required $\alpha$ and $x$ (because $\{f_i\}_{i\in{\mbf N}}$ weak$^*$ converges to $0$). Thus we have proved that operators $L_{N,\ell}$ are weak$^*$ continuous.
\end{proof}
Lemma \ref{lem3.3} implies | that there exists a bounded operator of finite rank $H_{N,\ell}$ on $G_b^{k,\omega}({\mbf R}^n)$ whose adjoint $H_{N,\ell}^*$ coincides with $L_{N,\ell}$.
\begin{Lm}\label{lem3.5}
The sequence of finite rank bounded operators $\{H_{N,N}\}_{N\in{\mbf N}} |
$ converges pointwise to the identity operator on $G_b^{k,\omega}({\mbf R}^n)$.
\end{Lm}
\begin{proof}
Let $g\in G_b^{k,\omega}({\mbf R}^n)$. Due to Corollary \ref{cor2.3}, given $\varepsilon>0$ there exist $J\in{\mbf N}$ and families $c_{j\alpha}\in {\ | mbf R}$, $x_{j\alpha}\in{\mbf R}^n$, $1\le j\le J$, $\alpha\in {\mbf Z}_+^n$, $0\le |\alpha|<k$, and
$d_{j\alpha}\in {\mbf R}$, $x_{j\alpha}, y_{j\alpha}\in{\mbf R}^n$, $x_{j\alpha}\ne y_{j\alpha}$, $1\le j\le J$, $\alpha\in {\mbf Z}_+^n$, $|\alpha|=k$, |
such that
\[
g=\sum_{j,\alpha}c_{j\alpha}\delta_{x_{j\alpha}}^\alpha+\sum_{j,\alpha}d_{j\alpha}\frac{\delta_{x_{j\alpha}}^\alpha-\delta_{y_{j\alpha}}^\alpha}{\omega(\|x_{j\alpha}-y_{j\alpha} \|)}+g''=:g'+g'',
\]
where
\[
\sum_{j,\alpha}|c_{j\alpha}|+\sum_{ | j,\alpha}|d_{j\alpha}|\le \|g\|_{G_b^{k,\omega}({\mbf R}^n)}\quad {\rm and}\quad \|g''\|_{G_b^{k,\omega}({\mbf R}^n)}<\frac{\varepsilon}{2(C_N+1)},
\]
see Lemma \ref{norm} for the definition of $C_N$.
Next, for each $f\in C_b^{k,\omega}({\mbf R}^n)$, $\| |
f\|_{C_b^{k,\omega}({\mbf R}^n)}=1$, we have by means of Lemma \ref{lem2.2},\smallskip
\begin{equation}\label{eq3.20}
\begin{array}{l}
\ \ \ \vspace*{-2mm}\\
\displaystyle
\bigl|f\bigl(H_{NN}\,g-g\bigr)\bigr|=\bigl|\bigl(L_{NN}f-f\bigr)(g)\bigr|\le \bigl| | \bigl(L_{NN}f-f\bigr)(g')\bigr|+ \bigl|\bigl(L_{NN}f-f\bigr)(g'')\bigr|\medskip\\
\displaystyle < \bigl|\bigl(E_{N}f_N-f\bigr)(g')\bigr|+\|L_{NN}-{\rm id}\|\cdot\frac{\varepsilon}{2(C_{N}+1)}\le\bigl|\bigl(E_{N}f_N-f\bigr)(g')\bigr|+\frac{\varepsilon}{2}.
|
\end{array}
\end{equation}
Let $N_0\in{\mbf N}$ be so large that all points $x_{j\alpha}, y_{j\alpha}$ as above belong to $\mathbb K_{N_0}^n$. Since $f_{N_0}=f$ on $\mathbb K_{N_0}^n$, for all $N\ge N_0$,
\[
\bigl(E_{N}f_N-f\bigr)(g')=\bigl(E_{N}f_N-f_N\bi | gr)(g').
\]
Hence, due to Lemma \ref{lem2.2}\,(c) for $z=x_{j\alpha}$ or $y_{j\alpha}$,
\[
\bigl|\bigl(E_{N}f_N-f\bigr)(\delta_z^\alpha)\bigr|\le \|E_{N}f_N-f_N\|_{C_b^{k}({\mbf R}^n)} \le c_N.
\]
This implies that for all $N\ge N_0$
\[
\bigl|\bigl(E_{N}f_ |
N-f\bigr)(g')\bigr|\le c_N\cdot\left(\sum_{j,\alpha}|c_{j\alpha}| +\left(\sum_{j,\alpha}2|d_{j\alpha}|\right)\cdot\max_{j,\alpha}\left\{\frac{1}{\omega(\|x_{j\alpha}-y_{j\alpha} \|)}\right\}\right).
\]
Choose $N_0'\ge N_0$ so large that for all $N\ge N_0' | $ the right-hand side of the previous inequality is less than $\frac{\varepsilon}{2}$. Then combining this with \eqref{eq3.20} we get for all $N\ge N_0'$,
\[
\|H_{NN}\,g-g\|_{G_b^{k,\omega}({\mbf R}^n)}<\varepsilon.
\]
This shows that for all $g\in G_b |
^{k,\omega}({\mbf R}^n)$
\[
\lim_{N\rightarrow\infty}H_{NN}\,g=g
\]
which completes the proof of the lemma.
\end{proof}
Let us finish the proof of the theorem for $S={\mbf R}^n$. We set
\begin{equation}\label{equ4.28}
T_{N}:=\left(1+c_{k,n}\cdot 4\sqr | t n\cdot (k+1)\cdot\lim_{t\rightarrow\infty}\,\frac{1}{\omega(t)}\right)\cdot\frac{H_{NN}}{C_N},
\end{equation}
see Lemma \ref{norm} for the definition of $C_N$.
Since $\{C_N\}_{N\in{\mbf N}}$ converges to the first factor in the definition of $T_N$, due t |
o Lemma \ref{lem3.5} $\{T_N\}_{N\in{\mbf N}}$ is the sequence of operators of finite rank on $G_b^{k,\omega}({\mbf R}^n)$ of norm at most $\lambda:=1+c_{k,n}\cdot 4\sqrt n\cdot (k+1)\cdot\lim_{t\rightarrow\infty}(1/\omega(t))$ converging pointwise to the | identity operator. In particular, this sequence converges uniformly to the identity operator on each compact subset of $G_b^{k,\omega}({\mbf R}^n)$. This shows that $G_b^{k,\omega}({\mbf R}^n)$ has the $\lambda$-approximation property with respect to the |
approximating sequence of operators $\{T_N\}_{N\in{\mbf N}}$.
The proof of Theorem \ref{te1.3} for $S={\mbf R}^n$ is complete.
\end{proof}
\subsection{Proof of Theorem \ref{te1.3}\,(2)}
\begin{proof}
In the case of $G_b^{k,\omega}(S)$, the required seque | nce of finite rank linear operators approximating the identity map is
$\bigl\{PT_N|_{G_b^{k,\omega}(S)}\bigr\}_{N\in{\mbf N}}$, where $T_N$ are linear operators defined by \eqref{equ4.28} and $P: G_b^{k,\omega}({\mbf R}^n)\rightarrow G_b^{k,\omega}(S)$ is |
the projection of Theorem \ref{teo1.6}. We have
\[
\|PT_N|_{G_b^{k,\omega}(S)}\|\le \|P\|\cdot\|T_N\|=:\|P\|\cdot\lambda(k,n,\omega).
\]
Choosing here $P$ corresponding to the extension operators of papers \cite{Gl} ($k=0$), \cite{BS2} ($k=1$) and \cite{Lu | } ($k\ge 2$) we obtain the required result.
The proof of Theorem \ref{te1.3} is complete.
\end{proof}
\subsection{Proof of Theorem \ref{teor1.10}}
\begin{proof}
Due to the result of Pe\l czy\'nski \cite{P} there are a separable Banach space $Y$ with a nor |
m one monotone basis $\{b_j\}_{j\in{\mbf N}}$, an isomorphic embedding $T:X\rightarrow Y$ with distortion $\|T\|\cdot\|T^{-1}\|\le 4\lambda$, and a linear projection $P:Y\rightarrow T(X)$ with $\|P\|\le 4\lambda$. For an operator $H\in\mathcal L(G_b^{k,\o | mega}({\mbf R}^n);X)$ we define
\[
\widetilde H:=T\cdot H\in \mathcal L(G_b^{k,\omega}({\mbf R}^n);Y).
\]
Then for each $x\in{\mbf R}^n$,
\[
\widetilde H(\delta_x^0)=\sum_{j=1}^\infty \tilde h_j(x)\cdot b_j
\]
for some $\tilde h_j(x)\in{\mbf R}$, $j\in{\m |
bf N}$.
Further, consider the family of bounded linear functionals $\{b_j^*\}_{j\in{\mbf N}}\subset Y^*$ such that $b_j^*(b_i)=\delta_{ij}$ (- the Kronecker delta) for all $i, j\in{\mbf N}$. As the basis $\{b_j\}_{j\in{\mbf N}}$ is monotone, $\|b_j^*\|\l | e 2$ for all $j\in{\mbf N}$. Since $b_j^*\circ\widetilde H\in \bigl(G_b^{k,\omega}({\mbf R}^n)\bigr)^*=C_b^{k,\omega}({\mbf R}^n)$, the functions $\tilde h_j$, $\tilde h_j(x):=(b_j^*\circ \widetilde H)(\delta_x^0)$, $x\in{\mbf R}^n$, belong to $C_b^{k,\ome |
ga}({\mbf R}^n)$ and
\begin{equation}\label{equ4.32a}
\|\tilde h_j\|_{C_b^{k,\omega}({\mbf R}^n)}\le 2\cdot\|T\|\cdot\|H\|\quad {\rm for\ all}\quad j\in{\mbf N}.
\end{equation}
In particular, $(b_j^*\circ\widetilde H)(\delta_x^\alpha)=D^\alpha (b_j^*\circ | \widetilde H)(\delta_x^0)=D^\alpha \tilde h_j(x)$ for all $\alpha\in{\mbf Z}_+^n$, $|\alpha|\le k$, $x\in{\mbf R}^n$, $j\in{\mbf N}$. This implies that for all such $\alpha$ and $x$,
\begin{equation}\label{equ4.32}
\widetilde H(\delta_x^\alpha)=\sum_{j=1}^ |
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