# On Whitney extensions and orthonormal expansions

Steven B. Damelin Department of Mathematics, University of Michigan, 530
Church Street, Ann Arbor, MI, USA. email<EMAIL_ADDRESS>
###### Abstract

The Whitney near extension problem for finite sets in $\mathbb{R}^{d},\,d\geq
2$ asks the following: Let $\phi:E\to\mathbb{R}^{d}$ be a near distortion on a
finite set $E\subset\mathbb{R}^{d}$ with certain geometry. How to decide
whether $\phi$ extends to a smooth, one to one and onto near distortion
$\Phi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ which agrees with $\phi$ on $E$ and
with Euclidean motions in $\mathbb{R}^{d}$. The Whitney near extension problem
for compact sets $E\subset U$ in open subsets $U$ of $\mathbb{R}^{n},\,n\geq
1$ asks the following: Let $U\subset R^{n}$ be open and let $E\subset U$ be a
compact set. Let $\phi:U\to\mathbb{R}^{n}$ be a smooth near isometry. How to
decide if there exists a smooth one-to-one and onto near isometry
$\Phi:\mathbb{R}^{n}\to\mathbb{R}^{n}$ which extends $\phi$ on $E$ and agrees
with Euclidean motions on $\mathbb{R}^{n}$. The classical Whitney extension
problem asks the following: Let $\phi:E\to\mathbb{R}$ be a map defined on an
arbitrary set $E\subset\mathbb{R}^{n}$. How can one decide whether $\phi$
extends to a map $\Phi:\mathbb{R}^{n}\to\mathbb{R}$ which agrees with $\phi$
on $E$ and is in $C^{m}(\mathbb{R}^{n}),\,m\geq 1$, the space of functions
from $\mathbb{R}^{n}$ to $\mathbb{R}$ whose derivatives of order $m$ are
continuous and bounded.

In this paper, we survey some of our work on the near Whitney extension
problem [2] in $\mathbb{R}^{n}$. Thereafter, we survey some of our work on
weighted $L_{p}(\mathbb{R}),\,1<p\leq\infty$ convergence of orthonormal
expansions in $\mathbb{R}$ [12] and present a result of [13]. The motivation
for doing this is motivated by interesting connections between Whitney
extension theorems, Taylor series and Fourier expansions. Finally, we raise
various open questions to study.

##  1 Whitney, Taylor

###  1.1 Whitney extension problem

Throughout this paper $d\geq 2$, $n,m\geq 1$ and $|.|$ denotes the Euclidean
norm on $\mathbb{R}^{n}$. Typically, most constants (most often denoted by
$c,c_{1},..$) are always small enough and are independent of $d,n$ and
possibly other quantities. The context will be clear. ***By small enough, we
mean bounded by a controlled small constant. The analogy of large enough is
clear.. If this is not explicitly stated, the context will be clear. The same
letter may denote different constants and/or functions at any given time. The
context will be clear. The classical Whitney extension problem asks the
following: Let $\phi:E\to\mathbb{R}$ be a map defined on an arbitrary set
$E\subset\mathbb{R}^{n}$. How can one decide whether $\phi$ extends to a map
$\Phi:\mathbb{R}^{n}\to\mathbb{R}$ which agrees with $\phi$ on $E$ and is in
$C^{m}(\mathbb{R}^{n})$, the space of functions from $\mathbb{R}^{n}$ to
$\mathbb{R}$ whose derivatives of order $m$ are continuous and bounded.

In order to state one answer to this problem from the paper [32], we assume
that $E$ is closed. We then ask for a theorem that says that it is possible to
extend a given real valued function with domain $E$ in such a way as to have
prescribed derivatives at the points of $E$. The set $E$ will in general, lack
a differentiable structure and to this end we need a careful statement of
Taylor’s theorem on $\mathbb{R}^{n}$ which provides an essential ingredient
for the aforementioned theorem’s statement and proof. The theorem we state is
that of Hassler Whitney [32] and is a partial converse to Taylor’s theorem.

###  1.2 Taylor to Whitney

A $n$-dimensional multi-index is an $n$-tuple of non-negative integers
$\alpha=(\alpha_{1},....,\alpha_{n})$. We write
$|\alpha|=\alpha_{1}+...+\alpha_{n}$, $\alpha!=\alpha_{1}!...\alpha_{n}!.$ For
$x\in\mathbb{R}^{n}$, we write
$x^{\alpha}=x_{1}^{\alpha_{1}}...x_{n}^{\alpha_{n}}$. Given a function
$f:\mathbb{R}^{n}\to\mathbb{R}$ in $C^{m}(\mathbb{R}^{n})$, we set

$D^{\alpha}f:=\frac{\partial^{|\alpha|}f}{\partial x^{\alpha_{1}}...\partial
x_{n}^{\alpha_{n}}},$

when well defined.

We have Taylor’s theorem: Let $f:\mathbb{R}^{n}\to\mathbb{R}$ in
$C^{m}(\mathbb{R}^{n})$ at a point $a\in\mathbb{R}^{n}$. Then there exist
functions $h_{\alpha}:\mathbb{R}^{n}\to\mathbb{R}$ with multi index $\alpha$
such that for $x\in\mathbb{R}^{n}$,

$f(x)=\sum_{|\alpha|\leq
m}\frac{D^{\alpha}f(a)}{\alpha!}(x-a)^{\alpha}+\sum_{|\alpha|=m+1}h_{\alpha}(x)(x-a)^{\alpha}$

and $\lim_{x\to a}h_{\alpha}(x)=0.$

Differentiating and replacing $h_{\alpha}$ as needed (still denoted by
$h_{\alpha}$), we obtain with $f_{\alpha}:=D^{\alpha}f$, for each multi index
$\alpha$,

  * (1)

$f_{\alpha}(x)=\sum_{|\beta|\leq
m-|\alpha|}\frac{f_{\alpha+\beta}(a)}{\beta!}(x-a)^{\alpha}+h_{\alpha}(x,a).$
with $h_{\alpha}=o(|x-a|)$ uniformly as $x\to a$.

We note that (1) may be regarded as purely a compatibility condition between
the functions $f_{\alpha}$ which must be satisfied in order for these
functions to be the coefficients of the Taylor series of the function $f$.
This important observation allows us to deal with the problem of the closed
set $E$, in general lacking a differentiable structure in the following
Whitney theorem of Hassler Whitney [32] (below) which follows in part from it.

Let $\phi_{\alpha}:E\to\mathbb{R}$ be a collection of functions defined on a
closed set $E\subset\mathbb{R}^{d}$ with all multi-indices $|\alpha|\leq m$
satisfying the compatibility condition (1) for all $x,a\in\mathbb{R}^{d}$.
Then there exists a function $\Phi:\mathbb{R}^{d}\to\mathbb{R}$ in
$C^{m}(\mathbb{R}^{d})$ such that the following hold:

  * •

$\Phi=\phi_{0}$.

  * •

$D^{\alpha}\Phi=\phi_{\alpha}$.

  * •

$\Phi$ is real-analytic in $\mathbb{R}^{d}\setminus E$.

Fourier series versus Taylor series

A holomorphic function in an anulus containing the unit circle has a Laurent
series about zero which generalizes the Taylor series of a holomorphic
function in a neighborhood of zero. When restricted to the unit circle, this
Laurent series gives a Fourier series of the corresponding periodic function.
This explains the connection between the Cauchy integral formula and the
integral defining the coefficients of a Fourier series. The connection between
the two provided by the Cauchy integral formula is therefore quite remarkable;
one takes an integral of $f$ over the unit circle and it tells you information
about the behavior of $f$ at the origin. But this is more a magic property of
holomorphic functions than anything else. A variant of this argument would be
the following: Assume that the Taylor expansion of a real valued function $f$:
$f(x)=\sum_{k=0}^{\infty}a_{k}x^{k}$ is convergent for some $|x|>1$. Then $f$
can be extended in a natural way into the complex plane by writing
$f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ with $|z|\leq 1$. Restricting to the unit
circle $|z|=1$, define $F(\phi):=f(e^{i\phi})$, that is, consider $f$ as a
function of the polar angle $\phi$. $F$ has period $2\pi$ and its Fourier
expansion is nothing else but $F(\phi)=\sum_{k=0}^{\infty}a_{k}e^{ik\phi}$
where the $a_{k}$ are the Taylor coefficients of the real valued map
$x\longmapsto f(x)$

Our ultimate goal from this paper is to begin an exploration of relations
between Witney extensions and orthonormal expansions in line with the above
discussion and the interesting paper [13]. To this end, moving forward we will
survey our work in [12, 2] and then state an interesting result of [13] (See
Section 5) and raise some pertinent open questions related to these ideas.

We begin with:

##  2 The near distorted Whitney extension problem

###  2.1 The problem

In this section, we provide a small taiste of some of the work in [2] and
refer the reader to the monograph [2] for the full adventure!

  * •

A map $A:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is an improper Euclidean motion
(rigid motion, distance preserving transformation) if there exist $M\in O(d)$
and a translation $x_{0}\in\mathbb{R}^{d}$ so that for every
$x\in\mathbb{R}^{d}$, $A(x)=Mx+x_{0}$.

  * •

If $M\in SO(d)$, then $A$ is a proper (orientation preserving) Euclidean
motion. Here, $O(d)$ and $SO(d)$ are respectively the orthogonal and special
orthogonal groups. A Euclidean motion can either be proper or improper.

  * •

A map $f:\mathbb{R}^{d}\to\mathbb{R}^{d}$, will be called a $c$-distortion if
there exists $c>0$ small enough depending on $d$ so that
$(1-c)|x-y|\leq|f(x)-f(y)|\leq|x-y|(1+c)$ for every $x,y\in\mathbb{R}^{d}$.
Note that $f$ is non-rigid.

The problems we are concerned with are:

  * (1)

Let $\phi:E\to\mathbb{R}^{d}$ be a smooth near distortion †††When we use the
term ”near distortion”, we mean $c$-distortion for small enough $c$. on a
finite set $E\subset\mathbb{R}^{d}$ with certain geometry. How to decide
whether $\phi$ extends to a smooth, one to one and onto near distortion
$\Phi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ which agrees with $\phi$ on $E$ and
with Euclidean motions in $\mathbb{R}^{d}$.

  * (2)

Let $U\subset R^{n}$ be open and let $E\subset U$ be a compact set with
certain geometry. Let $\phi:U\to\mathbb{R}^{n}$ be a smooth near isometry. How
to decide if there exists a smooth one-to-one and onto near isometry
$\Phi:\mathbb{R}^{n}\to\mathbb{R}^{n}$ which agrees with $\phi$ on $E$ and
agrees with Euclidean motions on $\mathbb{R}^{n}$.

In [2], we study (1-2) in detail. This is a fascinating problem with
connections to harmonic analysis, best approximation in algebraic varieties,
algebraic geometry in particular, tensor analysis, alignment, clustering, the
Procrustes alignment data problem, the manifold data learning problem and BMO
(the space of maps of bounded mean variation). See also [1, 12, 16, 17, 18,
19, 20].

###  2.2 Some variants

Problem (1) below is an example of (1) above:

######  Problem 1.

Let us be given a positive constant $c$ small enough depending on $d$. Does
there exist a positive constant $c^{\prime}$ small enough depending on $c$ so
that the following holds. Given two sets of $k\geq 1$ distinct points in
$\mathbb{R}^{d}$, $\left\\{y_{1},...,y_{k}\right\\}$ and
$\left\\{z_{1},...,z_{k}\right\\}$. Suppose for every $1\leq i,j\leq k$,

$(1-c^{\prime})<\frac{|z_{i}-z_{j}|}{|y_{i}-y_{j}|}\leq(1+c^{\prime}).$ (2.1)

  * (1)

Does there exist a smooth one-to-one and onto $c$-distortion
$\Phi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ which obeys $\Phi(y_{i})=z_{i},\,1\leq
i\leq k$? ‡‡‡Note that this is an interpolation problem as well a non-rigid
alignment data problem.

  * (2)

$\Phi$ agrees with Euclidean motions on $\mathbb{R}^{d}$.

  * (3)

Can one say something about how $c,c^{\prime},k,d$ are related?

######  Problem 2.

Let us be given a positive constant $c$ small enough depending on $d$. Does
there exist a positve constant $c^{\prime}$ small enough depending on $c$ so
that the following holds. Given two sets of $k\geq 1$ distinct points in
$\mathbb{R}^{d}$, $\left\\{y_{1},...,y_{k}\right\\}$ and
$\left\\{z_{1},...,z_{k}\right\\}$. Suppose for every $1\leq i,j\leq k$, (2.1)
holds with $c^{\prime}$.

  * (1)

Is it possible to find a Euclidean motion $A$ so that uniformly for every
$1\leq i\leq k$,

$(1-c)|y_{i}-z_{i}|\leq|A(y_{i})-A(z_{i})|\leq(1+c)|y_{i}-z_{i}|$

  * (2)

Can one say something about how $c,c^{\prime},k,d$ are related?

In the case of isometric extension, Theorem 2 is less subtle. It is well
known, see for example [31] that following holds..

Let $\left\\{y_{1},...,y_{k}\right\\}$ and $\left\\{z_{1},...,z_{k}\right\\}$
be two collections of $k\geq 1$ distinct points in $\mathbb{R}^{d}$. Suppose

$|z_{i}-z_{j}|=|y_{i}-y_{j}|,\,1\leq i,j\leq k.$

Then, there exists a Euclidean motion, $A:\mathbb{R}^{d}\to\mathbb{R}^{d}$
with

$A(y_{i})=z_{i},\,1\leq i\leq k.$ §§§In $\mathbb{R}^{d}$, every rigid map is a
Euclidean motion. The converse is clear.

###  2.3 Problem 1

Problem 1 and Problem 2 are a new way to view interpolation and non-rigid
alignment of data. See [2] for the full details.

Figure 1: Near isometric correspondence shape matching found between a dog and
a wolf. See the reference [24] Figure 2: Near isometric orrespondences found
between horses: See the reference [29]

We now begin to look at Problem 1 and Problem 2. We will translate "smooth"
into the idea of a $c$-distorted diffeomorphism.

Given $c$ small enough, a diffeomorphism $f:\mathbb{R}^{d}\to\mathbb{R}^{d}$
is a $c$-distorted diffeomorphism if $f$ is a $c$ distortion. ¶¶¶A
diffeomorphism $f:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a differentiable one-to-
one and onto function with a differentiable inverse.

We are now going to provide two examples of $c$-distorted diffeomorphisms.

###  2.4 Slow twists

######  Example 1.

Let $\varepsilon>0$ small enough and $x\in\mathbb{R}^{d}$. Let $T_{x}$ be the
block-diagonal matrix given by

$\begin{bmatrix}D_{1}(x)&0&0&0&0&0\\\ 0&D_{2}(x)&0&0&0&0\\\ 0&0&.&0&0&0\\\
0&0&0&.&0&0\\\ 0&0&0&0&.&0\\\ 0&0&0&0&0&D_{r}(x).\end{bmatrix}$

where for each $i$ either $D_{i}(x)i$ s the $1\times 1$ identity matrix or
else

$D_{i}(x)=\begin{bmatrix}\cos f_{i}(|x|)&\sin f_{i}(|x|)\\\ -\sin
f_{i}(|x|)&\cos f_{i}(|x|)\end{bmatrix}$

for a function $f_{i}$ of one variable satisfying:

$A:\,t|f^{\prime}_{i}(t)|<c\varepsilon,\,t\geq 0.$

Here $c$ is small enough*. Then,

$f_{ST}(x):=M^{T}T_{x}(Mx)$

for $x\in\mathbb{R}^{d}$ and for a fixed $M\in SO(d)$. is a $\varepsilon$
distorted diffeomorphism. We call $f_{ST}$ a Slow twist (in analogy to
rotations).

###  2.5 Slides

######  Example 2.

Let $\varepsilon>0$ be small enough and let $f:\mathbb{R}^{d}\to\mathbb{R}$ be
a smooth function satisfying the following condition**

$B:|f^{\prime}(t)|<c\varepsilon,\,t\geq 0.$

Here, $c$ is small enough. Consider the function $f_{SL}(t)=t+f(t)$. Then
$f_{SL}$ is $\varepsilon$-distorted and we call it a Slide (in analogy to
translations).

Here are two examples: Take a function $f:\mathbb{R}^{2}\to\mathbb{R}$ and
suppose (A) holds. Then

$T_{x}=\begin{bmatrix}\cos f(|x|)&\sin f(|x|)\\\ -\sin f(|x|)&\cos
f(|x|)\end{bmatrix}.$

Take a function $f:\mathbb{R}^{3}\to\mathbb{R}$ and suppose (A) holds. Then

$T_{x}=\begin{bmatrix}1&0&0\\\ 0&\cos(f(|x|))&\sin(f(|x|))\\\
0&-\sin(f(|x|))&\cos(f(|x|))\\\ \end{bmatrix}.$

###  2.6 Slow twists: action

Here we illustrate the concept of a Slow twist on $\mathbb{R}^{2}$. Given a
$\varepsilon>0$ and a function $f:\mathbb{R}^{2}\to\mathbb{R}$ so that (A)
holds with the function $f$. Define the Slow twist matrix $T_{x}$ for any
$x\in\mathbb{R}^{2}$ via

$T_{x}:=\begin{bmatrix}\cos f(|x|)&\sin f(|x|)\\\ -\sin f(|x|)&\cos f(|x|)\\\
\end{bmatrix}.$

Then given any pure rotation $M\in\text{SO}(2)$, the following function
$f_{St}(x):=M^{T}T_{x}Mx:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is a Slow twist.

In $2$ dimensions, the rotation $M$ does not affect anything since rotations
are commutative on $\mathbb{R}^{2}$. However, for higher dimensions this is
not the case, and hence we leave them in the formulas, but for now we always
fix $M$ to be the identity matrix. For a first set of illustrations, we will
look only at one application of a Slow twist with $f$ being an exponential
function with differing scaling parameter.

Figure 3: Initial points lying on the line $y=x$, and the application of a
Slow twist with $f(x)=\exp(-c|x|)$, with $c=10$ (top left), $c=1$ (top
middle), and $c=0.1$ top right).

For large values of $c$ depending on $d$ the twist is near isometric, and even
outside a small enough cube centered at the origin, the points are left
essentially fixed. On the other hand, as $c$ tends to $0$, the twist becomes
closer to a pure rotation near the origin. Nevertheless, at a far enough
distance, the Slow twist $f_{ST}$ will leave the points essentially unchanged.
Indeed, the next figures illustrate this:

Figure 4: Large scale for Slow twists with $f(x)=\exp(-c|x|)$. Left: $c=1$,
the twist leaves the points essentially static outside $[-5,5]^{2}$; Right:
$c=0.1$, the twist only starts to leave the points static outside about
$[-30,30]^{2}$.

###  2.7 Fast twists

Let us pause to consider what happens when the decay condition on the twist
function $f$ (A) is not satisfied; in this case we will dub the twist function
a Fast twist for reasons that will become apparent presently.

Figure 5: Fast twist with function $f(x)=|x|$.

Figure 6: Fast twist with $f(x)=|x|^{2}$ for a small enough interval $[-3,3]$
(left) and large interval $[-10,10]$.

One can see that when $f$ is the identity function, the rate of twisting is
proportional to the distance away from the origin, and hence there is no way
that the twist function will leave points fixed outside of any ball centered
at the origin. Likewise, one sees from that the Fast twist with function
$f(x)=|x|^{2}$ rapidly degenerates points into a jumbled mess.

###  2.8 Iterated Slow twists

Here we illustrate what happens when one iteratively applies a Slow twist to a
fixed initial point.

Figure 7: Iterated Slow twist with $f(x)=\exp(-0.5|x|)$. Shown is the initial
points along the line $y=x$, $f_{St}(x)$, $f_{St}\circ f_{St}(x)$, and
$f_{St}\circ f_{St}\circ f_{St}(x)$.

We see an illustration of the fact that the composition of Slow twists remains
a Slow twist, but the distortion changes slightly; indeed, notice that as we
take more iterations of the exponential Slow twist, we must go farther away
from the origin before the new twist leaves the points unchanged.

[loop,autoplay,scale=0.5]1./Twist_015.

###  2.9 Slides: action

We illustrate some simple examples of Slides on $\mathbb{R}^{2}$.

First consider equally spaced points on the line $y=-x$, and the Slide given
by the function

$f(t):=\begin{bmatrix}\frac{1}{1+|t_{1}|^{2}}\\\ \\\
\frac{1}{2}e^{-|t_{2}|}\end{bmatrix}.$

This is illustrated in Figure 8.

Figure 8: Slide with the function $f$ given above.

To give some more sophisticated examples, we consider first the Slide function

$f(t):=\begin{bmatrix}e^{-|t_{1}|}\\\ \\\ e^{-0.1|t_{2}|}\end{bmatrix},$

acting iteratively on uniform points along both the lines $y=x$ and $y=-x$.

Figure 9: Lines $y=x$ and $y=-x$ along with $f_{SL}(x)$, $f_{SL}\circ f_{SL}$
and $f_{SL}\circ f_{SL}\circ f_{SL}(x)$ for $f$.

Similarly, the following figure shows the Slide function

$f_{2}(t):=\begin{bmatrix}1-e^{-|t_{1}|}\\\ \\\
1-e^{-0.1|t_{2}|}\end{bmatrix}$

acting iteratively on uniform points along the lines $y=x$ and $y=-x$.

Figure 10: Lines $y=x$ and $y=-x$ along with $f_{SL}(x)$, $f_{SL}\circ f_{SL}$
and $f_{SL}\circ f_{SL}\circ f_{SL}(x)$ for $f_{2}$.

###  2.10 Slides at different distances

To illustrate the effect of the distance of points from the origin, we
illustrate here how Slides affect uniform points on circles of different
radii.

We use again the asymmetric sliding function

$f(t)=\begin{bmatrix}\frac{1}{1+|t_{1}|^{2}}\\\ \\\
\frac{1}{2}e^{-|t_{2}|}\end{bmatrix}.$

Figure 11: Circles under 3 iterated Slides with the function $g$ above,
beginning with a circle of radius 1 (top left), 2 (top right) and 4 (bottom).

We see that the farther out the points are; i.e., the larger the radius of the
initial circle, the less the effect of the Slide, which makes sense given the
definition and the fact that the Slides must be $\varepsilon$–distortions of
$\mathbb{R}^{2}$.

###  2.11 3D motions

Here we illustrate some of the motions above in $\mathbb{R}^{3}$.

Here, we construct a generic rotation matrix in $SO(3)$ by specifying
parameters $a,b,a_{1},d$ satisfying $a^{2}+b^{2}+a_{1}^{2}+d^{2}=1$, and the
rotation matrix $M$ is defined by

$M=\begin{bmatrix}a^{2}+b^{2}-a_{1}^{2}-d^{2}&2(ba_{1}-ad)&2(bd+aa_{1})\\\
2(ba_{1}+ad)&a^{2}-b^{2}+a_{1}^{2}-d^{2}&2(a_{1}d-ab)\\\ 2(bd-
aa_{1})&2(a_{1}d+ab)&a^{2}-b^{2}-a_{1}^{2}+d^{2}\\\ \end{bmatrix}.$

As a reminder, our Slow twist on $\mathbb{R}^{3}$ is thus $M^{T}_{x}(Mx)$.

######  Example 1.

Our first example is generated by the rotation matrix $M$ as above with
parameters $a=b=\frac{1}{\sqrt{3}}$ and $a_{1}=d=\frac{1}{\sqrt{6}}$, and the
Slow twist matrix $St$ as

$\begin{bmatrix}1&0&0\\\ 0&\cos(f(|x|))&\sin(f(|x|))\\\
0&-\sin(f(|x|))&\cos(f(|x|))\\\ \end{bmatrix},$

where $f(t)=e^{-\frac{t}{2}}$. Figures 12 and 13 show two views of the twisted
motions generated by these parameters.

Figure 12: Slow twist in $\mathbb{R}^{3}$. Figure 13: Alternate View of Slow
twist in $\mathbb{R}^{3}$.

[loop,autoplay,scale=0.5]1./3DTwist_05

###  2.12 3D Slides

Here we generate 1000 random points on the unit sphere in $\mathbb{R}^{3}$ and
allow them to move under a Slide formed by

$f(x)=x+\begin{bmatrix}e^{-0.5|x_{1}|}\\\ e^{-|x_{2}|}\\\
e^{-\frac{3}{2}|x_{3}|}\\\ \end{bmatrix}.$

Figure 14: Anisotropic Slide on the $2$–sphere.

From the definition of Slow twists and Slides, the following holds.

######  Theorem 2.1.

  * (1)

Let $\varepsilon>0$ be small enough. There exists $\delta_{1}$ small enough
depending on $\varepsilon$ so that the following holds. Let $M\in SO(d)$ and
let $c_{1}<\delta_{1}c_{2}$. Then there exists a $\varepsilon$\- distorted
diffeomorphism $f$ with $f(x)=M(x),\,|x|\leq c_{1}$ and $f(x)=x,\,|x|\geq
c_{2}.$

  * (2)

Let $\varepsilon>0$. There exists $\delta_{1}$ small enough depending on
$\varepsilon$ so that the following holds. Let $A(x):=M(x)+x_{0}$ be a proper
Euclidean motion and let $c_{3}<\delta_{1}c_{4}$, $|x_{0}|\leq
c_{5}\varepsilon c_{3}$. Then there exists a $\varepsilon$\- distorted
diffeomorphism $f$ with $f(x)=A(x),\,|x|\leq c_{3}$ and $f(x)=x,\,|x|\geq
c_{4}$.

  * (3)

Let $\varepsilon>0$. There exists $\delta_{1}$ small enough depending on
$\varepsilon$ such that the following holds. Let $c_{6}\leq\delta_{1}c_{7}$
and let $x,x^{\prime}\in\mathbb{R}^{d}$ with $|x-x^{\prime}|\leq
c_{8}\varepsilon c_{6}$ and $|x|\leq c_{6}$. Then, there exists a
$\varepsilon$-distorted diffeomorphism $f$ with:

  * (a)

$f(x)=x^{\prime}$.

  * (b)

$f(y)=y,\,|y|\geq c_{7}.$

Moreover, we have:

######  Theorem 2.2.

  * •

Let $E\subset\mathbb{R}^{d}$ be a finite set wth ${\rm diameter}(E)\leq 1$.
There exists $\varepsilon$-distorted diffeomorphism
$\Phi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and $c,c^{\prime}$ depending on
$\varepsilon$ with $c$ small enough and $c^{\prime}$ large enough so that:

  * •

$\Phi$ coincides with an improper Euclidean motion on
$\left\\{x\in\mathbb{R}^{d}:\,{\rm dist}(x,E)\geq c\right\\}$.

  * •

$\Phi$ coincides with an improper Euclidean motion $A_{z}$ on
$B(z,c^{\prime})$ for each $z\in E$.

  * •

$\Phi(z)=z$ for each $z\in E$.

We have then:

######  Theorem 2.3.

Let $\varepsilon>0$ be smalll enough. Then there exist $\delta$ and
$\hat{\delta}$ depending on $\varepsilon$ small enough such that the following
holds. Let $E\subset\mathbb{R}^{d}$ be a collection of distinct $k\geq 1$
points $E:=\left\\{y_{1},...,y_{k}\right\\}$. Suppose we are given a function
$\phi:E\to\mathbb{R}^{d}$ with

$|x-y||(1+\delta)^{-1}\leq|\phi(x)-\phi(y)|\leq(1+\delta)|x-y|,\,x,y\in E.$
(2.2)

  * (1)

If $k\leq d$, there exists a $\varepsilon$-distorted diffeomorphism
$\Phi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ so that:

  * (a)

$\Phi$ agrees with $\phi$ on $E$.

  * (b)

Suppose $y_{i_{0}}=\phi(y_{i_{0}})$ for one $i=i_{0},\,1\leq i\leq k$. Then
$\Phi(x)=x$, $|x-y_{i_{0}}|\geq\hat{\delta}^{-1/2}{\rm
diam}\left\\{y_{1},...,y_{k}\right\\}$.

  * (2)

There exists $\delta_{1}$ such that the following holds. Let
$E\subset\mathbb{R}^{d}$ be a collection of distinct $k\geq 1$ points
$E:=\left\\{y_{1},...,y_{k}\right\\}$. Suppose that (2.2) holds with
$\delta_{1}$. There exists a Euclidean motion $A$ with

$|\phi(x)-A(x)|\leq\varepsilon{\rm diam}(E),\,x\in E.$ (2.3)

If $k\leq d$, then $A$ can be taken as proper.

An important observation in Theorem 2.3 is that if $y_{i}=\phi(y_{i})$ for one
$i=i_{0},\,1\leq i_{0}\leq k$, the extension $\Phi$ agrees with a Euclidean
motion away from the set $E$. What this says is that if the function $\phi$
has a fixed point at one of the points of $E$, then the function $\Phi$ must
be essentially rigid away from the set $E$. Note the unfortunate restriction
of $k\leq d$ in Theorem 2.3(1). The good news is that the problem for this
restriction occurs for only some degenerate cases and once removed, an
explicit relation between the distortion constants $\varepsilon$ and $\delta$
can be given.

In summary: See [2] for more details:

  * •

Removing the restriction of $k$ in Theorem 2.3(1). Removal of Degenerate
Cases. We assume the following:

  * •

The diameter of the set $E$ is not too large. More precisely: ${\rm
diameter}(E)\leq 1$.

  * •

The points of the set $E$ cannot be too close to each other. More precisely:
There exists $0<\eta<1$ so that $|x-y|>\eta$ for every distinct $x,y\in E$.

  * •

The points of the set $E$ are close to a hyperplane in $\mathbb{R}^{d}$. More
precisely: For a set of $l+1$ points in $\mathbb{R}^{d}$, with $l\leq d$, say
$z_{0},…,z_{l}$ we define $V_{l}(z_{0},...,z_{l}):={\rm vol}_{l\leq d}({\rm
simplex}_{l})$ where ${\rm simplex}_{l}$ is the $l$-simplex with vertices at
the points $z_{0},...,z_{l}$. Thus $V_{l}(z_{0},...,z_{l})$ is the $l\leq
d$-dimensional volume of the $l$-simplex with vertices at the points
$z_{0},...,z_{l}$. We write $V_{l}(E)$ as the maximum of
$V_{l}(z_{0},...,z_{l})$ over all points $z_{0},z_{1},...,z_{l}$ in $E$.
Assume the following: There exists $0<\eta<1$ so that $V_{l}(E)<\eta$.

  * •

Another interpretation: What is required is that on any $d+1$ of the $k$
points which form vertices of a relatively voluminous simplex, the mapping
$\phi$ is orientation preserving.

  * •

We then have: $\delta=\exp\left(-\frac{c}{\varepsilon}\right)$.

Regarding (2) in Theorem 2.3 , it is shown in [2] using tools in approximation
theory and algebraic geometry that if the points $E$ are forced lie on ellipse
of radius 1, then one can take $\delta=c\varepsilon^{c}$.

We now work with compact sets in open sets in $\mathbb{R}^{n}$ and achieve the
optimal $\delta=c\varepsilon$.

We work with a $C^{1}$ map $\phi:U\to\mathbb{R}^{n}$ with
$U\subset\mathbb{R}^{n}$ be an open set. For a compact set $E$ in
$\mathbb{R}^{n}$, and $x\in{\mathbb{R}}^{n}$, we write for convenience
$d(x):={\rm dist}(x,E)$. Let $\varepsilon>0$.

We assume the following:

  * (a)

Geometry of $E$: For certain positive constants $c_{0}$, $C_{1}$, $c_{2}$
depending on $n$, the following holds: Let $x\in{\mathbb{R}}^{n}\backslash E$.
If $d(x)\leq c_{0}\,{\rm diam}(E)$, then there exists a ball $B(z,r)\subset E$
such that $|z-x|\leq c_{1}\,d(x)$ and $r\geq c_{2}\,d(x)$.

  * (b)

Geometry of $\phi$: For $x,y\in E$,
$|x-y|(1-\varepsilon)\leq|\phi(x)-\phi(y)|\leq(1+\varepsilon)|x-y|.$

  * (c)

Controlled constants: $\varepsilon$ is smaller than a small positive constant
determined by $c_{i}$, $i=0,1,2$ and $n$.

Here is the main result from [2].

######  Theorem 2.4.

Under the above assumptions, there exists a $C^{1}$ function
$\Phi:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ and a Euclidean motion
$A:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$, with the following properties:

  * •

$(1-c^{\prime}\varepsilon)\left|x-y\right|\leq\left|\Phi(x)-\Phi(y)\right|\leq(1+c^{\prime}\varepsilon)\left|x-y\right|$
for all $x,y\in{\mathbb{R}}^{n}$. $c^{\prime}$ determined by $c_{i},\,i=1,2,3$
and $n$.

  * •

$\Phi=\phi$ in a neighborhood of $E$.

  * •

$\Phi=A$ outside $\left\\{x\in{\mathbb{R}}^{n}:d(x)<c_{0}\right\\}$.

  * •

$\Phi:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is one-to-one and onto.

  * •

If $\phi\in C^{m}(U)$ for some given $m\geq 1$, then $\Phi\in
C^{m}\left({\mathbb{R}}^{n}\right)$.

  * •

If $\phi\in C^{\infty}(U)$, then $\Phi\in C^{\infty}({\mathbb{R}}^{n})$.

Moving forward we are now going to survey some of our work on weighted
$L_{p}(\mathbb{R}),\,1<p\leq\infty$ convergence of orthonormal expansions in
$\mathbb{R}$, see [12]. Our survey is motivated by our discussion above under
the title "Fourier series versus Taylor series", the work of [13] (orthonormal
expansions) and the work on Taylor series (Section 1.2) both of which imply
Whitney extension theorems.

##  3 Orthonormal expansions on the line $\mathbb{R}$

For a positive measure $d\alpha$ on the real line with infinitely many points
its support and with all power moments $\int x^{l}\alpha(t),\,l=0,1,2,...$
finite, the Gram-Schmidt process then generates orthonormal polynomials
$p_{n}(x)=p_{n}(d\alpha)(x)=\gamma_{n}x^{n}+...+\gamma_{n},$ satisfying $\int
p_{n}(t)p_{l}(t)d\alpha(t)=\delta_{ln}$. ∥∥∥Recall from the beginning that we
always have $n\geq 1$. Here,

$\delta_{l,n}:=\left\\{\begin{array}[]{ll}0,&l\neq n\\\
1,&l=n\end{array}\right.$

Here we set $p_{-1}=0$, $p_{0}=\left(\int d\alpha(t)dt\right)^{-1/2}$. The
orthonormal polynomials satisfy the recurrence.

$xp_{n}(x)=\alpha_{n}p_{n+1}(x)+\alpha_{n-1}p_{n-1}(x),\,x\in\mathbb{R}$

where $\alpha_{n}:=\frac{\gamma_{n-1}}{\gamma_{n}}$, $\gamma_{0}=1$ and ${\rm
lim}_{n\to\infty}\frac{\alpha_{n+1}}{\alpha_{n}}=1.$ Now for functions
$f:\mathbb{R}\to\mathbb{R}$ for which $f(t)t^{l},\,l\geq 0$ is integrable with
respect to $d\alpha$, we can form the formal orthornomal polynomial expansion
$S(f):=S(f,d\alpha)=\sum_{l=0}^{\infty}c_{l}p_{l}$ where for $1\geq
0,\,c_{l}:=\int fp_{l}d\alpha$ and by
$S_{n}(f):=S_{n}(f,d\alpha)=\sum_{j=0}^{n-1}c_{j}p_{j}$, the $(n-1)$th partial
sum of $S(f)$. We are interested in studying how to understand for fixed $f$,
convergence of $S_{n}(f)$ to $S(f)$ as $n\to\infty$.

##  4 Translation to weighted $L_{p}(\mathbb{R}),\,1<p\leq\infty$ convergence
of $S_{n}(f)$ to $S(f)$ as $n\to\infty$ for fixed admissible $f$.

We set $d\alpha(t)=w(t)^{2}dt$ with $w$ a positive even function on
$\mathbb{R}$ of smooth polynomial decay for large argument. For
$1<p\leq\infty$ and fixed weight $w$, we will always work with functions $f$
so that the norm below is finite. We will call such functions admissible.

$|fw|_{p(\mathbb{R})}:=\left\\{\begin{array}[]{ll}{\rm
sup}_{x\in\mathbb{R}}|fw(x)|,&p=\infty\\\
\left(\int_{\mathbb{R}}|(fw)(x)|^{p}dx\right)^{1/p},&1<p<\infty.\end{array}\right.$

Given $u,x\in\mathbb{R}$, we set

$\Delta_{u}(f)(x):=f(x+u)-f(x)$

and for a fixed $\gamma\in\mathbb{R}$, we will also define

$u_{\gamma}(y):=(1+|y|)^{\gamma},\,y\in\mathbb{R}.$

###  4.1 A class of admissible weights.

We work with the following class of admissible weights. See [22, 23, 12].

######  Definition 4.5.

A weight function $w={\rm exp}(-Q):\mathbb{R}\to(0,\infty)$ will be called
admissible if each of the following conditions below is satisfied:

  * (a)

$Q:={\rm log}(1/w)$ is continuously differentiable, even and satisfies
$Q(0)=0$;

  * (b)

$Q^{\prime}$ is nondecreasing in $\mathbb{R}$ with

$\lim_{x\to\infty}Q(x)=\lim_{x\to-\infty}Q(x)=\infty.$

Assume that there exists $\eta>1$ with

$\eta<\frac{xQ^{\prime}(x)}{Q(x)}\leq
C,\,x\in\mathbb{R}\backslash\left\\{0\right\\}.$

  * (c)

For every $\varepsilon>0$, there exists $\delta>0$ such that for every
$x\in\mathbb{R}\backslash\\{0\\}$,

$\int_{x-\delta|x|}^{x+\delta|x|}\frac{|Q^{\prime}(s)-Q^{\prime}(x)|}{|s-x|^{3/2}}ds\leq\varepsilon|Q^{\prime}(x)|.$

Definition (4.5), defines a general class of even weights for which our
results hold. ******Later, we will look at a paper [13] which shows a deep
connection between Whitney extensions and orthogonal expansions for Laguerre
orthogonal polynomials. The weak regularity and smoothness conditions on $w$
above are needed, for bounds on $p_{n}$, and its zeroes and are used heavily
in our proofs of the results below taken from [12]. Note that Definition (4.5)
does not require $Q^{\prime\prime}$ to exist. Instead, we require only a mild
local Lipshitz $1/2$ condition on $Q^{\prime}$. Note that $Q$ grows as a
polynomial for large argument and $w$ is of smooth polynomial decay for large
argument. We note, as an easily absorbed example, that $w_{\beta}(x):={\rm
exp}\left(-|x|^{\beta}\right),\,\beta>1,\,x\in\mathbb{R}$ is an example of an
admissible weight. The case $\beta=2$ is just the Hermite weight. See [22].

###  4.2 The numbers $a_{u}$ and $\alpha_{n}$

In studying weighted polynomial approximation for admissible weights $w$ on
the line, an important role is played by the scaled endpoints $\pm a_{u}$ of
the support of the equilibrium measure for $w^{2}$ (see for example [12, 22,
23]) and the asymptotic behavior of the quotient $\frac{\alpha_{n}}{a_{n}}$.
More precisely:

Given $u>0$, we define the real number $a_{u}$ by the positive root of the
equation

$u=\frac{2}{\pi}\int_{0}^{1}\frac{a_{u}tQ^{\prime}(a_{u}t)}{\sqrt{1-t^{2}}}dt,\,u>0.$

It is known, see [22, 23, 12], that $a_{u}$ is uniquely defined, strictly
increasing in $(0,\infty)$ with

$\lim_{u\to\infty}a_{u}=\infty$

and of polynomial growth for large argument. For example, for the weight,
$w_{\beta}(x):={\rm exp}\left(-|x|^{\beta}\right)$ above , it is known that

$a_{u}\sim u^{1/\beta}.$

One of the important properties of the number $a_{u}$ which allows the
analysis the convergence of $S_{n}(f)$ to $S(f)$ in $L_{p}(\mathbb{R})$ is the
fact that, see [22, 23, 12], that

$|Pw|_{\infty(\mathbb{R})}=|Pw|_{\infty[-a_{n},a_{n}]}$

and

$|(Pw)(x)|_{\infty(|x|\geq sa_{n})}\leq\exp(-cn)|Pw|_{\infty[-a_{n},a_{n}]}$

for every fixed $s>1$ and for every polynomial $P$ of degree at most $n$ with
$1<p<\infty$ analogues. By differentiation, it not difficult to see why the
above should hold for ${\rm exp}\left(-|x|^{\beta}\right)$. Indeed, these
"Infinite-finite" inequalities allow a shift in various weighted polynomial
approximations on $\mathbb{R}$ to approximations on sequences of compact
intervals. It is also known, see [22, 23, 12] that
$\lim_{n\to\infty}\frac{\alpha_{n}}{a_{n}}=1/2$.

###  4.3 Main Results in [12]

Following are the main results of [12].

####  4.3.1 Necessary results: $L_{\infty}(\mathbb{R})$

######  Theorem 4.6.

Let $w$ be an admissible weight and $B,b\in\mathbb{R}$ with $b<B$. Then for

$\displaystyle{\rm sup}_{n\in\Omega}|S_{n}(f)wu_{b}(x)|$ (4.4)
$\displaystyle\leq c|fwu_{B}|_{\infty(\mathbb{R})}$

to hold for some infinite subsequence $\Omega\subseteq\mathbb{N}$, for all
$x\in\mathbb{R}$ and admissible $f$ for which the right-hand side of (4.4) is
finite it is necessary that

$B>0$ (4.5)

and uniformly

$a_{n}^{b-{\rm min}\left\\{B,1\right\\}}n^{1/6}C_{B,n}=O(1)$ (4.6)

where

$C_{B,n}:=\left\\{\begin{array}[]{ll}1,&B\neq 1\\\ \log
n,&B=1.\end{array}\right.$

####  4.3.2 Sufficiency results: $L_{\infty}(\mathbb{R})$

######  Theorem 4.7.

Let $w$ be an admissible weight, $b\leq 0$ and assume (4.5) and (4.6). Let
$x\in\mathbb{R}$ and assume moreover that

$\frac{\alpha_{n+1}}{\alpha_{n}}=1+O\left(\frac{1}{n}\right),\qquad
n\to\infty.$ (4.7) ††††††(4.7) and (4.9) are known for classes of smooth
admissible weights and the weight $w_{2n}$, see [5, 11]

. Then there exists an infinite subsequence $\Omega\subseteq\mathbb{N}$ so
that for $n\in\Omega$

$\displaystyle|(S_{n}(f)wu_{b})(x)|$ (4.8) $\displaystyle\leq c\left[{\rm
log}n|fwu_{B}|_{\infty(\mathbb{R})}+\int_{-1}^{1}\left|\frac{[wu_{b}\Delta_{y}(f)](x)}{y}dy\right|\right]$

for admissible $f$ for which the right-hand side of (4.8) is finite. If in
addition, (4.9) holds below,

$\frac{\alpha_{n}}{a_{n}}=\frac{1}{2}\left[1+O\left(\frac{1}{n^{2/3}}\right)\right],\qquad
n\to\infty,$ (4.9)

then for $n\geq 1$

$\displaystyle|(S_{n}(f)wu_{b})(x)|$ (4.10) $\displaystyle\leq c\left[{\rm
log}n|fwu_{B}|_{\infty(\mathbb{R})}+\int_{-1}^{1}\left|\frac{[wu_{b}\Delta_{y}(f)](x)}{y}dy\right|\right]$

for all admissible $f$ for which (4.10) is finite.

######  Theorem 4.8.

Let $w$ be admissible, $x\in\mathbb{R}$, $b\leq 0$ and assume (4.5)-(4.7).
Then there exists an infinite subsequence $\Omega\subseteq\mathbb{N}$ such
that for $n\in\Omega$

$|S_{n}[f]wu_{b}(x)|\leq c\left[{\rm
log}n|fwu_{B}|_{\infty(\mathbb{R})}+|f^{\prime}wu_{b}|_{\infty(\mathbb{R})}\right]$
(4.11)

for admissible $f:\mathbb{R}\to\mathbb{R}$ for which the right side of (4.11)
is finite. Moreover, if (4.9) holds in addition, then for $n\geq 1$,

$|S_{n}[f]wu_{b}(x)|\leq c\left[{\rm
log}n|fwu_{B}|_{\infty(\mathbb{R})}+|f^{\prime}wu_{b}|_{\infty(\mathbb{R})}\right]$
(4.12)

for all admissible functions $f:\mathbb{R}\to\mathbb{R}$ for which the right-
hand side of (4.12) holds.

Note that (4.8) and (4.11) are for a subsequence whereas (4.10) and (4.12) is
for all $n\geq 1$.

Now we look at $L_{p}$ analogues of Theorems (4.6-4.7).

###  4.4 $L_{p}(\mathbb{R})$ results

######  Theorem 4.9.

Let $w$ be admissible, $b,B\in\mathbb{R}$ with $b\leq B$, $1<p<\infty$ and
$C_{B,n}$ as defined in (4.6). Then for

${\rm sup}_{n\in\Omega}|S_{n}[f]wu_{b}|_{p(\mathbb{R})}\leq
c|fwu_{B}|_{p(\mathbb{R})}$ (4.13)

to hold for some infinite subsequence $\Omega\subseteq\mathbb{N}$ and for all
admissible $f$ for which (4.13) is finite it is necessary that

$b<1-1/p,\,B>-1/p.$ (4.14)

In addition:

  * (a)

If $p<4/3$ then necessarily,

$a_{n}^{{\rm max}\left\\{b,-1/p\right\\}-B}n^{1/6(4/p-3)}C_{B,n}=O(1).$

  * (b)

If $p=4/3$ or $4$ then necessarily $b<B$.

  * (c)

If $p>4$ then necessarily

$a_{n}^{b-{\rm min}\left\\{B,1-1/p\right\\}}n^{1/6(1-4/p)}C_{B,n}=O(1).$

Moreover, (4.7), (4.14), (a-c) are sufficient to give (4.13) for all
admissible $f$ for which the right-hand side of (4.13) is finite. In addition,
if (4.9) also holds then

${\rm sup}_{n\geq 1}|S_{n}[f]wu_{b}|_{p(\mathbb{R})}\leq
c|fwu_{B}|_{p(\mathbb{R})}.$ (4.15)

and

$\lim_{n\to\infty}|(S_{n}[f]-f)wu_{b}|_{p(\mathbb{R})}=0$ (4.16)

for all continuous $f:\mathbb{R}\to\mathbb{R}$ satisfying

$\lim_{|x|\to\infty}|fwu_{B+\delta}(x)|=0$ (4.17)

for some $\delta>1$.

##  5 Orthonormal expansions to Whitney

In this section, we illustrate an interesting connection between certain
Whitney extensions and certain Laguerre polynomial orthonormal expansions
detailed in the paper [13]. By $\mathbb{R}_{+}^{d}$, we mean the usual
cartesian product space $(0,\infty)^{d}$, by $\overline{\mathbb{R}_{+}^{d}}$
its closure $[0,\infty)^{d}$ and similary define ${\mathbb{N}}_{0}^{d}$ with
${\mathbb{N}}_{0}:={\mathbb{N}}\cup\left\\{0\right\\}$.
$S(\mathbb{R}_{+}^{d})$ consists of all real valued functions $f\in
C^{\infty}(\mathbb{R}_{+}^{d})$ such that all derivatives
$D^{p}f,\,p\in{\mathbb{N}}_{0}^{d}$ extend to continuous functions on
$\overline{\mathbb{R}_{+}^{d}}$ and satisfying ${\rm
sup}_{x\in{\mathbb{R}}_{+}^{d}}x^{k}|D^{p}f(x)|<\infty,\,\forall
k,p\in{\mathbb{N}}_{0}^{d}.$ The space $S(\mathbb{R}^{d})$ is defined
similarly. The main result in the paper [13] deals with the existence of a
Whitney extension of the space $S(\mathbb{R}_{+}^{d})$ onto the space
$S(\mathbb{R}^{d})$ . A crucial tool in dealing with this problem are certain
Laguerre polynomial orthonormal expansions.

We recall for $n=0,1,2,...$, the functions
$P_{n}(x):=\frac{e^{x}}{n!}\left(\frac{d}{dx}\right)^{n}(e^{-x}x^{n}),\,x>0$
are the Laguerre polynomials, ${\cal P}_{n}(x):=P_{n}(x)e^{x/2}$ are the
Laguerre functions and the set $\left\\{{\cal P}_{n}(x),\,n=0,1,...\right\\}$
is an orthonormal basis for $L^{2}(0,\infty)$, the space of squared integrable
functions on $(0,\infty)$. [30].

In [13], the following is proved: Convergence of the Laguerre series in the
space $S(\mathbb{R}_{+}^{d})$ and this is then used as a tool to obtain the
existence of a Whitney extension of the space $S(\mathbb{R}_{+}^{d})$ onto the
space $S(\mathbb{R}^{d})$.

  * (1)

Convergence of the Laguerre series in the space $S(\mathbb{R}_{+}^{d})$: For
$f\in S(\mathbb{R}_{+}^{d})$, let
$a_{n}(f):=\int_{\mathbb{R}_{+}^{d}}f(x){\cal P}_{n}(x)dx.$ Then
$f(x)=\sum_{n\in{\mathbb{N}}_{0}^{d}}a_{n}(f){\cal P}_{n}(x)$ converges
absolutely for all $x\in S(\mathbb{R}_{+}^{d})$.

  * (2)

Existence of a Whitney extension of the space $S(\mathbb{R}_{+}^{d})$ onto the
space $S(\mathbb{R}^{d})$: The restriction mapping $f\to
f|_{\mathbb{R}_{+}^{d}}:\,S(\mathbb{R}^{d})\to S(\mathbb{R}_{+}^{d})$ is a
topological onto homomorphism. Moreover, the space $S(\mathbb{R}_{+}^{d})$ is
topologically isomorphic to the quotient space $S(\mathbb{R}^{d})/N$ where

$N:=\left\\{f\in S(\mathbb{R}^{d})\,|\,{\rm
supp}\,f\subseteq(\mathbb{R}^{d}\setminus\mathbb{R}_{+}^{d})\right\\}.$

##  6 Open Questions

  * (a)

It would be of interest to develop a framework for the ideas in Section (5) or
the near Whitney extension problem using the work in Section (4) replacing
Laguerre polynomials by the weighted orthogonal polynomials and subsequent
orthonormal expansion theory there.

  * (b)

It would also be of interest to do the same in (a) except using weighted
orthogonal polynomials with respect to different fast decreasing weights over
different domains, for example the even Erdos type weights over $\mathbb{R}$,
of faster than smooth polynomial decay with large argument, the even Pollazek
weights which are of faster than smooth polynomial decay close to the
endpoints of $(-1,1)$, classes of exponential (possibly non-symmetric weights)
on real intervals $(a,b)$ with $a<0<b$ and other classical orthogonal
polynomials on $\mathbb{R}^{n}$. See [6, 22, 25, 26]

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