\documentclass{procimm}

\title[On the H\"older continuity \ldots]{On the H\"older continuity in ridge function representation}

\author{Rashid A. Aliev}
\address{Baku State University, Baku, AZ 1148, Azerbaijan,}
\address{Institute of Mathematics and Mechanics, NAS of Azerbaijan, Baku, AZ 1141, Azerbaijan}
\email{aliyevrashid@mail.ru}

\author{Aysel A. Asgarova}
\address{Azerbaijan University of Languages, Baku, AZ 1014, Azerbaijan}
\email{asgarova2016@mail.ru}

\author{Vugar E. Ismailov}
\address{Institute of Mathematics and Mechanics, NAS of Azerbaijan, Baku, AZ 1141, Azerbaijan}
\email{vugaris@mail.ru}

\keywords{ridge function, H\"older continuity, Cauchy functional equation}

\subjclass[2010]{26B40, 39B22}

\begin{document}

\begin{abstract}
In this paper, we show that if a multivariate function is locally H\"older
continuous of some degree and represented by a sum of arbitrarily behaved
ridge functions, then, under a suitable condition, it can be also
represented by a sum of ridge functions, which are locally H\"older
continuous of the same degree.
\end{abstract}

\maketitle

\section{Introduction} \label{s1}

A \textit{ridge function} is a multivariate function of the form%
\[
g\left( \mathbf{a}{\cdot }\mathbf{x}\right) =g\left( a_{1}x_{1}+\ldots
+a_{m}x_{m}\right) ,
\]%
where $g:\mathbb{R}\rightarrow \mathbb{R}$, $\mathbf{a}=\left( {a_{1},...,a}%
_{m}\right) $ is a fixed vector (direction) in $\mathbb{R}^{m}\backslash
\left\{ \mathbf{0}\right\} ,$ $\mathbf{x}=(x_{1},...,x_{m})$ is the variable
and $\mathbf{a}{\cdot }\mathbf{x}$ is the usual inner product in $\mathbb{R}%
^{m}$. In the theory of partial differential equations, ridge functions have
been known under the name of \textit{plane waves (}see, e.g., \cite{69}).
The term \textquotedblleft ridge function" was devised by Logan and Shepp in
their pioneering paper \cite{97} dedicated to the mathematics of
computerized tomography (see also \cite{72,74,106,111}). After a 1981 paper
by Friedman and Stuetzle \cite{33} ridge functions started to appear also in
statistics, especially, in the theory of projection pursuit and projection
regression (see, e.g., \cite{27,33,42}). The general idea therein was to
reduce \textquotedblleft dimension" and thus bypass the \textquotedblleft
curse of dimensionality".

Ridge functions are used in many models in neural network theory. For
example, in one of the popular models called MLP (multilayer feedforward
perceptron) model, the simplest case considers functions of the form

\[
\sum_{i=1}^{r}c_{i}\sigma (\mathbf{w}^{i}\mathbf{\cdot x}-\theta _{i}).
\]%
Here the weights $\mathbf{w}^{i}$ are vectors in $\mathbb{R}^{m}$, the
thresholds $\theta _{i}$ and the coefficients $c_{i}$ are real numbers and \
the activation function $\sigma $ is a univariate function. Note that for
each $\theta \in \mathbb{R}$ and $\mathbf{w}\in \mathbb{R}^{m}\backslash \{%
\mathbf{0}\}$ the function

\[
\sigma (\mathbf{w\cdot x}-\theta )
\]%
is a ridge function. For a detailed survey of approximation theory of the
MLP model see \cite{119}.

Ridge functions are interesting also to approximation theorists. In
approximation theory, these functions are implemented as an effective and
convenient tool for approximating complicated multivariate functions (see,
e.g., \cite{I1,I2,I3,52,88,95,102,117,118}).

In this paper, we consider the problem of representation by sums of ridge
functions with $r,$ $r\geq 1,$ fixed directions. Let the directions $\mathbf{%
a}^{i}\in \mathbb{R}^{m}\backslash \left\{ \mathbf{0}\right\} ,$ $i=1,...,r,$
be given and pairwise linearly independent. The first problem arising here
is about the representability of a given multivariate function $f:\mathbb{R}%
^{m}\rightarrow \mathbb{R}$ as a sum of ridge functions with the directions $%
\mathbf{a}^{i},$ $i=1,...,r$. In other words, we want to know when the
function $f$ can be written in the form%
$$
f(\mathbf{x})=\sum_{i=1}^{r}g_{i}(\mathbf{a}^{i}\cdot \mathbf{x}).\eqno(1.1)
$$%
This problem has a simple solution if $f$ depends on two variables and has
partial derivatives up to $r$-th order. For the representation of $f(x,y)$
in the following form%
\[
f(x,y)=\sum_{i=1}^{r}g_{i}(a_{i}x+b_{i}y),
\]%
it is necessary and sufficient that

$$
\prod\limits_{i=1}^{r}\left( b_{i}\frac{\partial }{\partial x}-a_{i}\frac{%
\partial }{\partial y}\right) f=0.\eqno(1.2)
$$%
Note that the last assertion is valid also for continuous functions of two
variables provided that the derivatives are understood in the sense of
distributions. It should be remarked that this simple assertion is not
directly generalized to the case when $f$ depends on more than two variables.

Assume we know that a function $f(\mathbf{x})\ $can be represented in the
form (1.1). Assume in addition that $f$ is of the class $C^{k}(\mathbb{R}%
^{m})$. What can we say about $g_{i}$? Can we say that $g_{i}\in C^{k}(%
\mathbb{R})$? The case $r=1$ is obvious. In this case, if $f\in C^{k}(%
\mathbb{R}^{m})$, then for $\mathbf{c}\in \mathbb{R}^{m}$ satisfying $%
\mathbf{a}^{1}\cdot \mathbf{c}=1$ we have that $g_{1}(t)=f(t\mathbf{c)}$ is
in $C^{k}(\mathbb{R}).$ The same argument can be carried out for the case $%
r=2.$ In this case, since the vectors $\mathbf{a}^{1}$ and $\mathbf{a}^{2}$
are linearly independent, there exists a vector $\mathbf{c}\in \mathbb{R}%
^{m} $ satisfying $\mathbf{a}^{1}\cdot \mathbf{c}=1$ and $\mathbf{a}%
^{2}\cdot \mathbf{c}=0.$ Therefore, we obtain that the function $g_{1}(t)=f(t%
\mathbf{c)}-g_{2}(0)$ is in the class $C^{k}(\mathbb{R})$. Similarly, one
can verify that $g_{2}\in C^{k}(\mathbb{R})$ (see \cite{A1}).

The above question becomes quite difficult if the number of directions $%
r\geq 3$. For $r=3$, there are many smooth functions which decompose into
sums of very badly behaved ridge functions. This is a consequence of the
classical Cauchy Functional Equation (CFE). This equation is defined as%
$$
h(x+y)=h(x)+h(y),\text{ }h:\mathbb{R\rightarrow R}\text{,}\eqno(1.3)
$$%
which has a class of simple solutions $h(x)=cx,$ $c\in \mathbb{R}$. However,
it easily follows from the Hamel basis theory that CFE has also a large
class of badly behaved solutions. These solutions are called
\textquotedblleft badly behaved" because they are weird over reals. They
are, for example, not continuous at a point, not monotone at an interval,
not bounded at any set of positive measure (see, e.g., \cite{1}). Let $h_{1}$
be any such solution of the equation (1.3). Then the zero function can be
written as%
$$
0=h_{1}(x)+h_{1}(y)-h_{1}(x+y).\eqno(1.4)
$$%
Note that the functions involved in (1.4) are ridge functions with the
directions $(1,0)$, $(0,1)$ and $(1,1)$ respectively. This particular
example shows that for smoothness of the representation (1.1) one must
impose additional conditions on the functions $g_{i},$ $i=1,...,r.$

It was first proved by Buhman and Pinkus \cite{12} that if in (1.1) $f\in
C^{k}(\mathbb{R}^{m})$, $k\geq r-1$ and $g_{i}\in L_{loc}^{1}(\mathbb{R)}$
for each $i$, then $g_{i}\in C^{k}(\mathbb{R)}$ for $i=1,...,r.$ In \cite%
{Pinkus} Pinkus found a strong relationship between CFE and the problem of
smoothness in ridge function representation. He generalized extensively the
previous result of Buhman and Pinkus \cite{12}. He showed that the solution
is quite simple and natural if the functions $g_{i}$ are taken from a class $%
\mathcal{B}$ of real-valued functions $u$ defined on $\mathbb{R}$. By
definition, $u$ is in $\mathcal{B}$ if for any function $v\in C(\mathbb{R)}$
for which $u-v$ satisfies CFE, $u-v$ is linear, i.e. $u(x)-v(x)=cx,$ where $%
c\in \mathbb{R}$ (see \cite{Pinkus}). The result of Pinkus states that if in
(1.1) $f\in C^{k}(\mathbb{R}^{m})$ and each $g_{i}\in \mathcal{B}$, then
necessarily $g_{i}\in C^{k}(\mathbb{R)}$ for $i=1,...,r.$

The above representation problem was also considered by Konyagin and
Kuleshov \cite{K1,K2} and by Kuleshov \cite{K3}. They mainly analyze the
continuity of the representation, that is, the question if and when
continuity of $f$ in (1.1) guarantees the continuity of $g_{i}$. There are
also other results concerning the smoothness of ridge function
representation generalizing the above result of Pinkus (see \cite{K3}). The
results in \cite{K1,K2,K3} involve certain subsets (convex open sets, convex
bodies, etc.) of $\mathbb{R}^{m}$ instead of only $\mathbb{R}^{m}$ itself.

The results of Pinkus \cite{Pinkus} gave rise to the following natural and
important problem. Assume in the representation (1.1) $f\in C^{k}(\mathbb{R}%
^{m}),$ but the functions $g_{i}$ are arbitrarily behaved (that is, we allow
very badly behaved functions). Can we write $f$ as a sum $%
\sum_{i=1}^{r}f_{i}(\mathbf{a}^{i}\cdot \mathbf{x})$ but with the $f_{i}\in
C^{k}(\mathbb{R})$, $i=1,...,r$? This problem was posed in \cite{12} and
\cite{P}. In \cite{A1}, Aliev and Ismailov obtained a partial solution to this
problem. Their solution comprises the cases in which $k>1$ and $r-1$
directions of given $r$ directions are linearly independent. Note that such
condition on directions is satisfied by default if we are given three
directions, as it is assumed that all the directions are pairwise linearly
independent. The representation problem in the case of three directions was
initially considered in \cite{A3}. For bivariate functions having the degree
of smoothness $k\geq r-2,$ the problem was completely solved in \cite{A2}.

Kuleshov \cite{K3} generalized Aliev and Ismailov's result \cite{A1} to the other possible cases of $k$. That is, he proved that if a function $f\in C^{k}(\mathbb{R}%
^{m})$, where $k \geq 0$, is of the form (1.1) and $r-1$-tuple of the given set of $r$ directions $\mathbf{a}^{i}$ forms a linearly independent system, then in (1.1) the functions $g_i$ can be replaced with functions $f_{i}\in C^{k}(\mathbb{R})$, $i=1,...,r$ (see \cite[Theorem 3]{K3}). In
\cite{A0}, we proved this result using completely different ideas. Note that our proof contains a theoretical method for constructing the mentioned functions $f_{i}\in C^{k}(\mathbb{R})$ (see \cite[Theorem 2.1, Theorem 2.2]{A0}). Using this method, we also estimated the modulus of continuity of $f_{i}$ in terms of the modulus
of continuity of $f$ (see \cite[Remark 2]{A0}).

In this paper, we continue investigations on ridge function representations.
The question considered here is as follows. Assume $f$ is H\"{o}lder
continuous on compact subsets of $\mathbb{R}^{m}$ and possesses the
representation (1.1). Can we replace $g_{i}$ with H\"{o}lder continuous
functions of the same degree as $f$? We answer this question positively in
the case when $r-1$ directions of given $r$ directions are linearly
independent. First we prove this when the space dimension $m=2$ and the
number of directions $r=3.$ Then based on the multidimensional techniques
exploited in our previous paper \cite{A0}, we formulate the main result in
the general case. Note that the main result of this paper (see Theorem 2.2)
cannot be obtained directly from the above mentioned results.


\section{Main result}

\label{s2}

We start this section with the following well-known definition.

\begin{definition} We say that a function $F:\mathbb{R}^{m}\rightarrow
\mathbb{R}$ is locally H\"older continuous with degree $\alpha $, $0<\alpha
\leq 1$, if for any compact set $K\subset \mathbb{R}^{m}$ there is a number $%
M=M\left( F;\alpha ;K\right) >0$ such that for any $\mathbf{x}=\left(
x_{1},...,x_{m}\right) \in K$ and $\mathbf{y}=\left( y_{1},...,y_{m}\right)
\in K$ the inequality $\left\vert F\left( \mathbf{x}\right) -F\left( \mathbf{%
y}\right) \right\vert \leq M\cdot \sum_{i=1}^{m}\left\vert
x_{i}-y_{i}\right\vert ^{\alpha }$ holds.
\end{definition}

The class of locally H\"older continuous functions with degree $\alpha $ is
denoted by $H_{\alpha }^{\left( loc\right) }\left( \mathbb{R}^{m}\right) .$
For a function $F\in H_{\alpha }^{\left( loc\right) }\left( \mathbb{R}%
^{m}\right) $ and a compact set $K\subset \mathbb{R}^{m}$ put
\[
H\left( F;\alpha ;K\right) =\sup \left\{ \left\vert F\left( \mathbf{x}%
\right) -F\left( \mathbf{y}\right) \right\vert \cdot \left[
\sum_{i=1}^{m}\left\vert x_{i}-y_{i}\right\vert ^{\alpha }\right] ^{-1}:\,%
\mathbf{x},\mathbf{y}\in K,\,\mathbf{x}\neq \mathbf{y}\right\} .
\]

The following lemma plays a key role in the proof of our main result.

\begin{lemma} Assume a function $F\in H_{\alpha }^{\left( loc\right) }\left(
\mathbb{R}^{2}\right) $, $0<\alpha \leq 1,$ has the form
\[
F\left( x,y\right) =h\left( x\right) +h\left( y\right) -h\left( x+y\right) ,
\]%
where $h:\mathbb{R}\rightarrow \mathbb{R}$ is an arbitrarily behaved
function. Then there exists a function $f\in H_{\alpha }^{\left( loc\right)
}\left( \mathbb{R}\right) $ such that
\[
F\left( x,y\right) =f\left( x\right) +f\left( y\right) -f\left( x+y\right) .
\]%
\end{lemma}

\textbf{Proof.} Consider the functions
\[
G\left( x,y\right) =F\left( x,y\right) -F\left( 0,0\right) ,~g\left(
x\right) =h\left( x\right) -F\left( 0,0\right) .
\]%
Then the function $G\left( x,y\right) $ also belongs to the class $H_{\alpha
}^{\left( loc\right) }\left( \mathbb{R}^{2}\right) $ and we have the
equalities
\begin{equation}
G\left( x,y\right) =g\left( x\right) +g\left( y\right) -g\left( x+y\right)
,~G\left( 0,0\right) =0.  \label{GrindEQ__1_}
\end{equation}%
It follows from \eqref{GrindEQ__1_} that
\[
g\left( 0\right) =g\left( 0\right) +g\left( 0\right) -g\left( 0\right)
=G\left( 0,0\right) =0
\]%
and for any $x\in \mathbb{R}$
\begin{equation}
G\left( x,0\right) =g\left( x\right) +g\left( 0\right) -g\left( x\right)
=g\left( 0\right) =0,  \label{GrindEQ__2_}
\end{equation}%
\begin{equation}
G\left( 0,x\right) =g\left( 0\right) +g\left( x\right) -g\left( x\right)
=g\left( 0\right) =0,  \label{GrindEQ__3_}
\end{equation}%
\begin{equation}
G\left( x,x\right) =g\left( x\right) +g\left( x\right) -g\left( 2x\right)
=2g\left( x\right) -g\left( 2x\right) .  \label{GrindEQ__4_}
\end{equation}

Since the equality \eqref{GrindEQ__4_} is equivalent to the equality
\[
g\left( x\right) =\frac{1}{2}g\left( 2x\right) +\frac{1}{2}G\left(
x,x\right) ,
\]%
we can write that
\[
g\left( \frac{1}{2}\right) =\frac{1}{2}g\left( 1\right) +\frac{1}{2}G\left(
\frac{1}{2},\frac{1}{2}\right) ,
\]%
\[
g\left( \frac{1}{4}\right) =\frac{1}{4}g\left( 1\right) +\frac{1}{4}G\left(
\frac{1}{2},\frac{1}{2}\right) +\frac{1}{2}G\left( \frac{1}{4},\frac{1}{4}%
\right) ,
\]%
{\dots }{\dots }{\dots }{\dots }{\dots }{\dots }{\dots }{\dots }{\dots }{%
\dots }{\dots }{\dots }{\dots }{\dots }{\dots }.
\begin{equation}
g\left( \frac{1}{2^{n}}\right) =\frac{1}{2^{n}}g\left( 1\right) +\frac{1}{%
2^{n}}G\left( \frac{1}{2},\frac{1}{2}\right) +\frac{1}{2^{n-1}}G\left( \frac{%
1}{4},\frac{1}{4}\right) +...+\frac{1}{2}G\left( \frac{1}{2^{n}},\frac{1}{%
2^{n}}\right) ,n\in N.  \label{GrindEQ__5_}
\end{equation}

We obtain from \eqref{GrindEQ__2_} that for any $k\in \mathbb{N}$
\[
\left\vert G\left( \frac{1}{2^{k}},\frac{1}{2^{k}}\right) \right\vert
=\left\vert G\left( \frac{1}{2^{k}},\frac{1}{2^{k}}\right) -G\left( \frac{1}{%
2^{k}},0\right) \right\vert \leq \left( \frac{1}{2^{k}}\right) ^{\alpha
}\cdot H\left( G;\alpha ;\left[ -1,1\right] ^{2}\right)
\]%
\begin{equation}
=\frac{1}{2^{k\alpha }}\cdot H\left( F;\alpha ;\left[ -1,1\right]
^{2}\right) =\frac{c_{0}}{2^{k\alpha }},  \label{GrindEQ__6_}
\end{equation}%
where $c_{0}=H\left( F;\alpha ;\left[ -1,1\right] ^{2}\right) $. It follows
from \eqref{GrindEQ__5_} and \eqref{GrindEQ__6_} that
\[
\left\vert g\left( \frac{1}{2^{n}}\right) \right\vert \leq \frac{\left\vert
g\left( 1\right) \right\vert }{2^{n}}+\frac{1}{2^{n}}\cdot \frac{c_{0}}{%
2^{\alpha }}+\frac{1}{2^{n-1}}\cdot \frac{c_{0}}{2^{2\alpha }}+...+\frac{1}{2%
}\cdot \frac{c_{0}}{2^{n\alpha }}
\]%
\[
\leq \frac{\left\vert g\left( 1\right) \right\vert }{2^{n}}+\frac{c_{0}}{%
2^{n\alpha +1}}\left[ 1+\frac{1}{2^{\alpha -1}}+...+\frac{1}{2^{\left(
n-2\right) \left( \alpha -1\right) }}+\frac{1}{2^{\left( n-1\right) \left(
\alpha -1\right) }}\right]
\]
\begin{equation}
\leq \frac{\left\vert g\left( 1\right) \right\vert }{2^{n}}+\frac{c_{1}}{%
2^{n\alpha +1}}\leq \frac{c_{2}}{2^{n\alpha }},  \label{GrindEQ__7_}
\end{equation}%
where $c_{1}$ and $c_{2}$ are some constants independent of $n\in \mathbb{N}$%
. Consider a regular irreducible dyadic fraction $\frac{m}{2^{n}}\in \left[
\frac{1}{2^{k}},\frac{1}{2^{k-1}}\right) $ and represent it in the binary
system:
\[
\frac{m}{2^{n}}=\frac{1}{2^{k}}+\frac{1}{2^{k+p_{1}}}+...+\frac{1}{%
2^{k+p_{s}}}+\frac{1}{2^{n}}.
\]%
We obtain from \eqref{GrindEQ__1_} that
\[
g\left( \frac{m}{2^{n}}\right) =g\left( \frac{1}{2^{k}}+\frac{1}{2^{k+p_{1}}}%
+...+\frac{1}{2^{k+p_{s}}}+\frac{1}{2^{n}}\right)
\]%
\[
=g\left( \frac{1}{2^{k}}\right) +g\left( \frac{1}{2^{k+p_{1}}}+...+\frac{1}{%
2^{k+p_{s}}}+\frac{1}{2^{n}}\right) -G\left( \frac{1}{2^{k}},\frac{1}{%
2^{k+p_{1}}}+...+\frac{1}{2^{k+p_{s}}}+\frac{1}{2^{n}}\right)
\]%
\[
=g\left( \frac{1}{2^{k}}\right) +g\left( \frac{1}{2^{k+p_{1}}}\right)
+...+g\left( \frac{1}{2^{k+p_{s}}}\right) +g\left( \frac{1}{2^{n}}\right)
\]
\[
-G\left( \frac{1}{2^{k}},\frac{1}{2^{k+p_{1}}}+...+\frac{1}{2^{k+p_{s}}}+%
\frac{1}{2^{n}}\right)
\]%
\begin{equation}
-G\left( \frac{1}{2^{k+p_{1}}},\frac{1}{2^{k+p_{2}}}+...+\frac{1}{2^{k+p_{s}}%
}+\frac{1}{2^{n}}\right) -...-G\left( \frac{1}{2^{k+p_{s}}},\frac{1}{2^{n}}%
\right) .  \label{GrindEQ__8_}
\end{equation}%
It follows from \eqref{GrindEQ__2_} and \eqref{GrindEQ__3_} that for any $%
a>0 $ and for any $x,y\in \left[ -a,a\right] ^{2}$
\begin{equation}
\left\vert G\left( x,y\right) \right\vert =\left\vert G\left( x,y\right)
-G\left( 0,y\right) \right\vert \leq H\left( F;\alpha ;\left[ -a,a\right]
^{2}\right) \cdot \left\vert x\right\vert ^{\alpha },  \label{GrindEQ__9_}
\end{equation}%
\begin{equation}
\left\vert G\left( x,y\right) \right\vert =\left\vert G\left( x,y\right)
-G\left( x,0\right) \right\vert \leq H\left( F;\alpha ;\left[ -a,a\right]
^{2}\right) \cdot \left\vert y\right\vert ^{\alpha }.  \label{GrindEQ__10_}
\end{equation}%
Then considering \eqref{GrindEQ__7_} and \eqref{GrindEQ__9_} we obtain from %
\eqref{GrindEQ__8_} that
\[
\left\vert g\left( \frac{m}{2^{n}}\right) \right\vert \leq \left\vert
g\left( \frac{1}{2^{k}}\right) \right\vert +\left\vert g\left( \frac{1}{%
2^{k+p_{1}}}\right) \right\vert +...+\left\vert g\left( \frac{1}{2^{k+p_{s}}}%
\right) \right\vert +\left\vert g\left( \frac{1}{2^{n}}\right) \right\vert
\]%
\[
+\left\vert G\left( \frac{1}{2^{k}},\frac{1}{2^{k+p_{1}}}+...+\frac{1}{%
2^{k+p_{s}}}+\frac{1}{2^{n}}\right) \right\vert +\left\vert G\left( \frac{1}{%
2^{k+p_{1}}},\frac{1}{2^{k+p_{2}}}+...+\frac{1}{2^{k+p_{s}}}+\frac{1}{2^{n}}%
\right) \right\vert +...
\]%
\[
+\left\vert G\left( \frac{1}{2^{k+p_{s}}},\frac{1}{2^{n}}\right) \right\vert
\leq \frac{c_{2}}{2^{k\alpha }}+\frac{c_{2}}{2^{\left( k+p_{1}\right) \alpha
}}+...+\frac{c_{2}}{2^{\left( k+p_{s}\right) \alpha }}+\frac{c_{2}}{%
2^{n\alpha }}
\]%
\begin{equation}
+\frac{c_{0}}{2^{k\alpha }}+\frac{c_{0}}{2^{\left( k+p_{1}\right) \alpha }}%
+...+\frac{c_{0}}{2^{\left( k+p_{s}\right) \alpha }}\leq \frac{c_{3}}{%
2^{k\alpha }}\leq c_{3}\cdot \left( \frac{m}{2^{n}}\right) ^{\alpha },
\label{GrindEQ__11_}
\end{equation}%
where $c_{3}$ is some constant.

Now we construct an auxiliary function $g^{\ast }=g^{\ast }\left( x\right) $
as follows. For any $m\in \mathbb{Z}$ and $n\in \mathbb{N}$ set $g^{\ast
}\left( \frac{m}{2^{n}}\right) =g\left( \frac{m}{2^{n}}\right) $; for a
number $x$ having in the binary system the representation $x=p_{0}+\frac{%
p_{1}}{2}+...+\frac{p_{n}}{2^{n}}+...$ set
\begin{equation}
g^{\ast }\left( x\right) =\mathop{\lim }\limits_{n\rightarrow \infty
}g\left( p_{0}+\frac{p_{1}}{2}+...+\frac{p_{n}}{2^{n}}\right) .
\label{GrindEQ__12_}
\end{equation}%
We must prove that there exists a finite limit on the right-hand side of %
\eqref{GrindEQ__12_}. It follows from \eqref{GrindEQ__1_}, %
\eqref{GrindEQ__10_} and \eqref{GrindEQ__11_} that for any $n\in \mathbb{N}$
and $k\in \mathbb{N}$
\[
\left\vert g\left( p_{0}+\frac{p_{1}}{2}+...+\frac{p_{n}}{2^{n}}\right)
-g\left( p_{0}+\frac{p_{1}}{2}+...+\frac{p_{n}}{2^{n}}+...+\frac{p_{n+k}}{%
2^{n+k}}\right) \right\vert
\]%
\[
\leq \left\vert g\left( \frac{p_{n+1}}{2^{n+1}}+...+\frac{p_{n+k}}{2^{n+k}}%
\right) \right\vert +\left\vert G\left( p_{0}+\frac{p_{1}}{2}+...+\frac{p_{n}%
}{2^{n}},\frac{p_{n+1}}{2^{n+1}}+...+\frac{p_{n+k}}{2^{n+k}}\right)
\right\vert
\]%
\begin{equation}
\leq \left( c_{3}+c_{0}\right) \cdot \left\vert \frac{p_{n+1}}{2^{n+1}}+...+%
\frac{p_{n+k}}{2^{n+k}}\right\vert ^{\alpha }\leq \frac{c_{3}+c_{0}}{%
2^{n\alpha }}.  \label{GrindEQ__13_}
\end{equation}%
We obtain from \eqref{GrindEQ__13_} that the sequence $\left\{ g\left( p_{0}+%
\frac{p_{1}}{2}+...+\frac{p_{n}}{2^{n}}\right) \right\} _{n=1}^{\infty }$ is
fundamental and, therefore, there exists a finite limit on the right-hand
side of \eqref{GrindEQ__12_}.

Now we show that for any $x,y\in \mathbb{R}$
\begin{equation}
G\left( x,y\right) =g^{\ast }\left( x\right) +g^{\ast }\left( y\right)
-g^{\ast }\left( x+y\right) .  \label{GrindEQ__14_}
\end{equation}%
Let in the binary system
\[
x=p_{0}+\frac{p_{1}}{2}+...+\frac{p_{n}}{2^{n}}+...,y=q_{0}+\frac{q_{1}}{2}%
+...+\frac{q_{n}}{2^{n}}+...,
\]%
\[
x+y=r_{0}+\frac{r_{1}}{2}+...+\frac{r_{n}}{2^{n}}+...
\]%
Denote
\[
x^{\left( n\right) }=p_{0}+\frac{p_{1}}{2}+...+\frac{p_{n}}{2^{n}},y^{\left(
n\right) }=q_{0}+\frac{q_{1}}{2}+...+\frac{q_{n}}{2^{n}}+...,
\]%
\[
\left( x+y\right) ^{\left( n\right) }=r_{0}+\frac{r_{1}}{2}+...+\frac{r_{n}}{%
2^{n}}+...,~n\in \mathbb{N}.
\]%
Considering \eqref{GrindEQ__1_} we can write that
\begin{equation}
G\left( x^{\left( n\right) },y^{\left( n\right) }\right) =g\left( x^{\left(
n\right) }\right) +g\left( y^{\left( n\right) }\right) -g\left( x^{\left(
n\right) }+y^{\left( n\right) }\right) .  \label{GrindEQ__15_}
\end{equation}%
Passing to the limit in \eqref{GrindEQ__15_} and taking into account the
continuity of the function $G\left( x,y\right) $, we have
\begin{equation}
G\left( x,y\right) =g^{\ast }\left( x\right) +g^{\ast }\left( y\right) -%
\mathop{\lim }\limits_{n\rightarrow \infty }g\left( x^{\left( n\right)
}+y^{\left( n\right) }\right) .  \label{GrindEQ__16_}
\end{equation}%
Since $\left( x+y\right) ^{\left( n\right) }-x^{\left( n\right) }-y^{\left(
n\right) }$ equals to $0$ or $\frac{1}{2^{n}}$, it follows from %
\eqref{GrindEQ__1_}, \eqref{GrindEQ__7_} and \eqref{GrindEQ__10_} that
\[
\left\vert g\left( \left( x+y\right) ^{\left( n\right) }\right) -g\left(
x^{\left( n\right) }+y^{\left( n\right) }\right) \right\vert \leq \left\vert
g\left( \left( x+y\right) ^{\left( n\right) }-x^{\left( n\right) }-y^{\left(
n\right) }\right) \right\vert
\]%
\[
+\left\vert G\left( x^{\left( n\right) }+y^{\left( n\right) },\left(
x+y\right) ^{\left( n\right) }-x^{\left( n\right) }-y^{\left( n\right)
}\right) \right\vert \leq \left\vert g\left( \frac{1}{2^{n}}\right)
\right\vert +\left\vert G\left( x^{\left( n\right) }+y^{\left( n\right) },%
\frac{1}{2^{n}}\right) \right\vert
\]%
\[
\leq \frac{c_{2}}{2^{n\alpha }}+H\left( F;\alpha ;\left[ -\left\vert
x\right\vert -\left\vert y\right\vert -1,\left\vert x\right\vert +\left\vert
y\right\vert +1\right] ^{2}\right) \cdot \frac{1}{2^{n\alpha }}.
\]%
It follows that
\[
\mathop{\lim }\limits_{n\rightarrow \infty }g\left( x^{\left( n\right)
}+y^{\left( n\right) }\right) =\mathop{\lim }\limits_{n\rightarrow \infty
}g\left( \left( x+y\right) ^{\left( n\right) }\right) =g^{\ast }\left(
x+y\right) .
\]%
Thus from \eqref{GrindEQ__16_} we obtain \eqref{GrindEQ__14_}.

Now we show that $g^{\ast }\in H_{\alpha }^{\left( loc\right) }\left(
\mathbb{R}\right) $. First we prove that for any $\delta \in \left(
0,1\right) $
\begin{equation}
\left\vert g^{\ast }\left( \delta \right) \right\vert \leq c_{3}\cdot \delta
^{\alpha }.  \label{GrindEQ__17_}
\end{equation}%
Indeed, let $\delta =\frac{p_{1}}{2}+...+\frac{p_{n}}{2^{n}}+...\in \left(
0,1\right) $. Then it follows from \eqref{GrindEQ__11_} that
\begin{equation}
\left\vert g\left( \frac{p_{1}}{2}+...+\frac{p_{n}}{2^{n}}\right)
\right\vert \leq c_{3}\cdot \left( \frac{p_{1}}{2}+...+\frac{p_{n}}{2^{n}}%
\right) ^{\alpha }.  \label{GrindEQ__18_}
\end{equation}%
Passing to the limit in \eqref{GrindEQ__18_} we obtain \eqref{GrindEQ__17_}.

It follows from \eqref{GrindEQ__14_}, \eqref{GrindEQ__17_} and %
\eqref{GrindEQ__10_} that for any $a>0$, $\delta \in \left( 0,1\right) $ and
for any $x,y\in \left[ -a,a\right] ^{2}$, $0\leq x-y\leq \delta $, we have
\[
\left\vert g^{\ast }\left( x\right) -g^{\ast }\left( y\right) \right\vert
\leq \left\vert g^{\ast }\left( x-y\right) \right\vert +\left\vert G\left(
y,x-y\right) \right\vert
\]%
\[
\leq c_{3}\cdot \left\vert x-y\right\vert ^{\alpha }+H\left( F;\alpha ;\left[
-a,a\right] ^{2}\right) \cdot \left\vert x-y\right\vert ^{\alpha }.
\]%
This means that $g^{\ast }\in H_{\alpha }^{\left( loc\right) }\left( \mathbb{%
R}\right) $.

Consider the function
\[
f\left( x\right) =g^{\ast }\left( x\right) +F\left( 0,0\right) .
\]%
Note that $f\left( x\right) $ also belongs to $H_{\alpha }^{\left(
loc\right) }\left( \mathbb{R}\right) $ and we have the equality
\[
F\left( x,y\right) =G\left( x,y\right) +F\left( 0,0\right) =g^{\ast }\left(
x\right) +g^{\ast }\left( y\right) -g^{\ast }\left( x+y\right) +F\left(
0,0\right)
\]%
\[
=f\left( x\right) +f\left( y\right) -f\left( x+y\right) .
\]%
The lemma has been proved.

Now we are ready to prove the following theorem.

\begin{theorem} Assume a function $F\in H_{\alpha }^{\left( loc\right)
}\left( \mathbb{R}^{2}\right) $, $0<\alpha \leq 1$, has the form%
\begin{equation}
F\left( x,y\right) =h_{1}\left( a_{1}x+b_{1}y\right) +h_{2}\left(
a_{2}x+b_{2}y\right) +h_{3}\left( a_{3}x+b_{3}y\right) ,
\label{GrindEQ__19_}
\end{equation}%
where the $\left( a_{i},b_{i}\right) $, $i=1,2,3$, are pairwise linearly
independent directions in $\mathbb{R}^{2}$ and $h_{i}:\mathbb{R}\rightarrow
\mathbb{R}$, $i=1,2,3$. Then there exist functions $f_{i}\in H_{\alpha
}^{\left( loc\right) }\left( \mathbb{R}\right) $, $i=1,2,3,$ such that
\[
F\left( x,y\right) =f_{1}\left( a_{1}x+b_{1}y\right) +f_{2}\left(
a_{2}x+b_{2}y\right) +f_{3}\left( a_{3}x+b_{3}y\right) .
\]%
\end{theorem}

\textbf{Proof.} Since the vectors $\left( a_{i},b_{i}\right) $, $i=1,2,3$
are pairwise linearly independent, we can apply a nonsingular linear
transformation $S:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ of the
coordinates such that $S\left( \left( a_{1},b_{1}\right) \right) =\left(
1,0\right) $ and $S\left( \left( a_{2},b_{2}\right) \right) =\left(
0,1\right) $. Therefore without loss of generality we may assume that $%
\left( a_{1},b_{1}\right) =\left( 1,0\right) $, $\left( a_{2},b_{2}\right)
=\left( 0,1\right) $ and herewith $a_{3}\neq 0$, $b_{3}\neq 0$. Thus we
prove the theorem if we prove that for any function $F\in H_{\alpha
}^{\left( loc\right) }\left( \mathbb{R}^{2}\right) $ of the form
\begin{equation}
F\left( x,y\right) =h_{1}\left( x\right) +h_{2}\left( y\right) +h_{3}\left(
a_{3}x+b_{3}y\right)  \label{GrindEQ__20_}
\end{equation}%
there exists $f_{i}\in H_{\alpha }^{\left( loc\right) }\left( \mathbb{R}%
\right) $, $i=1,2,3,$ such that

\begin{equation}
F\left( x,y\right) =f_{1}(x)+f_{2}(y)+f_{3}\left( a_{3}x+b_{3}y\right) .
\label{GrindEQ__21_}
\end{equation}

Since $a_{3}\neq 0$, $b_{3}\neq 0,$ it follows from \eqref{GrindEQ__20_}
that
\begin{equation}
F\left( \frac{x}{a_{3}},\frac{y}{b_{3}}\right) =h_{1}\left( \frac{x}{a_{3}}%
\right) +h_{2}\left( \frac{y}{b_{3}}\right) +h_{3}\left( x+y\right) ,
\label{GrindEQ__22_}
\end{equation}%
\begin{equation}
F\left( \frac{x}{a_{3}},0\right) =h_{1}\left( \frac{x}{a_{3}}\right)
+h_{2}\left( 0\right) +h_{3}\left( x\right) ,  \label{GrindEQ__23_}
\end{equation}%
\begin{equation}
F\left( 0,\frac{y}{b_{3}}\right) =h_{1}\left( 0\right) +h_{2}\left( \frac{y}{%
b_{3}}\right) +h_{3}\left( y\right) .  \label{GrindEQ__24_}
\end{equation}

Consider the function
\[
H\left( x,y\right) =F\left( \frac{x}{a_{3}},0\right) +F\left( 0,\frac{y}{%
b_{3}}\right) -F\left( \frac{x}{a_{3}},\frac{y}{b_{3}}\right) -h_{1}\left(
0\right) -h_{2}\left( 0\right) ,
\]%
Note that the function $H\left( x,y\right) $ belongs to the class $H_{\alpha
}^{\left( loc\right) }\left( \mathbb{R}^{2}\right) $ and from %
\eqref{GrindEQ__22_} - \eqref{GrindEQ__24_} it follows that
\[
H\left( x,y\right) =h_{3}\left( x\right) +h_{3}\left( y\right) -h_{3}\left(
x+y\right) .
\]%
Applying Lemma 2.1, we obtain that there exists a function $f\in H_{\alpha
}^{\left( loc\right) }\left( \mathbb{R}\right) $ such that
\[
H\left( x,y\right) =f\left( x\right) +f\left( y\right) -f\left( x+y\right) .
\]%
Introduce the following functions
\[
f_{1}\left( x\right) =F\left( x,0\right) -h_{1}\left( 0\right) -f\left(
a_{3}x\right) ,~f_{2}\left( x\right) =F\left( 0,x\right) -h_{2}\left(
0\right) -f\left( b_{3}x\right) ,~f_{3}\left( x\right) =f\left( x\right) .
\]%
It is not difficult to see that $f_{i}\in H_{\alpha }^{\left( loc\right)
}\left( R\right) $, $i=1,2,3$, and
\[
f_{1}\left( x\right) +f_{2}\left( y\right) +f_{3}\left( a_{3}x+b_{3}y\right)
\]%
\[
=\left[ F\left( x,0\right) -h_{1}\left( 0\right) -f\left( a_{3}x\right) %
\right] +\left[ F\left( 0,y\right) -h_{2}\left( 0\right) -f\left(
b_{3}y\right) \right] +f\left( a_{3}x+b_{3}y\right)
\]%
\[
=F\left( x,0\right) +F\left( 0,y\right) -h_{1}\left( 0\right) -h_{2}\left(
0\right) -\left[ f\left( a_{3}x\right) +f\left( b_{3}y\right) -f\left(
a_{3}x+b_{3}y\right) \right]
\]%
\[
=F\left( x,0\right) +F\left( 0,y\right) -h_{1}\left( 0\right) -h_{2}\left(
0\right) -H\left( a_{3}x,b_{3}y\right)
\]%
\[
=F\left( x,0\right) +F\left( 0,y\right) -h_{1}\left( 0\right) -h_{2}\left(
0\right) -\left[ F\left( x,0\right) +F\left( 0,y\right) -F\left( x,y\right)
-h_{1}\left( 0\right) -h_{2}\left( 0\right) \right]
\]%
\[
=F\left( x,y\right) .
\]%
Thus we obtain that \eqref{GrindEQ__21_} holds. The theorem has been
proved.

\bigskip

Using the multidimensional techniques exploited in our previous paper \cite{A0},
Theorem 2.1 can be proven in a more general case. Since the proof for such a generalization is purely technical, we only formulate the final result.

\bigskip

\begin{theorem} Assume we are given $r$ directions $\mathbf{a}%
^{i}$, $i=1,...,r$, in $\mathbb{R}^{m}\backslash \{\mathbf{0}\}$ and $r-1$
of them are linearly independent.\ Assume that a function $F\in H_{\alpha
}^{\left( loc\right) }\left( \mathbb{R}^{m}\right) $, $0<\alpha \leq 1$,  is
of the form
$$
F(\mathbf{x})=\sum_{i=1}^{r}g_{i}(\mathbf{a}^{i}\cdot \mathbf{x}).
$$%
Then $F$ can be represented also in the form
$$
F(\mathbf{x})=\sum_{i=1}^{r}f_{i}(\mathbf{a}^{i}\cdot \mathbf{x}),
$$%
where the functions $f_{i}\in H_{\alpha }^{\left( loc\right) }\left(
\mathbb{R}\right) $, $i=1,...,r$.
\end{theorem}

\bigskip

\textbf{Acknowledgements.} This research was supported by the Science Development Foundation under the President of the Republic of Azerbaijan (grant no. EIF/MQM/Elm-Tehsil-1-2016- 1(26)-71/08/01).

\smallskip

\begin{thebibliography}{99}
\bibitem{1} J. Acz\'{e}l, \textit{Functional Equations and their Applications%
}, Academic Press, New York, 1966.

\bibitem{A0} R.A. Aliev, A.A. Asgarova and V.E. Ismailov, A note on
continuous sums of ridge functions, \textit{J. Approx. Theory} \textbf{237} (2019),
210--221.

\bibitem{A1} R.A. Aliev and V.E. Ismailov, On a smoothness problem in ridge
function representation, \textit{Adv. in Appl. Math.} \textbf{73} (2016), 154--169.

\bibitem{A2} R.A. Aliev and V.E. Ismailov, On the representation by
bivariate ridge functions, arXiv preprint https://arxiv.org/abs/1606.07940

\bibitem{A3} R.A. Aliev, V.E. Ismailov, T.M. Shahbalayeva, On the
representation by sums of ridge functions, \textit{Proc. Inst. Math. Mech.
Natl. Acad. Sci. Azerb.} \textbf{41} (2015), no. 2, 106--118.

\bibitem{12} M.D. Buhmann and A. Pinkus, Identifying linear combinations of
ridge functions, \textit{Adv. in Appl. Math.} \textbf{22} (1999), 103-118.

\bibitem{27} D.L. Donoho and I.M. Johnstone, Projection-based approximation
and a duality method with kernel methods, \textit{Ann. Statist.} \textbf{17} (1989),
58-106.

\bibitem{33} J.H. Friedman and W. Stuetzle, Projection pursuit regression,
\textit{J.Amer. Statist. Assoc.} \textbf{76} (1981), 817-823.

\bibitem{42} P.J. Huber, Projection pursuit, \textit{Ann. Statist.} \textbf{13}
(1985), 435-475.

\bibitem{I1} V.E. Ismailov, Approximation by sums of ridge functions with
fixed directions. (Russian); translated from \textit{Algebra i Analiz} \textbf{28}
(2016), no. 6, 20--69 \textit{St. Petersburg Math. J.} \textbf{28} (2017), no. 6,
741-772.

\bibitem{I2} V.E. Ismailov, A note on the equioscillation theorem for best
ridge function approximation, \textit{Expo. Math.} \textbf{35} (2017), no. 3,
343--349.

\bibitem{I3} V.E. Ismailov, Approximation by ridge functions and neural
networks with a bounded number of neurons. \textit{Appl. Anal.} \textbf{94} (2015),
no. 11, 2245--2260.

\bibitem{52} V.E. Ismailov, On the best $L_{2}$ approximation by ridge
functions, \textit{Appl. Math. E-Notes} \textbf{7} (2007), 71-76.

\bibitem{69} F. John, \textit{Plane Waves and Spherical Means Applied to
Partial Differential Equations}, Interscience, New York, 1955.

\bibitem{72} I. Kazantsev, Tomographic reconstruction from arbitrary
directions using ridge functions, \textit{Inverse Problems} \textbf{14} (1998),
635-645.

\bibitem{74} I. Kazantsev, I. Lemahieu, Reconstruction of elongated
structures using ridge functions and natural pixels, \textit{Inverse
Problems }\textbf{16} (2000), 505-517.

\bibitem{K1} S.V. Konyagin, A.A. Kuleshov, On the continuity of finite sums
of ridge functions. (Russian); translated from \textit{Mat. Zametki} \textbf{98}
(2015), no. 2, 308--309 \textit{Math. Notes} \textbf{98} (2015), no. 1-2, 336--338.

\bibitem{K2} S.V. Konyagin, A.A. Kuleshov, On some properties of finite sums
of ridge functions defined on convex subsets of $\mathbb{R}^{n}$. (Russian)
\textit{Tr. Mat. Inst. Steklova} \textbf{293} (2016), Funktsionalnye Prostranstva,
Teoriya Priblizheni\u{\i}, Smezhnye Razdely Matematicheskogo An., 193--200.

\bibitem{88} A. Kroo, On approximation by ridge functions, \textit{Constr.
Approx.} \textbf{13} (1997), 447-460.

\bibitem{K3} A.A. Kuleshov, On some properties of smooth sums of ridge
functions. (Russian) \textit{Tr. Mat. Inst. Steklova} \textbf{294} (2016),
Sovremennye Problemy Matematiki, Mekhaniki i Matematichesko\u{\i} Fiziki.
II, 99--104.

\bibitem{95} V.Ya. Lin and A. Pinkus, Fundamentality of ridge functions,
\textit{J. Approx. Theory} \textbf{75} (1993), 295-311.

\bibitem{97} B.F. Logan and L.A. Shepp, Optimal reconstruction of a function
from its projections, \textit{Duke Math.J.} \textbf{42} (1975), 645-659.

\bibitem{102} V.E. Maiorov, On best approximation by ridge functions,
\textit{J.Approx. Theory} \textbf{99} (1999), 68-94.

\bibitem{106} R.B. Marr, On the reconstruction of a function on a circular
domain from a sampling of its line integrals. \textit{J. Math. Anal. Appl.}
\textbf{45} (1974), 357--374.

\bibitem{111} F. Natterer, \textit{The Mathematics of Computerized Tomography%
}, Wiley, New York, 1986.

\bibitem{117} P.P. Petrushev, Approximation by ridge functions and neural
networks, \textit{SIAM J.Math. Anal.} \textbf{30} (1998), 155-189.

\bibitem{P} A. Pinkus, \textit{Ridge Functions}, Cambridge Tracts in
Mathematics, 205. Cambridge University Press, Cambridge, 2015.

\bibitem{Pinkus} A. Pinkus, Smoothness and uniqueness in ridge function
representation, \textit{Indag. Math. (N.S.)} \textbf{24} (2013), no. 4, 725--738.

\bibitem{118} A. Pinkus, Approximating by ridge functions, in: \textit{%
Surface Fitting and Multiresolution Methods}, (A.Le M\'{e}haut\'{e}, C.Rabut
and L.L.Schumaker, eds), Vanderbilt Univ.Press (Nashville), 1997, 279-292.

\bibitem{119} A. Pinkus, Approximation theory of the MLP model in neural
networks, \textit{Acta Numerica} \textbf{8} (1999), 143-195.
\end{thebibliography}

\end{document}
