On the representation by bivariate ridge functions
UDC 517.5 We consider the problem of representation of a bivariate function by sums of ridge functions. It is shown that if a function of a certain smoothness class is represented by a sum of finitely many arbitrarily behaved ridge functions, then it can also be represented by a sum of ridge functio...
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Institute of Mathematics, NAS of Ukraine
2021
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| author | Aliev, R. A. Asgarova, A. A. Ismailov, V. E. Aliev, Rashid Asgarova, Aysel Ismailov, Vugar Aliev, R. A. Asgarova, A. A. Ismailov, V. E. |
| author_facet | Aliev, R. A. Asgarova, A. A. Ismailov, V. E. Aliev, Rashid Asgarova, Aysel Ismailov, Vugar Aliev, R. A. Asgarova, A. A. Ismailov, V. E. |
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| description | UDC 517.5
We consider the problem of representation of a bivariate function by sums of ridge functions. It is shown that if a function of a certain smoothness class is represented by a sum of finitely many arbitrarily behaved ridge functions, then it can also be represented by a sum of ridge functions of the same smoothness class. As an example, this result is applied to a homogeneous constant coefficient partial differential equation. |
| doi_str_mv | 10.37863/umzh.v73i5.263 |
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DOI: 10.37863/umzh.v73i5.263
UDC 517.5
R. A. Aliev (Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku and Baku State Univ., Azerbaijan),
A. A. Asgarova (Azerbaijan Univ. Languages, Baku),
V. E. Ismailov (Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku)
ON THE REPRESENTATION BY BIVARIATE RIDGE FUNCTIONS*
ПРО ЗОБРАЖЕННЯ ГРЕБЕНЕВИМИ ФУНКЦIЯМИ ДВОХ ЗМIННИХ
We consider the problem of representation of a bivariate function by sums of ridge functions. It is shown that if a function
of a certain smoothness class is represented by a sum of finitely many arbitrarily behaved ridge functions, then it can
also be represented by a sum of ridge functions of the same smoothness class. As an example, this result is applied to a
homogeneous constant coefficient partial differential equation.
Розглядається задача зображення двовимiрної функцiї сумами гребеневих функцiй. Показано, що коли функцiю
певного класу гладкостi зображено скiнченною сумою гребеневих функцiй довiльної поведiнки, її також можна
зобразити сумою гребеневих функцiй того ж класу гладкостi. Як приклад цей результат застосовано до однорiдного
диференцiального рiвняння з частинними похiдними i зi сталими коефiцiєнтами.
1. Introduction. Last 30 years have seen a growing interest in the study of special multivariate
functions called ridge functions. This interest is due to applicability of such functions in various
research areas. A ridge function is a multivariate function of the form
g(\bfa \cdot \bfx ) = g(a1x1 + . . .+ amxm),
where g : \BbbR \rightarrow \BbbR and \bfa = (a1, . . . , am) is a fixed vector (direction) in \BbbR m\setminus \{ \bfzero \} . These functions and
their linear combinations find applications in computerized tomography (see, e.g., [16, 21, 24]), in
statistics (especially, in the theory of projection pursuit and projection regression; see, e.g., [4, 6]) and
in the theory of neural networks (see, e.g., [7, 9, 11, 23, 28]). Ridge functions are also widely used
in modern approximation theory as an effective and convenient tool for approximating complicated
multivariate functions (see, e.g., [8, 13, 20, 22, 25]). For more on ridge functions and application
areas, see the book [26] and survey papers [10, 12, 19].
It should be remarked that ridge functions have been used in the theory of partial differential
equations under the name of plane waves (see, e.g., [15]). In general, linear combinations of ridge
functions with fixed directions occur in the study of hyperbolic constant coefficient partial differential
equations. For example, assume that (\alpha i, \beta i), i = 1, . . . , r, are pairwise linearly independent vectors
in \BbbR 2. Then the general solution to the homogeneous equation
r\prod
i=1
\biggl(
\alpha i
\partial
\partial x
+ \beta i
\partial
\partial y
\biggr)
u(x, y) = 0 (1.1)
are all functions of the form
* 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).
c\bigcirc R. A. ALIEV, A. A. ASGAROVA, V. E. ISMAILOV, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5 579
580 R. A. ALIEV, A. A. ASGAROVA, V. E. ISMAILOV
u(x, y) =
r\sum
i=1
vi(\beta ix - \alpha iy) (1.2)
for arbitrary univariate functions vi, i = 1, . . . , r, from the class Cr(\BbbR ).
Note that the solution of Eq. (1.1) is the sum of bivariate ridge functions. Sums of bivariate ridge
functions also occur in basic mathematical problems of computerized tomography. For example,
Logan and Shepp [15] (the term “ridge function” was coined by them) considered the problem of
reconstructing a given but unknown function f(x, y) from its integrals along certain lines in the plane.
More precisely, let D be the unit disk in the plane and a function f(x, y) be square integrable and
supported on D. We are given projections Pf (t, \theta ) (integrals of f along the lines x \mathrm{c}\mathrm{o}\mathrm{s} \theta +y \mathrm{s}\mathrm{i}\mathrm{n} \theta = t)
and looking for a function g = g(x, y) of minimum L2 norm, which has the same projections as f :
Pg(t, \theta j) = Pf (t, \theta j), j = 0, 1, . . . , n - 1, where angles \theta j generate equally spaced directions,
i.e., \theta j =
j\pi
n
, j = 0, 1, . . . , n - 1. The authors of [15] showed that this problem of tomography
is equivalent to the problem of L2-approximation of the function f by sums of bivariate ridge
functions with equally spaced directions (\mathrm{c}\mathrm{o}\mathrm{s} \theta j , \mathrm{s}\mathrm{i}\mathrm{n} \theta j), j = 0, 1, . . . , n - 1. They gave a closed-
form expression for the unique function g(x, y) and showed that the unique polynomial P (x, y) of
degree n - 1 which best approximates f in L2(D) is determined from the above n projections of f
and can be represented as a sum of n bivariate ridge functions.
In this paper, we are interested in the problem of smoothness in representation by sums of
bivariate ridge functions with finitely many fixed directions. Assume we are given n pairwise
linearly independent directions (ai, bi), i = 1, . . . , n, in \BbbR 2 and a function F : \BbbR 2 \rightarrow \BbbR of the form
F (x, y) =
n\sum
i=1
gi(aix+ biy). (1.3)
Assume in addition that F is of a certain smoothness class, what can we say about the smoothness of
gi? The case n = 1 is obvious. In this case, if F \in Ck(\BbbR 2), then for a vector (c, d) \in \BbbR 2 satisfying
a1c + b1d = 1 we have that g1(t) = F (ct, dt) is in Ck(\BbbR ). The same argument can be carried out
for the case n = 2. In this case, since the vectors (a1, b1) and (a2, b2) are linearly independent, there
exists a vector (c, d) \in \BbbR 2 satisfying a1c+ b1d = 1 and a2c+ b2d = 0. Therefore, we obtain that the
function g1(t) = F (ct, dt) - g2(0) is in the class Ck(\BbbR ). Similarly, one can verify that g2 \in Ck(\BbbR ).
The picture drastically changes if the number of directions n \geq 3. For n = 3, there are ulti-
mately smooth functions which decompose into sums of very badly behaved ridge functions. This
phenomena comes from the classical Cauchy functional equation. This equation,
f(x+ y) = f(x) + f(y), f : \BbbR \rightarrow \BbbR , (1.4)
looks very simple and has a class of simple solutions f(x) = cx, c \in \BbbR . Nevertheless, it easily
follows from the Hamel basis theory that the Cauchy functional equation has also a large class of
wild solutions. These solutions are called “wild” because they are extremely pathological 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., [1]). Let g be any wild solution of the equation (1.4). Then the zero
function can be represented as
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
ON THE REPRESENTATION BY BIVARIATE RIDGE FUNCTIONS 581
0 = g(x) + g(y) - g(x+ y). (1.5)
Note that the functions involved in (1.5) are bivariate ridge functions with the directions (1, 0), (0, 1)
and (1, 1), respectively. This example shows that for smoothness of the representation (1.3) one must
impose additional conditions on the representing functions gi, i = 1, . . . , n.
Such additional conditions are recently found by Pinkus [27]. He proved that for a large class
of representing functions gi, the representation is smooth. That is, if apriori assume that in the
representation (1.3), the functions gi belong to a certain class of “well behaved functions”, then they
have the same degree of smoothness as the function F. As the mentioned class of “well behaved
functions” one may take, e.g., the set of functions that are continuous at a point, bounded on one
side on a set of positive measure, monotonic at an interval, Lebesgue measurable, etc. (see [27]).
Konyagin and Kuleshov [17] proved that in (1.3) the functions gi inherit smoothness properties of F
(without additional assumptions on gi) if and only if the directions \bfa i are linearly independent. Note
that the results of Pinkus and also Konyagin and Kuleshov are valid not only in bivariate but also in
multivariate case. There are also other results on ridge function representation, which involve certain
convex subsets of the m-dimensional space (see [17, 18]).
In this paper, we study a different aspect of the problem of representation by ridge functions.
Assume in the representation (1.3) F \in Ck(\BbbR 2) but the functions gi are arbitrary. That is, we allow
very badly behaved functions (for example, not continuous at any point). Can we write F as a sum\sum n
i=1
fi(aix+ biy) but with the fi \in Ck(\BbbR ), i = 1, . . . , n? We see that the answer to this question
is positive as expected. For the sake of convenience we state the result over \BbbR 2, but in fact it holds
over any open set in \BbbR 2.
Note that the above problem is not elementary as it seems. There are cases when representation
with good functions is not possible. Such situations happen over closed sets with no interior. In [14],
Ismailov and Pinkus presented an example of a function of the form
F (x, y) = g1(a1x+ b1y) + g2(a2x+ b2y),
that is bounded and continuous on the union of two straight lines but such that both g1 and g2 are
necessarily discontinuous, and thus cannot be replaced with continuous functions f1 and f2.
The result of this paper can be applied to a higher order partial differential equation in two
variables if its solution is given by a sum of sufficiently smooth plane waves (see, for example,
Eq. (1.1)). Based on our theorem below, in this case, one can demand only smoothness of the sum
and dispense with smoothness of the plane wave summands.
2. Main results. We start this section with the following theorem.
Theorem 2.1. Assume that (ai, bi), i = 1, . . . , n, are pairwise linearly independent vectors in
\BbbR 2 and a function F \in Ck(\BbbR 2) has the form
F (x, y) =
n\sum
i=1
gi(aix+ biy), (2.1)
where gi are arbitrary univariate functions and k \geq n - 2. Then F can be represented also in the
form
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
582 R. A. ALIEV, A. A. ASGAROVA, V. E. ISMAILOV
F (x, y) =
n\sum
i=1
fi(aix+ biy).
Here, the functions fi \in Ck(\BbbR ), i = 1, . . . , n.
Proof. Since the vectors (an - 1, bn - 1) and (an, bn) are linearly independent, there is a non-
singular linear transformation S : (x, y) \rightarrow (x
\prime
, y
\prime
) such that S : (an - 1, bn - 1) \rightarrow (1, 0) and S :
(an, bn) \rightarrow (0, 1). Thus, without loss of generality we may assume that the vectors (an - 1, bn - 1) and
(an, bn) coincide with the coordinate vectors e1 = (1, 0) and e2 = (0, 1), respectively. Therefore, to
prove the theorem it is enough to show that if a function F \in Ck(\BbbR 2) is expressed in the form
F (x, y) =
n - 2\sum
i=1
gi(aix+ biy) + gn - 1(x) + gn(y)
with arbitrary gi, then there exist functions fi \in Ck(\BbbR ), i = 1, . . . , n, such that F is expressed also
in the form
F (x, y) =
n - 2\sum
i=1
fi(aix+ biy) + fn - 1(x) + fn(y).
By \Delta
(\delta )
l f we denote the increment of a function f in a direction l = (l\prime , l\prime \prime ). That is,
\Delta
(\delta )
l f(x, y) = f(x+ l\prime \delta , y + l\prime \prime \delta ) - f(x, y).
We also use the notation
\partial f
\partial l
which denotes the derivative of f in the direction l.
It is easy to check that the increment of a ridge function g(ax+by) in a direction perpendicular to
(a, b) is zero. Let l1, . . . , ln - 2 be unit vectors perpendicular to the vectors (a1, b1), . . . , (an - 2, bn - 2)
correspondingly. Then for any set of numbers \delta 1, . . . , \delta n - 2 \in \BbbR we have
\Delta
(\delta 1)
l1
. . .\Delta
(\delta n - 2)
ln - 2
F (x, y) = \Delta
(\delta 1)
l1
. . .\Delta
(\delta n - 2)
ln - 2
\bigl[
gn - 1(x) + gn(y)
\bigr]
. (2.2)
Denote the left-hand side of (2.2) by S(x, y). That is, set
S(x, y)
def
= \Delta
(\delta 1)
l1
. . .\Delta
(\delta n - 2)
ln - 2
F (x, y).
Then from (2.2) it follows that, for any real numbers \delta n - 1 and \delta n,
\Delta (\delta n - 1)
e1 \Delta (\delta n)
e2 S(x, y) = 0,
or, in expanded form,
S(x+ \delta n - 1, y + \delta n) - S(x, y + \delta n) - S(x+ \delta n - 1, y) + S(x, y) = 0.
Putting in the last equality \delta n - 1 = - x, \delta n = - y, we obtain
S(x, y) = S(x, 0) + S(0, y) - S(0, 0).
This means that
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
ON THE REPRESENTATION BY BIVARIATE RIDGE FUNCTIONS 583
\Delta
(\delta 1)
l1
. . .\Delta
(\delta n - 2)
ln - 2
F (x, y) = \Delta
(\delta 1)
l1
. . .\Delta
(\delta n - 2)
ln - 2
F (x, 0)+
+\Delta
(\delta 1)
l1
. . .\Delta
(\delta n - 2)
ln - 2
F (0, y) - \Delta
(\delta 1)
l1
. . .\Delta
(\delta n - 2)
ln - 2
F (0, 0).
By the hypothesis of the theorem, the derivative
\partial n - 2
\partial l1 . . . \partial ln - 2
F (x, y) exists at any point (x, y) \in
\in \BbbR 2. Thus, it follows from the above formula that
\partial n - 2F
\partial l1 . . . \partial ln - 2
(x, y) = h1,1(x) + h2,1(y), (2.3)
where h1,1(x) =
\partial n - 2
\partial l1 . . . \partial ln - 2
F (x, 0) and h2,1(y) =
\partial n - 2
\partial l1 . . . \partial ln - 2
F (0, y) - \partial n - 2
\partial l1 . . . \partial ln - 2
F (0, 0).
Note that h1,1 and h2,1 belong to the class Ck - n+2(\BbbR ).
By h1,2 and h2,2 denote the antiderivatives of h1,1 and h2,1 satisfying the condition h1,2(0) =
= h2,2(0) = 0 and multiplied by the numbers 1/(e1 \cdot l1) and 1/(e2 \cdot l1) correspondingly. That
is,
h1,2(x) =
1
e1 \cdot l1
x\int
0
h1,1(z)dz,
h2,2(y) =
1
e2 \cdot l1
y\int
0
h2,1(z)dz.
Here, e \cdot l denotes the scalar product between vectors e and l. Obviously, the function
F1(x, y) = h1,2(x) + h2,2(y)
obeys the equality
\partial F1
\partial l1
(x, y) = h1,1(x) + h2,1(y). (2.4)
From (2.3) and (2.4) we obtain
\partial
\partial l1
\biggl[
\partial n - 3F
\partial l2 . . . \partial ln - 2
- F1
\biggr]
= 0.
Hence, for some ridge function \varphi 1,1(a1x+ b1y),
\partial n - 3F
\partial l2 . . . \partial ln - 2
(x, y) = h1,2(x) + h2,2(y) + \varphi 1,1(a1x+ b1y). (2.5)
Here, all the functions h2,1, h2,2(y), \varphi 1,1 \in Ck - n+3(\BbbR ).
Set the functions
h1,3(x) =
1
e1 \cdot l2
x\int
0
h1,2(z)dz,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
584 R. A. ALIEV, A. A. ASGAROVA, V. E. ISMAILOV
h2,3(y) =
1
e2 \cdot l2
y\int
0
h2,2(z)dz,
\varphi 1,2(t) =
1
(a1, b1) \cdot l2
t\int
0
\varphi 1,1(z)dz.
Note that the function
F2(x, y) = h1,3(x) + h2,3(y) + \varphi 1,2(a1x+ b1y)
obeys the equality
\partial F2
\partial l2
(x, y) = h1,2(x) + h2,2(y) + \varphi 1,1(a1x+ b1y). (2.6)
From (2.5) and (2.6) it follows that
\partial
\partial l2
\biggl[
\partial n - 4F
\partial l3 . . . \partial ln - 2
- F2
\biggr]
= 0.
The last equality means that, for some ridge function \varphi 2,1(a2x+ b2y),
\partial n - 4F
\partial l3 . . . \partial ln - 2
(x, y) = h1,3(x) + h2,3(y) + \varphi 1,2(a1x+ b1y) + \varphi 2,1(a2x+ b2y). (2.7)
Here, all the functions h1,3, h2,3, \varphi 1,2, \varphi 2,1 \in Ck - n+4(\BbbR ).
Note that in the left-hand sides of (2.3), (2.5) and (2.7) we have the mixed directional derivatives
of F and the order of these derivatives is decreased by one in each consecutive step. Continuing the
above process, until it reaches the function F, we obtain the desired result.
Theorem 2.1 is proved.
Theorem 2.1 can be applied to Eq. (1.1) as follows.
Corollary 2.1. Assume a function u \in Cr(\BbbR 2) is of the form (1.2) with arbitrarily behaved vi.
Then u is a solution to the equation (1.1).
Note that the method exploited in the proof of Theorem 2.1 enables us to construct the functions
fi, i = 1, . . . , n, by induction. First let us accept some notation. By (\widetilde ap,\widetilde bp), p = 1, . . . , n - 2, denote
the images of vectors (ap, bp) under the linear transformation which takes the vectors (an - 1, bn - 1)
and (an, bn) to the unit vectors e1 = (1, 0) and e2 = (0, 1), respectively. Clearly,
\widetilde ap = apbn - anbp
an - 1bn - anbn - 1
, \widetilde bp = an - 1bp - apbn - 1
an - 1bn - anbn - 1
, p = 1, . . . , n - 2.
Consider the vectors
lp =
\left( \widetilde bp\sqrt{} \widetilde a2p +\widetilde b2p ,
- \widetilde ap\sqrt{} \widetilde a2p +\widetilde b2p
\right) , p = 1, . . . , n - 2.
Note that for p = 1, . . . , n - 2, the vectors lp are perpendicular to the vectors (\widetilde ap,\widetilde bp), respectively.
Consider also the function, which is generated by the above liner transformation
F \ast (x, y) = F
\biggl(
bnx - bn - 1y
an - 1bn - anbn - 1
,
anx - an - 1y
anbn - 1 - an - 1bn
\biggr)
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
ON THE REPRESENTATION BY BIVARIATE RIDGE FUNCTIONS 585
Corollary 2.2. The functions fi, i = 1, . . . , n, in Theorem 2.1 can be constructed inductively by
the formulas
fp = \varphi p,n - p - 1, p = 1, . . . , n - 2,
fn - 1 = h1,n - 1, fn = h2,n - 1.
Here,
h1,1(t) =
\partial n - 2
\partial l1 . . . \partial ln - 2
F \ast (t, 0),
h2,1(t) =
\partial n - 2
\partial l1 . . . \partial ln - 2
F \ast (0, t) - \partial n - 2
\partial l1 . . . \partial ln - 2
F \ast (0, 0),
h1,k+1(t) =
1
e1 \cdot lk
t\int
0
h1,k(z)dz, k = 1, . . . , n - 2,
h2,k+1(t) =
1
e2 \cdot lk
t\int
0
h2,k(z)dz, k = 1, . . . , n - 2,
and
\varphi p,1(t) =
\partial n - p - 2F \ast
\partial lp+1 . . . \partial ln - 2
\Biggl( \widetilde apt\widetilde a2p +\widetilde b2p ,
\widetilde bpt\widetilde a2p +\widetilde b2p
\Biggr)
- h1,p+1
\Biggl( \widetilde apt\widetilde a2p +\widetilde b2p
\Biggr)
-
- h2,p+1
\Biggl( \widetilde bpt\widetilde a2p +\widetilde b2p
\Biggr)
-
p - 1\sum
s=1
\varphi s,p - s+1
\Biggl( \widetilde as\widetilde ap +\widetilde bs\widetilde bp\widetilde a2p +\widetilde b2p t
\Biggr)
,
p = 1, . . . , n - 2
\biggl(
for p = n - 2,
\partial n - p - 2F \ast
\partial lp+1 . . . \partial ln - 2
:= F \ast
\biggr)
,
\varphi p,k+1(t) =
1
(\widetilde ap,\widetilde bp) \cdot lk+p
t\int
0
\varphi p,k(z)dz, p = 1, . . . , n - 3, k = 1, . . . , n - p - 2.
The validity of above formulas for the functions h1,k and h2,k, k = 1, . . . , n - 1, is obvious.
The formulas for \varphi p,1 and \varphi p,k+1 can be obtained from (2.5), (2.7) and the subsequent (assumed but
not written) equations if we put x = \widetilde apt/(\widetilde a2p +\widetilde b2p) and y = \widetilde bpt/(\widetilde a2p +\widetilde b2p).
Remark 2.1. If in Theorem 2.1 k \geq n - 1, then the functions fi, i = 1, . . . , n, can be constructed
(up to polynomials) by the method discussed in Buhmann and Pinkus [3]. This method is based on
the fact that for a direction \bfc = (c1, . . . , cm) orthogonal to a given direction \bfa \in \BbbR m\setminus \{ \bfzero \} , the
operator
D\bfc =
m\sum
s=1
cs
\partial
\partial xs
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
586 R. A. ALIEV, A. A. ASGAROVA, V. E. ISMAILOV
acts on m-variable ridge functions g(\bfa \cdot \bfx ) as follows:
D\bfc g(\bfa \cdot \bfx ) = (\bfc \cdot \bfa ) g\prime (\bfa \cdot \bfx ).
Thus, if in our case for fixed r \in \{ 1, . . . , n\} , vectors lk, k \in \{ 1, . . . , n\} , k \not = r, are perpendicular
to the vectors (ak, bk), then
n\prod
k=1
k \not =r
DlkF (x, y) =
n\prod
k=1
k \not =r
Dlk
n\sum
i=1
fi(aix+ biy) =
=
n\sum
i=1
\left( n\prod
k=1
k \not =r
((ai, bi) \cdot lk)
\right) f
(n - 1)
i (aix+ biy) =
=
n\prod
k=1
k \not =r
((ar, br) \cdot lk) f (n - 1)
r (arx+ bry).
Now fr can be easily constructed from the above formula (up to a polynomial of degree at most
n - 2). Note that this method is not valid if in Theorem 2.1 the function F is of the class Cn - 2(\BbbR 2).
However, in this case, Corollary 2.2 is applicable.
Remark 2.2. Some polynomial terms appear while attempting to obtain a smoothness result in
multivariate case. In [2], we proved that if a function f(x1, . . . , xn) of a certain smoothness class
is represented by a sum of r arbitrarily behaved ridge functions, then, under suitable conditions, it
can be represented by a sum of ridge functions of the same smoothness class and some n-variable
polynomial of a certain degree. The appearance of a polynomial term is mainly related to the fact
that in \BbbR n, n \geq 3, there are many directions orthogonal to a given direction. Note that a polynomial
term also appears in verifying if a given function of n variables (n \geq 3) is a sum of ridge functions
(see [5]). However, paralleling the above theorem, we conjecture that if a multivariate function of a
certain smoothness class is represented by a sum of arbitrarily behaved ridge functions, then it can
also be represented by a sum of ridge functions of the same smoothness class.
Remark 2.3. For k \geq n - 1, Theorem 2.1 can be obtained from Theorem 3.1 of [2]. Indeed,
according to Theorem 3.1 [2], the function F in Theorem 2.1 can be represented in the form
F (x, y) =
n\sum
i=1
fi(aix+ biy) + P (x, y), (2.8)
where the functions fi \in Ck(\BbbR ) and P (x, y) is a polynomial of total degree at most n - 1. But it is
known that a bivariate polynomial of degree n - 1 is decomposed into a sum of ridge polynomials
with any given n pairwise linearly independent directions (see, e.g., [21]). That is, in (2.8)
P (x, y) =
n\sum
i=1
pi(aix+ biy),
where pi are univariate polynomials of degree at most n - 1. However, in the setting considered here,
Theorem 2.1 is more informative. It covers the extra case k = n - 2 and allows one to construct the
representing functions fi by induction.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
ON THE REPRESENTATION BY BIVARIATE RIDGE FUNCTIONS 587
Unfortunately, we do not yet know if the lower assumed degree of smoothness n - 2 in Theo-
rem 2.1 can be reduced. We think that the final and complete solution to the smoothness problem
considered here requires essentially different approach. Nevertheless, we can strengthen our result
by considering Hölder continuous functions.
We say that a function F : \BbbR m \rightarrow \BbbR , m \geq 1, is locally Hölder continuous with degree \alpha ,
0 < \alpha \leq 1, if for any compact set K \subset \BbbR m there is a number M = M (K) > 0 such that for any
\bfx = (x1, . . . , xm) \in K and \bfy = (y1, . . . , ym) \in K the inequality
| F (\bfx ) - F (\bfy )| \leq M \cdot
m\sum
i=1
| xi - yi| \alpha
holds. By Ck,\alpha (\BbbR m) we denote the class of functions in Ck(\BbbR m), kth order partial derivatives of
which are locally Hölder continuous with degree \alpha .
The following theorem is valid.
Theorem 2.2. Assume that (ai, bi), i = 1, . . . , n, are pairwise linearly independent vectors in
\BbbR 2 and a function F \in Ck,\alpha (\BbbR 2) has the form
F (x, y) =
n\sum
i=1
gi(aix+ biy),
where gi are arbitrary univariate functions and k \geq n - 2. Then F can be represented also in the
form
F (x, y) =
n\sum
i=1
fi(aix+ biy).
Here, the functions fi \in Ck,\alpha (\BbbR ), i = 1, . . . , n.
The proof of Theorem 2.2 can be easily obtained from Theorem 2.1. Indeed, to prove The-
orem 2.2 it is only needed to repeat the proof of Theorem 2.1, emphasizing that the functions
appearing in the right-hand sides of formulas (2.3), (2.5), (2.7), etc. belong to certain classes
Cs,\alpha (\BbbR ) with step by step increasing indicator of smoothness s. More precisely, the functions h1,1,
h2,1 \in Ck - n+2,\alpha (\BbbR ), the functions h1,2, h2,2, \varphi 1,1 \in Ck - n+3,\alpha (\BbbR ), the functions h1,3, h2,3, \varphi 1,2,
\varphi 2,1 \in Ck - n+4,\alpha (\BbbR ), etc.
Theorem 2.1 implies that if a function F \in Ck(\BbbR 2) has the form (2.1) and k \geq n - 2,then all the
partial derivatives of F up to order k are representable as a sum of ridge functions with the given
directions (ai, bi), i = 1, . . . , n. Note that the validity of Theorem 2.1 for other possible k strongly
depends on answers to the following two questions.
Question 1. Assume a function F \in Ck(\BbbR 2) is of form (2.1). Are the first order partial deriva-
tives \partial F/\partial x and \partial F/\partial y representable as a sum of arbitrarily behaved ridge functions with the
directions (ai, bi)?
Question 2. Assume a function F \in C(\BbbR 2) is of form (2.1). Is it true that F can be represented
also in the form
\sum n
i=1
fi(aix+ biy) with continuous fi?
Indeed, a positive answer to Question 1 would mean, by induction, that all the partial derivatives
up to order k and hence any mixed directional derivative \partial kF/\partial l1 . . . \partial lk are represented by a sum\sum n
i=1
gi(aix+ biy), where gi are arbitrary univariate functions. Once we could answer Question 2
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
588 R. A. ALIEV, A. A. ASGAROVA, V. E. ISMAILOV
affirmatively, we would immediately obtain that any derivative \partial kF/\partial l1 . . . \partial lk is also written in the
form
\sum n
i=1
fi(aix+ biy) with continuous fi. Then by choosing the directions l1, . . . , lk orthogonal
to the first k directions (ai, bi), i = 1, . . . , n, and applying the above method (see the proof of
Theorem 2.1) we could conclude that F has representation
\sum n
i=1
fi(aix+ biy), where fi \in Ck(\BbbR ).
It should be remarked that Question 2 is a part of the more general question posed in [26, p. 14].
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Received 11.09.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
|
| id | umjimathkievua-article-263 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:18Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/cd/4f348223a1b3a48fe7841739876997cd.pdf |
| spelling | umjimathkievua-article-2632025-03-31T08:48:07Z On the representation by bivariate ridge functions On the representation by bivariate ridge functions On the representation by bivariate ridge functions Aliev, R. A. Asgarova, A. A. Ismailov, V. E. Aliev, Rashid Asgarova, Aysel Ismailov, Vugar Aliev, R. A. Asgarova, A. A. Ismailov, V. E. Cauchy functional equation ridge function plane wave representation smoothness Cauchy functional equation ridge function plane wave representation smoothness UDC 517.5 We consider the problem of representation of a bivariate function by sums of ridge functions. It is shown that if a function of a certain smoothness class is represented by a sum of finitely many arbitrarily behaved ridge functions, then it can also be represented by a sum of ridge functions of the same smoothness class. As an example, this result is applied to a homogeneous constant coefficient partial differential equation. УДК 517.5 Про зображення гребеневими функцiями двох змiнних Розглядається задача зображення двовимiрної функцiї сумами гребеневих функцiй. Показано, що коли функцiю певного класу гладкостi зображено скiнченною сумою гребеневих функцiй довiльної поведiнки, її також можна зобразити сумою гребеневих функцiй того ж класу гладкостi. Як приклад цей результат застосовано до однорiдного диференцiального рiвняння з частинними похiдними i зi сталими коефiцiєнтами. Institute of Mathematics, NAS of Ukraine 2021-05-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/263 10.37863/umzh.v73i5.263 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 5 (2021); 579 - 588 Український математичний журнал; Том 73 № 5 (2021); 579 - 588 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/263/9011 Copyright (c) 2021 Vugar Ismailov |
| spellingShingle | Aliev, R. A. Asgarova, A. A. Ismailov, V. E. Aliev, Rashid Asgarova, Aysel Ismailov, Vugar Aliev, R. A. Asgarova, A. A. Ismailov, V. E. On the representation by bivariate ridge functions |
| title | On the representation by bivariate ridge functions |
| title_alt | On the representation by bivariate ridge functions On the representation by bivariate ridge functions |
| title_full | On the representation by bivariate ridge functions |
| title_fullStr | On the representation by bivariate ridge functions |
| title_full_unstemmed | On the representation by bivariate ridge functions |
| title_short | On the representation by bivariate ridge functions |
| title_sort | on the representation by bivariate ridge functions |
| topic_facet | Cauchy functional equation ridge function plane wave representation smoothness Cauchy functional equation ridge function plane wave representation smoothness |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/263 |
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