On the representation by linear superpositions

0.16 MB·English

by  Vugar Ismailov

ON THE REPRESENTATION BY LINEAR
SUPERPOSITIONS 1
Vugar E. Ismailov
Mathematics and Mechanics Institute
5 Azerbaijan National Academy of Sciences
1
Az-1141, Baku, Azerbaijan
0
2 e-mail: [email protected]

n
a Abstract. In a number of papers, Y. Sternfeld investigated the problems of repre-
J
sentation of continuous and bounded functions by linear superpositions. In particular, he
1
2 proved that if such representation holds for continuous functions, then it holds for bounded

functions. We consider the same problem without involving any topology and establish a
]
A rather practical necessary and sufficient condition for representability of an arbitrary func-
F tion by linear superpositions. In particular, we show that if some representation by linear
.
h superpositions holds for continuous functions, then it holds for all functions. This will lead
t
a us to the analogue of the well-known Kolmogorov superposition theorem for multivariate
m
functions on the d-dimensional unit cube.
[

1 Mathematics Subject Classifications: 26B40, 41A05, 41A63
v Keywords: Linear superposition; Closed path; Ridge function
8
6
2
5 1. Introduction
0
.
1 Let X be any set and h : X → R, i = 1,...,r, be arbitrarily fixed functions. Consider
0 i
5 the set
1
: r
v B(X) = B(h ,...,h ;X) = g (h (x)), x ∈ X, g : R → R, i = 1,...,r (1.1)
i 1 r i i i
X ( )
i=1
X
r
a
Members of this set will be called linear superpositions (see [35]). We are going to answer
the question: what conditions on X guarantee that each function on X will be in the set
B(X)? The simplest case X ⊂ Rd, r = d and h are the coordinate functions has been
i
solved in [16]. See also [5,15] for the case r = 2.
By B (X) and B (X) denote the right hand side of (1.1) with continuous and bounded
c b
g : R → R, i = 1,...,r, respectively. Our starting point is the well-known superposition
i
theorem of Kolmogorov [17]. It states that for the unit cube Id, I = [0,1], d ≥ 2, there
exists 2d+1 functions {s }2d+1 ⊂ C(Id) of the form
q q=1
d
s (x ,...,x ) = ϕ (x ), ϕ ∈ C(I), p = 1,...,d, q = 1,...,2d+1 (1.2)
q 1 d pq p pq
p=1
X
1 This researchhas been supported by INTAS under Grant YSF-06-1000015-6283
1
such that each function f ∈ C(Id) admits the representation
2d+1
f(x) = g (s (x)), x = (x ,...,x ) ∈ Id, g ∈ C(R). (1.3)
q q 1 d q
q=1
X
In our notation, (1.3) means that B (s ,...,s ;Id) = C(Id). This surprising and deep
c 1 2d+1
result, which solved (negatively) Hilbert’s 13-th problem, was improved and generalized
in several directions. It was first observed by Lorentz [21] that the functions g can be
q
replaced by a single continuous function g. Sprecher [29] showed that the theorem can be
proven with constant multiples of a single function ϕ and translations. Specifically, ϕ in
pq
(1.2) can be chosen as λpϕ(x +εq), where ε and λ are some positive constants. Fridman
p
[9] succeeded in showing that the functions ϕ can be constructed to belong to the class
pq
Lip(1). Vitushkin and Henkin [35] showed that ϕ cannot be taken to be continuously
pq
differentiable.
Ostrand [25] extended the Kolmogorov theorem to general compact metric spaces. In
particular, he proved that for each compact d-dimensional metric space X there exist con-
tinuous real functions {α }2d+1 ⊂ C(X) such that B (α ,...,α ;X) = C(X). Sternfeld
i i=1 c 1 2d+1
[32] showed that the number 2d + 1 cannot be reduced for any d-dimensional space X.
Thus the number of terms in the Kolmogorov superposition theorem is the best possible.
Some papers of Sternfeld have been devoted to the representation of continuous and
bounded functions by linear superpositions. Let C(X) and B(X) denote the space of
continuous and bounded functions on some set X respectively (in the first case, X is
supposed to be a compact metric space). Let F = {h} be a family of functions on X. F is
called a uniformly separating family (u.s.f.) if there exists a number 0 < λ ≤ 1 such that
for each pair {x }m , {z }m of disjoint finite sequences in X, there exists some h ∈ F
j j=1 j j=1
so that if from the two sequences {h(x )}m and {h(z )}m in h(X) we remove a maximal
j j=1 j j=1
number of pairs of points h(x ) and h(z ) with h(x ) = h(z ), there remains at least
j1 j2 j1 j2
λm points in each sequence (or , equivalently, at most (1 − λ)m pairs can be removed).
Sternfeld [31] proved that for a finite family F = {h ,...,h } of functions on X, being a
1 r
u.s.f. is equivalent to the equality B (h ,...,h ;X) = B(X), and that in the case where
b 1 r
X is a compact metric space and the elements of F are continuous functions on X, the
equality B (h ,...,h ;X) = C(X) implies that F is a u.s.f. Thus, in particular, Sternfeld
c 1 r
obtained that the formula (1.3) is valid for all bounded functions, where g are bounded
q
functions depending on f (see also [15, p.21]).
Let X be a compact metric space. The family F = {h} ⊂ C(X) is said to be a measure
separating family (m.s.f.) if there exists a number 0 < λ ≤ 1 such that for any measure µ
in C(X)∗, the inequality kµ◦h−1k ≥ λkµk holds for some h ∈ F. Sternfeld [33] proved
that B (h ,...,h ;X) = C(X) if and only if the family {h ,...,h } is a m.s.f. In [31], it has
c 1 r 1 r
been shown that if r = 2, then the properties u.s.f. and m.s.f. are equivalent. Therefore,
the equality B (h ,h ;X) = B(X) is equivalent to B (h ,h ;X) = C(X). But for r > 2,
b 1 2 c 1 2
these two properties are no longer equivalent. That is, B (h ,...,h ;X) = B(X) does not
b 1 r
always imply B (h ,...,h ;X) = C(X) (see [33]).
c 1 r
Our purpose is to consider the above mentioned problem of representation by linear
2
superpositions without involving any topology (that of continuity or boundedness). More
precisely, we want to characterize those sets X for which B(h ,...,h ;X) = T(X), where
1 r
T(X) is the space of all functions on X. This will be done in terms of closed paths, the
explicit and practically convenient objects. We show that nonexistence of closed paths in
X is equivalent to the equality B(X) = T(X). In particular, we will obtain that B (X) =
c
C(X) implies B(X) = T(X). Therefore, the formula (1.3) is valid for all multivariate
functions over the unite cube Id, where g are univariate functions depending on f. We will
q
also present an example due to Khavinson [15] showing that even in the case r = 2, the
equality B(h ,h ;X) = T(X) does not imply B (h ,h ;X) = C(X). At the end we will
1 2 c 1 2
make some observations around the problems of representation and interpolation by ridge
functions, which are widely used in multivariate approximation theory.
2. Closed paths
In the sequel, by δ we will denote the characteristic function of a set A ⊂ R. That is,
A
1, if y ∈ A
δ (y) =
A 0, if y ∈/ A.
(cid:26)
The following definition is based on the ideas set forth in the works [1] and [16].
Definition 2.1. Given a set X and functions h : X → R, i = 1,...,r. A set of points
i
{x ,...,x } ⊂ X is called to be a closed path with respect to the functions h ,...,h (or,
1 n 1 r
concisely, a closed path if there is no confusion), if there exists a vector λ = (λ ,...,λ )
1 n
with the nonzero real coordinates λ , i = 1,...,n, such that
i
n
λ δ = 0, i = 1,...,r. (2.1)
j hi(xj)
j=1
X
Let for i = 1,...,r, the set {h (x ), j = 1,...,n} have k different values. Then it is not
i j i
r
difficult to see that Eq. (2.1) stands for a system of k homogeneous linear equations
i=1 i
in unknowns λ ,...,λ . If this system has any solution with the nonzero components, then
1 n
P
the given set {x ,...,x } is a closed path. In the last case, the system has also a solution
1 n
m = (m ,...,m ) with the nonzero integer components m , i = 1,...,n. Thus, in Definition
1 n i
2.1, the vector λ = (λ ,...,λ ) can be replaced by a vector m = (m ,...,m ) with m ∈
1 n 1 n i
Z\{0}.
For example, the set l = {(0,0,0), (0,0,1), (0,1,0), (1,0,0), (1,1,1)} is a closed path
in R3 with respect to the functions h (z ,z ,z ) = z , i = 1,2,3. The vector λ in Definition
i 1 2 3 i
2.1 can be taken as (−2,1,1,1,−1).
In the case r = 2, the picture of closed path becomes more clear. Let, for example,
h and h be the coordinate functions on R2. In this case, a closed path is the union of
1 2
some sets A with the property: each A consists of vertices of a closed broken line with
k k
the sides parallel to the coordinate axis. These objects (sets A ) have been exploited in
k
3
practically all works devoted to the approximation of bivariate functions by univariate
functions, although under the different names (see, for example, [15, chapter 2]). If X
and the functions h and h are arbitrary, the sets A can be described as a trace of
1 2 k
some point traveling alternatively in the level sets of h and h , and then returning to its
1 2
primary position. It should be remarked that in the case r > 2, closed paths do not admit
such a simple geometric description. We refer the reader to Braess and Pinkus [1] for the
description of closed paths when r = 3 and h (x) = ai·x, x ∈R2, ai ∈ R2\{0}, i = 1,2,3.
i
Let T(X) denote the set of all functions on X. With each pair hp,λi, where p =
{x ,...,x } is a closed path in X and λ = (λ ,...,λ ) is a vector known from Definition
1 n 1 n
2.1, we associate the functional
n
G : T(X) → R, G (f) = λ f(x ).
p,λ p,λ j j
j=1
X
In the following, such pairs hp,λi will be called closed path-vector pairs of X. It is clear
that the functional G is linear. Besides, G (g) = 0 for all functions g ∈ B(h ,...,h ;X).
p,λ p,λ 1 r
Indeed, assume that (2.1) holds. Given i ≤ r, let z = h (x ) for some j. Hence,
i j
λ = 0and λ g (h (x )) = 0. Asummationyields G (g ◦h ) = 0.
j (hi(xj)=z) j j (hi(xj)=z) j i i j p,λ i i
r
Since G is linear, we obtain that G ( g ◦h ) = 0.
P p,λ P p,λ i=1 i i
The following lemma characterizes the set B(h ,...,h ;X) under some restrictions and
1 r
P
will be used in the proof of Theorem 2.5 below.
Lemma 2.2. Let X have closed paths h (X)∩h (X) = ∅, for all i,j ∈ {1,...,r}, i 6= j.
i j
Then a function f : X → R belongs to the set B(h ,...,h ;X) if and only if G (f) = 0
1 r p,λ
for any closed path-vector pair hp,λi of X.
Proof. The necessity is obvious, since the functional G annihilates all members of
p,λ
B(h ,...,h ;X). Let us prove the sufficiency. Introduce the notation
1 r
Y = h (X), i = 1,...,r;
i i
Ω = Y ∪...∪Y .
1 r
Consider the following subsets of Ω:
L = {Y = {y ,...,y } : if there exists x ∈ X such that h (x) = y , i = 1,...,r} (2.2)
1 r i i
Note that L is a set of some certain subsets of Ω. Each element of L is a set Y =
{y ,...,y } ⊂ Ω with the property that there exists x ∈ X such that h (x) = y , i = 1,...,r.
1 r i i
In what follows, all the points x associated with Y by (2.2) will be called (∗)-points of Y. It
is clear that the number of such points depends on Y as well as on the functions h ,...,h ,
1 r
and may be greater than 1. But note that if any two points x and x are (∗)-points of Y,
1 2
thennecessarily theset {x , x } formsaclosed pathwiththe associatedvector λ = (1;−1).
1 2
In this case, by the condition of the sufficiency, f(x ) = f(x ). Let now Y∗ be the set of all
1 2
(∗)-points of Y. Since we have already known that f(Y∗) is a single number, we can define
the function
4
t : L → R, t(Y) = f(Y∗).
Or, equivalently, t(Y) = f(x), where x is an arbitrary (∗)-point of Y.
k
Consider nowaclassS offunctionsoftheform r δ ,where k isapositive integer,
j=1 j Dj
r are real numbers and D are elements of L, j = 1,...,k. We fix neither the numbers
j j
P
k, r , nor the sets D . Clearly, S is a linear space. Over S, we define the functional
j j
k k
F : S → R, F r δ = r t(D ).
j Dj j j
!
j=1 j=1
X X
First of all, we must show that this functional is well defined. That is, the equality
k1 k2
rj′δD′ = rj′′δD′′
j j
j=1 j=1
X X
always implies the equality
k1 k2
r′t(D′) = r′′t(D′′).
j j j j
j=1 j=1
X X
In fact, this is equivalent to the implication
k k
r δ = 0 =⇒ r t(D ) = 0, for all k ∈ N, r ∈ R, D ⊂ L. (2.3)
j Dj j j j j
j=1 j=1
X X
Supposethattheleft-handsideoftheimplication(2.3)besatisfied. EachsetD consists
j
of r real numbers yj,...,yj, j = 1,...,k. By the hypothesis of the lemma, all these numbers
1 r
are different. Therefore,
r
δ = δ , j = 1,...,k. (2.4)
Dj yj
i
i=1
X
Eq. (2.4) together with the left-hand side of (2.3) gives
r k
r δ = 0. (2.5)
j yj
i
i=1 j=1
XX
Since the sets {y1,y2,...,yk}, i = 1,...,r, are pairwise disjoint, we obtain from (2.5) that
i i i
k
r δ = 0, i = 1,...,r. (2.6)
j yj
i
j=1
X
Let now x ,...,x be some (∗)-points of the sets D ,...,D respectively. Since by (2.2),
1 k 1 k
j
y = h (x ), for i = 1,...,r and j = 1,...,k, it follows from (2.6) that the set {x ,...,x }
i i j 1 k
5
k
is a closed path. Then by the condition of the sufficiency, r f(x ) = 0. Hence
j=1 j j
k
r t(D ) = 0. We have proved the implication (2.3) and hence the functional F is well
j=1 j j P
defined.
P
Note that the functional F is linear (this can be easily seen from its definition). Let F′
be a linear extension of F to the following space larger than S:
k
S′ = r δ ,
j ωj
( )
j=1
X
where k ∈ N, r ∈ R, ω ⊂ Ω. As in the above, we do not fix the parameters k, r and ω .
j j j j
Define the functions
g : Y → R, g (y ) d=ef F′(δ ), i = 1,...,r.
i i i i yi
Let x be an arbitrary point in X. Obviously, x is a (∗)-point of some set Y = {y ,...,y } ⊂
1 r
L. Thus,
r r
f(x) = t(Y) = F(δ ) = F δ = F′ δ =
Y yi yi
! !
i=1 i=1
X X
r r r
F′(δ ) = g (y ) = g (h (x)).
yi i i i i
i=1 i=1 i=1
X X X
Definition 2.3. A closed path p = {x ,...,x } is said to be minimal if p does not
1 n
contain any closed path as its proper subset.
For example, the set l = {(0,0,0), (0,0,1), (0,1,0), (1,0,0), (1,1,1)} considered
above is a minimal closed path with respect to the functions h (z ,z ,z ) = z , i = 1,2,3.
i 1 2 3 i
Adding the point (0,1,1) to l, we will have a closed path, but not minimal. The vector λ
associated with l∪{(0,1,1)} can be taken as (3,−1,−1,−2,2,−1).
A minimal closed path p = {x ,...,x } has the following obvious properties:
1 n
(a) The vectorλ associatedwith p by Eq. (2.1) is unique up to multiplication by a constant;
n
(b) If in (2.1), |λ | = 1, then all the numbers λ , j = 1,...,n, are rational.
j=1 j j
Thus, a minimPal closed path p uniquely (up to a sign) defines the functional
n n
G (f) = λ f(x ), |λ | = 1.
p j j j
j=1 j=1
X X
Lemma 2.4. The functional G is a linear combination of functionals G ,...,G ,
p,λ p1 pk
where p ,...,p are minimal closed paths in p.
1 k
6
Proof. Let hp,λi be a closed path-vector pair of X, where p = {x ,...,x } and λ =
1 n
(λ ,...,λ ). Assume that p is not minimal. Let p = {y ,...,y } be a minimal closed
1 n 1 1 s
path in p and
s s
G (f) = ν f(y ), |ν | = 1.
p1 j j j
j=1 j=1
X X
To prove the lemma, it is enough to show that G is a linear combination of G and
p,λ p1
some functional G , where l is a closed path in X and a proper subset of p. Without loss
l,θ
of generality, we may assume that y = x . Put
1 1
λ
1
t = .
1
ν
1
Then the functional G −t G has the form
p,λ 1 p1
k
G −t G = θ f(z ),
p,λ 1 p1 j j
j=1
X
where z ∈ p, θ 6= 0, j = 1,...,k. Clearly, the set l = {z ,...,z } is a closed path with
j j 1 k
the associated vector θ = (θ ,...,θ ). Thus, we obtain that G = G −t G . Note that
1 k l,θ p,λ 1 p1
since x ∈/ l, the closed path l is a proper subset of p.
1
Theorem 2.5. 1) Let X have closed paths. A function f : X → R belongs to the space
B(h ,...,h ;X) if and only if G (f) = 0 for any minimal closed path p ⊂ X with respect to
1 r p
the functions h ,...,h .
1 r
2) Let X has no closed paths. Then B(h ,...,h ;X) = T(X).
1 r
Proof. 1) The necessity is clear. Let us prove the sufficiency. On the strength of Lemma
2.4, it is enough to prove that if G (f) = 0 for any closed path-vector pair hp,λi of X,
p,λ
then f ∈ B(X).
Consider a system of intervals {(a ,b ) ⊂ R}r such that (a ,b )∩(a ,b ) = ∅ for all
i i i=1 i i j j
the indices i,j ∈ {1,...,r}, i 6= j. For i = 1,...,r, let τ be one-to-one mappings of R onto
i
(a ,b ). Introduce the following functions on X:
i i
′
h (x) = τ (h (x)), i = 1,...,r.
i i i
It is clear that any closed path with respect to the functions h ,...,h is also a closed
1 r
path with respect to the functions h′,...,h′, and vice versa. Besides, h′(X)∩h′(X) = ∅,
1 r i j
for all i,j ∈ {1,...,r}, i 6= j. Then by Lemma 2.2,
f(x) = g′(h′(x))+···+g′(h′(x)),
1 1 r r
where g′,...,g′ are univariate functions depending on f. From the last equality we obtain
1 r
that
7
f(x) = g′(τ (h (x)))+···+g′(τ (h (x))) = g (h (x))+···+g (h (x)).
1 1 1 r r r 1 1 r r
That is, f ∈ B(X).
2) Let f : X → R be an arbitrary function. First suppose that h (X) ∩ h (X) = ∅,
i j
for all i,j ∈ {1,...,r}, i 6= j. In this case, the proof is similar to and even simpler than
that of Lemma 2.2. Indeed, the set of all (∗)-points of Y consists of a single point, since
otherwise we would have a closed path with two points, which contradicts the hypothesis
of the 2-nd part of our theorem. Further, the well definition of the functional F becomes
obvious, since the left-hand side of (2.3) also contradicts the nonexistence of closed paths.
Thus, as in the proof of Lemma 2.2, we can extend F to the space S′ and then obtain the
desired representation for the function f . Since f is arbitrary, T(X) = B(X).
Using the techniques from the proof of the 1-st part of our theorem, one can easily
generalize the above argument to the case when the functions h ,...,h have arbitrary
1 r
ranges.
Theorem 2.6. B(h ,...,h ;X) = T(X) if and only if X has no closed paths.
1 r
Proof. The sufficiency immediately follows from Theorem 2.5. To prove the necessity,
assumethatX hasaclosedpathp = {x ,...,x }. Letλ = (λ ,...,λ )beavector associated
1 n 1 n
with p by Eq. (2.1). Consider a function f on X with the property: f (x ) = 1, for indices
0 0 i
i such that λ > 0 and f (x ) = −1, for indices i such that λ < 0. For this function,
i 0 i i
G (f ) 6= 0. Then by Theorem 2.5, f ∈/ B(X). Hence B(X) 6= T(X). The contradiction
p,λ 0 0
shows that X does not admit closed paths.
TheconditionwhetherX haveclosedpathsornot,dependsbothonX andthefunctions
h ,...,h . In the following sections, we see that if h ,...,h are ”nice” functions (smooth
1 r 1 r
functions with the simple structure. For example, ridge functions) and X ⊂ Rd is a ”rich”
set (for example, the set with interior points), then X has always closed paths. Thus
the representability by linear combinations of univariate functions with the fixed ”nice”
multivariate functions requires at least that X should not possess interior points. The
picture is quite different when the functions h ,...,h are not ”nice”. Even in the case
1 r
when they are continuous, we will see that many sets of Rd (the unite cube, any compact
subset of that, or even the whole space Rd itself) may have no closed paths. If disregard
the continuity, there exists even one function h such that every multivariate function is
representable as g ◦h over any subset of Rd.
3. The analogue of the Kolmogorov superposition theorem for
multivariate functions
Let X be a set and h : X → R, i = 1,...,r, be arbitrarily fixed functions. Consider
i
a class A(X) of functions on X with the property: for any minimal closed path p ⊂ X
with respect to the functions h ,...,h (if it exists), there is a function f in A(X) such
1 r 0
8
that G (f ) 6= 0. Such classes will be called ”permissible” function classes. Clearly, C(X)
p 0
and B(X) are both permissible function classes (in case of C(X), X is considered to be a
normal topological space).
Theorem 3.1. Let A(X) be a permissible function class. If B(h ,...,h ;X) = A(X),
1 r
then B(h ,...,h ;X) = T(X).
1 r
The proof is simple and based on the material of the previous section. Assume for a
moment that X admit a closed path p. The functional G annihilates all members of the
p
set B(h ,...,h ;X). By the above definition of permissible function classes, A(X) contains
1 r
a function f such that G (f ) 6= 0. Therefore, f ∈/ B(h ,...,h ;X). We see that the
0 p 0 0 1 r
equality B(h ,...,h ;X) = A(X) is impossible if X has a closed path. Thus X has no
1 r
closed paths. Then by Theorem 2.6, B(h ,...,h ;X) = T(X).
1 r
Remark. In the ”if part” of Theorem 3.1, instead of B(h ,...,h ;X) and A(X) one
1 r
can take B (h ,...,h ;X) and C(X) (or B (h ,...,h ;X) and B(X)) respectively.
c 1 r b 1 r
The main advantage of Theorem 3.1 is that we need not check directly if the set X has
no closed paths, which in many cases may turn out to be very tedious task. Using this
theorem, we can extend free-of-chargetheexisting superposition theorems for classes B(X)
or C(X) (or some other permissible function classes) to all functions defined on X. For
example, this theorem allows us to obtain the analogue of the Kolmogorov superposition
theorem for all multivariate functions defined on the unit cube.
Corollary 3.2. Let d ≥ 2 and s , q = 1,...,2d+1, be the Kolmogorov functions (1.2).
q
Then each function f : Id → R can be represented by the formula (1.3), where g are
q
univariate functions depending on f.
It should be remarked that Sternfeld [31], in particular, obtained that the formula (1.3)
is valid for functions f ∈ B(Id) provided that g are bounded functions depending on f
q
(see [15, chapter 1] for more detailed information and interesting discussions).
Let X be a compact metric space and h ∈ C(X), i = 1,...,r. The result of Sternfeld
i
(see Introduction) and Theorem 3.1 give us the implications
B (h ,...,h ;X) = C(X) ⇒ B (h ,...,h ;X) = B(X) ⇒ B(h ,...,h ;X) = T(X). (3.1)
c 1 r b 1 r 1 r
The first implication is invertible when r = 2 (see [31]). We want to show that the
second is not invertible even in the case r = 2. The following interesting example is due to
Khavinson [15, p.67].
Let X ⊂ R2 consists of a broken line whose sides are parallel to the coordinate axis
and whose vertices are
1 1 1 1 1 1
(0;0),(1;0),(1;1),(1+ ;1),(1+ ;1+ ),(1+ + ;1+ ),...
22 22 22 22 32 22
9
We add to this line the limit point of the vertices (π2, π2). Let r = 2 and h ,h be
6 6 1 2
the coordinate functions. Then the set X has no closed paths with respect to h and
1
h . By Theorem 2.6, every function f on X is of the form g (x ) + g (x ), (x ,x ) ∈ X.
2 1 1 2 2 1 2
Now construct a function f on X as follows. On the link joining (0;0) to (1;0) f (x ,x )
0 0 1 2
continuously increases from 0 to 1; on the link from (1;0) to (1;1) it continuously decreases
from 1 to 0; on the link from (1;1) to (1+ 1 ;1) it increases from 0 to 1; on the link from
22 2
(1+ 1 ;1) to (1+ 1 ;1+ 1 ) it decreases from 1 to 0; on the next link it increases from 0
22 22 22 2
to 1, etc. At the point (π2, π2) set the value of f equal to 0. Obviously, f is a continuous
3 6 6 0 0
functions and by the above argument, f (x ,x ) = g (x )+g (x ). But g and g cannot
0 1 2 1 1 2 2 1 2
be chosen as continuous functions, since they get unbounded as x and x tends to π2.
1 2 6
Thus, B(h ,h ;X) = T(X), but at the same time B (h ,h ;X) 6= C(X) (or, equivalently,
1 2 c 1 2
B (h ,h ;X) 6= B(X)).
b 1 2
We have seen that the unit cube in Rd has no closed paths with respect to some 2d+1
continuous functions (namely, the Kolmogorov functions s (1.2)). From the result of
q
Ostrand [25] (see Introduction) it follows that d-dimensional compact sets X also have no
closed paths with respect to some 2d+1 continuous functions on X. One may ask if there
exists a finite family of functions {h : Rd → R}n such that any subset of Rd does not
i i=1
admit closed paths with respect to this family? The answer is positive. This follows from
the result of Demko [7]: there exist 2d+1 continuous functions ϕ ,...,ϕ defined on Rd
1 2d+1
such that every bounded continuous function on Rd is expressible in the form 2d+1g◦ϕ
i=1 i
for some g ∈ C(R). This theorem together with Theorem 2.6 yield that every function
on Rd is expressible in the form 2d+1g ◦ ϕ for some g : R → R, i = 1P,...,2d + 1.
i=1 i i i
We do not yet know if g here or in Corollary 3.2 can be replaced by a single univariate
i
P
function. We also don’t know if the number 2d + 1 can be reduced so that the whole
space of Rd (or any d-dimensional compact subset of that, or at least the unit cube Id)
has no closed paths with respect to some continuous functions ϕ ,...,ϕ : Rd → R, where
1 k
k < 2d+1. One of the basic results of Sternfeld [32] says that the dimension of a compact
metric space X equals d if and only if there exist functions ϕ ,...,ϕ ∈ C(X) such
1 2d+1
that B (ϕ ,...,ϕ ;X) = C(X) and for any family {ψ }k ⊂ C(X), k < 2d + 1, we
c 1 2d+1 i i=1
have B (ψ ,...,ψ ;X) 6= C(X). In particular, from this result it follows that the number
c 1 k
of terms in the Kolmogorov superposition theorem cannot be reduced. But since the
equalities B (X) = C(X) and B(X) = T(X) are not equivalent, the above question on the
c
nonexistence of closed paths in Rd with respect to less than 2d+1 continuous functions is
far from trivial.
If disregard the continuity, one can construct even one function ϕ : Rd → R such that
the whole space Rd will not possess closed paths with respect to ϕ and therefore, every
function f : Rd → R will admit the representation f = g ◦ ϕ with some univariate g
depending on f. Our argument easily follows from Theorem 2.6 and the result of Sprecher
[30]: for any natural number d, d ≥ 2, there exist functions h : I → R, p = 1,...,d, such
p
that every function f ∈ C(Id) can be represented in the form
10