ON POLYNOMIAL APPROXIMATION OF FUNCTIONS ON HILBERT SPACE

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ON POLYNOMIAL APPROXIMATION OF FUNCTIONS ON HILBERT SPACE

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Mat. Sbornik Math. USSR SbornikTom 92 (134)(1973), No. 2 Vol. 21 (1973), No. 2

ON POLYNOMIAL APPROXIMATION OF FUNCTIONS ON HILBERT SPACE

UDC 513-881

A. S. NEMIROVSKII AND S. M. SEMENOV

Abstract. In this paper the question is considered of uniform approximation bypolynomials and smooth functions of uniformly continuous functions given on the unitball of countable-dimensional Hubert space.

Bibliography: 8 items.

Introduction

According to the classical Stone-Weierstrass theorem every continuous function on

a ball in η-dimensional space is the uniform limit of smooth functions (even polyno-

mials). Compactness of a ball in «-dimensional space plays a basic role in the proof

of this fact. In infinite-dimensional spaces balls are not compact and therefore theo-

rems of Stone-Weierstrass type do not immediately extend to this case.

One of the first approximation theorems in infinite-dimensional spaces was

Frechet's theorem asserting, essentially, that every continuous function on a sepa-

rable real Banach space can be represented as a pointwise limit of polynomials (L3J?

Russian p. 96).

In this theorem the question is pointwise approximation, and Frechet's construc-

tion does not give (and under the indicated general assumptions cannot give) a uniform

approximation.

With respect to uniform approximation, a theorem of Bonic and Frampton is known

(see [2], §2), which for simplicity we here state for Hubert space, although it has

been proved for a wider class of (Banach) spaces.

Let Κ be a closed subset of a separable real Hilbert space Η and suppose f is

a continuous function on K. Then f can be approximated uniformly on Κ by the re-

strictions of smooth functions on H.

Here the approximating functions have finite, but in general not bounded, Frechet

derivatives.

Another variant of the approximation theorem asserts that (see [5]) every uniformly

continuous and bounded function on a closed subset of a separable real Banach space

AMS (MOS) subject classifications (1970). Primary 41A10, 41A65.

Copyright © 1974, American Mathematical Society

255

256 A. S. NEMIROVSKIi AND S. M. SEMENOV

can be uniformly approximated by the restrictions of Gateaux differentiable functions

with bounded derivative.

In both cases the approximating functions do not form sufficiently natural classes.

By natural classes of approximations it is evidently reasonable to mean classes of

functions with bounded Frechet derivatives or, even better, the class of polynomials.

Thus the questions arise of (1) uniform approximation of uniformly continuous

functions by functions having bounded derivatives up to a given order, and (2) uniform

approximation of functions having a certain order of smoothness by functions of a

higher order of smoothness or even polynomials. The present paper presents results

obtained in this direction. As a rule we restrict ourselves to the case of functions

given on a real separable Hubert space.

The general answers to all the questions posed above are negative. Approxima-

tion of continuous functions by functions with one (bounded) Frechet derivative is in

fact possible (§1), but in general there are no approximations of higher smoothness

(§7).Furthermore, we shall construct examples of a) a function having a first Frechet

derivative that satisfies a Lipschitz condition but which is not approximable by func-

tions with two uniformly continuous derivatives (§7), and b) a function having uni-

formly continuous derivatives of all orders but not admitting polynomial approxima-

tions (§5).

The positive content of the paper is connected in an essential way with problems

of polynomial approximation. We shall construct a class of functions, relatively simple

in structure, admitting polynomial approximation—the class of regular functions. Their

definition (§6) is given in terms of continuity with respect to a certain topology on

Hubert space that is weaker than the usual one (but stronger than the weak topology).

The class of such functions turns out to be a nonseparable algebra closed with re-

spect to uniform convergence.

The results on polynomial approximation can be applied, for example, to the the-

ory of the Laplace-Levy operator, where they supply all the basic assertions concern-

ing correct solvability of boundary value problems (in a suitable extension of the class

of regular functions). The presentation of these questions, however, is beyond the

scope of this paper, and the authors intend to devote a separate paper to them.

Notation

Below we shall employ the following notation.

All spaces that are domains of definition of functions are infinite-dimensional

real separable Hubert space, if nothing to the contrary is mentioned. Η always de-

notes Hubert space. V (respectively S ) denotes a ball (sphere) in this space:

Vx.r={l\\x-l\<r}, 5 * . Γ = < ξ | | | * - ξ | = Γ};

V0 . is usually denoted simply by V.

For any set W and function /: W -> F, where F is a Banach space, we set

ON POLYNOMIAL APPROXIMATION OF FUNCTIONS ON HILBERT SPACE 257

if the right side makes sense.

If W C Ε is an open subset of the Banach space Ε and F is a Banach space,

then for each η = 0, 1, · · · ,<» Dn{W, F) denotes the space of η times Frechet differ-

entiable functions on W with values in F; for / € Dn{W, F) Dkf(x), k < n, denotes

the mapping that assigns to χ the &th differential of / at x, which is a mapping into

the space of continuous ^-linear operators from Ε to F furnished with the natural

norm. The value of the differential at a vector h € Ε will be denoted by {Dkf{x), h)·

Dn(W, F) denotes the subspace of Dn(W, F) consisting of all mappings each deriva-

tive of which is bounded on any bounded subset of W; Dn{W, F) denotes the subspace

of D"(W, F) consisting of the mappings each derivative of which is uniformly contin-

uous on bounded subsets of W.

If U C int U (more complicated cases will not arise here), then the spaces

Dn(U, F), DîU, F) and Dn((J, F) can be defined naturally: they consist precisely of

the continuous mappings of U into F whose restrictions to int U belong to the cor-

responding spaces and whose derivatives (in the case of D™(U, F)) can be extended to

continuous functions on U.

We shall assume that D^{U, F) and D^(U, F) have the topology of uniform con-

vergence together with derivatives up to the corresponding order on bounded subsets

of U. We shall omit the symbol F if F is the field of real (complex) numbers.

§1. Uniform approximation by functions of class D^{H)

In this section we establish that every uniformly continuous function on a ball in

Hubert space Η is the uniform limit of restrictions to the ball of functions in DU(H)

having a derivative that satisfies a Lipschitz condition. First we prove this asser-

tion for convex functions and then extend it to the general case.

We need the simple geometric

Lemma 1. Let Κ be a closed convex subset in Η and let Κ = Κ + VQ , r > 0.

We denote by p(x, Κ ) the usual distance from the point χ to the set Κ . Then out-

side any set K^ , rQ > r, the function p(x, Κ ) has a bounded derivative satisfying a

Lipschitz condition.

We omit the proof of this lemma.

Theorem 1. Let f be a uniformly continuous unimodal function on the ball V.

Then for any e > 0 there exists a function g e D^(H) with derivative satisfying a

Lipschitz condition, such that ||/— g\\v < e.

Proof. For definiteness suppose j/(x)| < 1 on V. We construct a partition of the

interval [- 1, l] into subintervals Δ^, | z | < M, of length not exceeding <r/2, and let

τ{ be the midpoint of Δ?.. Choose δ > 0 so small that the distance from τ. to Δ. for

i ^ / is greater that 8. Let

258 A. S. NEMIROVSKII AND S. M. SEMENOV

•Qt = Γ(Δ/), 5, = <*€ V| /(*)<τ,>.

Then the sets S. are convex and closed. Choose α > 0 so small that for any x,

y e V satisfying ||x - y\\ < α we have \f(x) - f{y)\ < 8·

Let S\ = Si + VQ a/2. We now construct functions φί on Η such that 1) φ.(χ) > 0

on H; 2) ΣΜ_ ΜφΧχ) > 1 on V; 3) if φ ix) / 0, then \fix) - r | < e for χ e V; 4) all

the functions 0. lie in D \H) and their derivatives satisfy a Lipschitz condition.

To effect this construction note that Q. C S. . and piQ ••> S. .) > a/4 for \i\ <

Λί — 1. Indeed, the inclusion is obvious and the inequality follows from the fact that if

x e Qi and y € S{_i, with | | JC- y|| < a/4, then by the construction of S._. there

exists y' e ·$"·_ j such that \\y - y'\\ < a /2 and hence | | x - y'\\ < a. But f(y') < r. l

and fix) € Δ?. so that |/(ΛΓ) - fiy')\ > 8, which is impossible by the choice of a.

We now apply Lemma 1 to each of these sets 5".. By this lemma the functions

r\x) = pix, 5\) are continuously differentiable outside S. and have a derivative satis-

fying a Lipschitz condition outside any r-neighborhood of the corresponding sets. If

dit) is a smooth function on the line, equal to zero for t < a/8 and to 1 for t > a/4,

with 0 < dit) < 1, then the functions τ {χ) = Oir-ix)) satisfy requirement 4).

Now it is possible to set <£_M(x) = 1 - r _Mix), <frMix) = r M , ( x ) and φ-ix) =ri-Vx'^ ~ r Γχ'' ^ΟΓ IZ1 S Λ1 - 1· ^ is easy to verify properties 1)—4) for this family

of functions.

Since Σ . _ _ Μ φ ix) > 1 on V and the right side is uniformly continuous on H, it is

possible to find a function (fkx) equal to zero on the ball V, nonnegative, satisfying

condition 4) and such that Σ Μ ..φ{χ) + φix) > 1 in Η.ι = —ΛΙ ιι ' —

Using these functions we construct a partition of unity on V by setting

.<*(*)= Τ . \i\<M.

φ (*) -f- 2 *i Μ

It is clear that the φ. satisfy condition 4). Now it is possible to take the approx-

imation for / in the form gix) — Σ,Μ_ ^τ .φΧχ). This g evidently satisfies the smooth-

ness conditions required in the theorem, and in a straightforward way it can be veri-

fied that it approximates / o n V by no worse than e. The theorem is proved.

Reduction of the general case to that of convex functions (!) is based on the fol-

lowing theorem.

Theorem 1 . The polynomials in functions convex and uniformly continuous on V

are dense in the space D (V).

Proof. Let u be the smallest uniformly closed algebra of functions on V con-

taining all convex functions of the class D (V); we shall prove that U = D iv). First

of all we can see that for any r, 0 < r < 1, the restrictions of functions from u to SQ

The idea of this reduction is due to V. D. Mil man.


are dense in the space of uniformly continuous functions on SQ ^. Note that if / e u

and φ is a continuous function on the line, then the composition φ ° f also lies in (t.

Now let r be fixed and suppose / is a uniformly continuous function on SQ f.

To prove that / is a uniform limit of restrictions to SQ r of functions in u it

suffices to show that if Q C SQ is a closed subset and e > 0, then there ex-

ists a function g € £? such that g|g = 1 and g(x) - 0 for all χ e So ^ such that

p(x, Q) > e (the rest of the argument is the same as in the proof of Theorem 1). Let us

construct the function g. Let Q be the closed convex hull of Q. Then the function

p(x, Q) is uniformly continuous and convex in V so that p{x, Q) e u. Since the Hu-

bert norm is uniformly convex, for some Ν - N(r) and all χ € SQ ^ we have N(r)[p(x, Q)Y2

> p(x, Q) > p(x, Q)· From this inequality it follows that if 6{t) is a function on the

line, continuous, equal to 1 at the origin and zero outside a sufficiently small neigh-

borhood of the origin, then the function g{x) = θ ο p(x, Q) [S the desired one.

We now complete the proof of the theorem. Let f > 0 and / £ D {v). We construct

a function g e Cl such that ||/— g\\ < e. By what we have already proved, for any r,

0 < r < 1, there exists a function g € u such that ||/ — g \\s < e/2. By virtue of the

uniform continuity of our functions there exists an interval Δ . 3 r such that for all

χ e V with ||x|| e Δ^ we have ||/(x) - gj<x)\\ < e. Let Δ^ ,· · · , Δ be a finite cover-

ing of the interval [θ, l] extracted from the covering {Δ }, and let θ At), - · · , 6N(t) be

a continuous partition of unity on [0, l] subordinate to this covering. Then the func-

tion g{x) = ζ,^_^^ \χ)θ\\\χ\\) is obviously the desired one. The theorem is proved.

From Theorems 1 and 1 the result promised at the beginning of this section evi-

dently follows.

Corollary. The restrictions to V of the functions of the class D (//) having a

derivative that satisfies a Lipschitz condition are dense in the space D0(H).

Remark. As we shall show below (see §7), the class Dl(H) in this corollary can-

not be replaced by D2(H).

Remark. Arguments similar to those used in the proof of Theorems 1 and l ' show

that if a Banach space admits a norm that is uniformly convex and uniformly differen-

tiable outside a neighborhood of the origin, then the restrictions of functions in D1{E)

to the unit ball V of Ε are dense in D°(V).

§2. Polynomials on a Banach space; definitions and properties

Before passing to the question of polynomial approximations, we establish some

properties of polynomials in infinitely many variables. Let us make more precise the

notion of a polynomial mapping of a Banach space Ε into a Banach space F.

Theorem 2. Let f: Ε -* F. The following properties of f are equivalent:

1) / is continuous, and for any a and h in Ε the function f(a + bi) is a poly-

nomial in t;

2) / is bounded on some open set, and for any a and b in Ε the functions

260 Α · s · NEMIROVSKII AND S. M. SEMENOV

f(a + bt) are polynomials in t whose degrees are bounded;

3) / can be uniquely represented in the form fix) = Σ" /.(χ, · · · , χ), where the

fix.,· · · , ΛΤ.) are continuous i-linear forms on Ε with values in F.

Proof. The implication 3) -> 2) is obvious. We prove 2) -> 1). For definiteness

suppose / is bounded in the ball of radius r > 0 with center at the origin. Consider

the family of polynomials / (/) = fite), where e e SQ 1# These polynomials have uni-

formly bounded degrees and are uniformly bounded on some interval with center at the

origin. Hence it follows that they are uniformly bounded on any (finite) interval of t,

i.e. / is bounded on any ball. Let us verify that / is continuous. We shall even show

that / satisfies a Lipschitz condition in any ball V . Consider the functions

/ 4 0 / * Γ (*/-*)).

These functions are polynomials in / of degrees not exceeding n, and they are

bounded uniformly in x, y 6 V on every (finite) interval Δ of ί (since for / € Δ the

point χ + t(y - #)/||y - x|| belongs to some fixed ball V(A)).

Then the indicated family of functions is "equi-Lipschitz" on the interval

[0, div')], where d(v') is the diameter of the ball V1; hence / satisfies a Lipschitz

condition in V . In [6] it was shown that from 1) follows 2), while from 1) and 2) fol-

lows 3), as shown in [ l], §3.1. The theorem is proved.

Remark. If dim Ε < °°, then the continuity requirement can be omitted from 1).

Definition. The mapping /: Ε -> F is called polynomial (or simply a polynomial)

if it has one of the equivalent properties 1), 2), 3) enumerated in Theorem 2.

The degree of the polynomial / is the order of the highest nonzero form /. in the

decomposition given in 3).

A polynomial is called homogeneous if only one term in its decomposition is dif-

ferent from zero.

If dim Ε < oo, then our definition of a polynomial reduces to the usual one, i.e.

Let us enumerate some simple properties of polynomials.

1. If F is a Banach algebra, then the space of polynomials forms an algebra with

respect to pointwise multiplication.

2. The restriction of a polynomial to a subspace is also a polynomial whose de-

gree does not exceed the degree of the original.

3. If /: Ε -» F is a polynomial mapping and deg f - n, then

a) / € D~(E, F),

b) D fix) is a polynomial mapping of Ε into the corresponding space of degree

η - k; Dkf(x) = 0 for k > n;

c) the estimate ||Dfe/(x)||y < C^(«, fOII/Hy holds in any ball VQ γ.

In the sequel we shall also need the following theorems.

Theorem 3. Let f : Ε -* F be a polynomial mapping with deg / < Ν and


χ e E> !!/„(*) - A^llII/JI ν - c for e v e r y n * a n d s u P P o s e f : E ~ * F is such t h a t > f°T

-» 0 as η -> °°·

Then f: Ε -* F is a polynomial with deg / < N.

Proof. It is obvious that / satisfies condition 2) of Theorem 2 and its degree with

respect to any line does not exceed N.

Theorem 4. Let f : Ε -» R be a polynomial with deg fn < Ν and \{n\v < C for

every n. If Ε is separable, then it is possible to extract a subsequence {/ } that con-

verges pointwise to a polynomial.

Proof. Let X be a countable dense subset of E. By property 3c) the fn are

uniformly bounded and equi-Lipschitz in any ball. Therefore, applying the diagonal

process, it is possible to extract a subsequence {/ } that converges pointwise on X.

Since {/ } is equi-Lipschitz, it will converge pointwise on E.

Setting fix) = lim, _o o/ (*) and applying Theorem 3, we obtain the required as-

sertion.

§3. Some algebras generated by polynomials

Let V be the closed unit ball in H, and let Ck(V) {k = 0, 1, · · · ,<») be the

smallest uniformly closed algebra of functions on V containing all real polynomials of

degree not exceeding k. The simplest of such algebras is the algebra C Av) generated

by the bounded linear functionals on H. Since the bounded linear functionals are con-

tinuous in the weak topology of the Hubert ball, and the latter is compact in the weak

topology, by the Stone-Weierstrass theorem C^(v) coincides with the algebra of all

weakly continuous functions on V. It is obvious that CΑν) ^ C 2(v). In fact, among

quadratic forms on Η the only weakly continuous ones are those generated by a com-

pletely continuous selfadjoint operator, while the form (x, x), for example, is not

weakly continuous.

It is obvious that CîV) C ^ + 1 ^ ) . It is natural to explore whether this chain of

algebras stabilizes. Note that in the finite-dimensional case C^V) already coincides

with the algebra of all continuous functions on V (the Stone-Weierstrass theorem). In

the infinite-dimensional case the situation is otherwise.

Theorem 5 (see [4]). // dim Η = <χ>, then the chain ί ^ ί ν ) } ^ does not stabilize,

i.e. the equality C^(v) = C^ s(V) does not hold for any k when s > 0.

We present an outline of the proof, omitting technical points. An orthonormal

basis e~ = le^l^lj is fixed; we shall denote the coordinates of a vector χ in this ba-

sis by x^ For any set π of positive integers let

t= 0 for

We introduce the notion of standard polynomial (relative to the basis e~) in H. Namely,

a real polynomial p is called standard if for any Ν and χ = Σ^_ χ .e., χ. ^ 0, and any

262 Α · s · NEMIROVSKII AND S. M. SEMENOV

x' = î-ixje•/{) such that /(ζ) > j(k) when i > k, we have p(x) = p(x'). In other words,

a standard polynomial must not change when its argument is replaced by any vector

with the same sequence of nonzero coordinates (the places where they stand are not

essential, only the order of their succession is important).

By repeated application of the diagonal process to the coefficients of a given poly-

nomial it is proved that for any f > 0 and any polynomial p there exist a standard poly-

nomial p( of the same degree and an infinite set π of positive integers such that

The theorem is now easily proved. Namely, the space of standard polynomials of

degree not exceeding η is finite dimensional (they are uniquely recoverable from their

restrictions to the subspace spanned by the first η vectors of the basis e"). If we as-

sume that Cnîy) = Cn(V) for all / > 0, then every homogeneous form \fjm(x) = 2°^*™

can be approximated by polynomials of degree not exceeding n; if p., · · · , p, e C (v)

and R\t~, · · · , t^) is a polynomial such that

then there exist nv and standard polynomials ρ^ντ · · , p,v oi degree not exceeding η

such that \p v — p \ <γ on Η f] V. For sufficiently small γ we obtain

Ι Ψ™ — * ( f t * · · · , Ρ*γ)1 ν Γ ΐ Η < 2 ε .

Since the polynomials on the left are standard, this inequality holds on all of V.

On the other hand, using the finite dimensionality of the space of standard polynomials

of degree not exceeding n, it is easy to find two points Xj and * 2

a n d a n integer

m> η such that Xj and x2 cannot be separated by any standard polynomials of degree

not exceeding n, but can be separated by the form ψ"1. Therefore the inequality above

is impossible for small e, which proves the theorem.

Remark. The result of Theorem 5 remains valid when Hubert space is replaced by

any space / 1 < p < °°. On the contrary, for the unit ball in the space cQ it turns

out that CAv ) = CîV ) is precisely the space of uniformly continuous (in the

weak topology of cQ) real functions on V . These results can be proved by the same

methods as Theorem 5.

Definitive results on the structure of natural subalgebras of C2(V) have also been

announced in [7]·

Theorem 6. Suppose ί^α} and \B Λ are two systems of selfadjoint bounded opera-

tors in Η and let C1[{A(Z}] and CjQfi J ] be the smallest uniformly closed algebras of

functions on V containing C^V) and the families jU f tx, x)\ and {{ΒβΧ, χ)}, respec-

tively, of quadratic forms. Then CjtJAj] = (^[{βο}] holds if and only if L(\Aj) =

L\{Bg\), where L({Aa}) and L(\Ba}) are the closed linear hulls of the images of the

operators of the systems {Aa\ and \Bg\ under projection of the ring Horn (H, H) onto

the quotient modulo the ideal of completely continuous operators.


From this theorem it follows, in particular, that if the quadratic form (Bx, X) can

be uniformly approximated on V with arbitrary precision by polynomials in the forms

(Α^χ, %),. . . , (A x, x) with coefficients in C^{V), then

where Τ is a completely continuous operator and λ^ 6 R.

§4. Symmetric polynomials

In this section we shall consider the structure of a particular subalgebra of the

algebra of polynomials on Hubert space, namely the subalgebra of symmetric polyno-

mials, which will play an essential role in the sequel.

Let ~e = {e-Γ .̂* be an orthonormal basis in H. By the symmetric group of Η with

respect to the basis e" is meant the subgroup Σ— of the group of orthogonal operators

in Η algebraically generated by the transpositions of coordinates in the basis e". A

set Μ C Η is called symmetric if it is invariant with respect to the action of the opera-

tors in Σ - . It is obvious that the closure of a symmetric set in any metric on Η in

which the operators of Σ_ are continuous is also symmetric.

Definition. A function /: Μ -» R defined on a symmetric set Μ is called symmet-

ric if f(o{x)) = fix) for any χ e Μ and ο e Σ_.

Let Η be the subspace of Η spanned by the vectors e^, · · · , e and let Φ be

the set of vectors with only finitely many nonzero coordinates in the basis e". It is

obvious that Φ = U •>,// . Define functions ψ171: W1 —* R and \bm\ Φ—»R by the formulas

ft oo

i # (*!,.... Xn) = 2 *r, ym(x)= 2 *r,

where m = 1, 2,· · · . These functions are symmetric polynomials on Rw and Φ re-

spectively; moreover for m > 2 the ψ71, naturally defined by the same formulas on H,

are polynomials on H.

Proposition 1. For m < η the functions ψ1^, · · · , t/r™ are algebraically indepen-

dent, i.e. if

where Qiy 1 ? · · · , y ) is a polynomial, then Q Ξ 0.

Proof. Define a mapping Fm: R" -> Rm by the formulas

This is naturally a smooth mapping. Compute the Jacobian matrix of the mapping Fm

1 . . . 1

2* r . . . 2xn

264 Α · S. NEMIROVSKII AND S. M. SEMENOV

It is easy to see that the upper left m χ m minor of this matrix is different from zero at

points for which the coordinates are distinct. By the theorem on the rank of the range

V of Fm it has a nonempty interior.

Therefore, if QiF™ix)) = 0, then Q\^ = 0. Consequently Q = 0.

Theorem 7. Let f: Φ -> R be a symmetric polynomial on Φ of degree m. Then

there exists a polynomial Q(yl,· · · •> ym) such that fix) = Qitf/1ix), - · • , if/m{x)) for all

χ e Φ.

Proof. For any η the polynomial f\H is the usual symmetric polynomial of de-

gree not exceeding m. It is well known that there exists a polynomial Q such that

/ (PnX) = Qn (tfn (P*), . . . , < (P«*)),

where Ρ η: Η -> Hn is the natural projection. Note that Ψ\^Ρ ̂ χ) = ̂ k^k^ ^ o r ^ - n

and φΚΡ,χ) = ώ{(Ρ,χ). We shall show that Q , = Q for all / - 0, 1, In fact,K, KM fZ Til -f-t "2

let

Then for χ - Ρ χ we havem

f {PmX) = Qm+l (ψ1» (Ρ«*) , . . - , ψ™ (PmX)),

f (PmX) = Qm (ψ« ( P ^ ) , . . . , tfn (Prr.X))

by the preceding remark.

Since, by Proposition 1, ψm, · · · , ψ7^ are algebraically independent on R777, Qm+i

= Q . Consequently the equality fix) = Q (^Hx))· · · > ifsmix)) holds on all of Φ, and

such a representation is natural.

Corollary. The space of symmetric polynomials on Φ of degree not exceeding m

is finite dimensional.

Proof. By Theorem 7, fix) = Q iif/lix),- · · , ijjmixj) when deg f < m. Consequently

the mapping f \-* f ο Ρ^ is an isomorphism of the space of symmetric polynomials of

degree not exceeding m, defined on Φ, onto the analogous space of polynomials on

R7", which is obviously finite dimensional.

Lemma 2. Suppose the set Κ C R772 is such that for any open ί/C K. the set

p~liU)is unbounded, where Κγ = piK) and p: Rm -> R 7 " " 1 is the orthoprojection of the

arithmetic Euclidean spaces ((x., · · · , χ ) Η (χ , . . . , χ )) and int /C, ̂ 0. If ther I m \ m — I 1 ^ '

polynomial Qiy ^ · · · » y ) is bounded on K, then dQ/dy, Ξ 0.

Proof. Assume the contrary. Then

k

Q (#i, · · ·, ym) = 2/=0


with 1 < k < deg Q and p, ^ 0. In turn there exists a point a £ int p{K) such that

pk(a) ^ 0. By assumption there exists a sequence \aj,as € K, with p(as) -> a as s-»oc

and |(« ) J -> o· when s -><*>. Then

A:

2 > C Ι ρ* (α)| |(as)i|*->oo as s-*oo.

This contradicts the boundedness of Q on K. This contradiction proves the lemma.

Theorem 8. Let f: Η -> R fee fl symmetric polynomial of degree m. Then there ex-

ists a polynomial Q(y.,. . . , y j) swcA

orz H.

Proof. Let / = f\ · Then by Theorem 7 there exists a polynomial Q(y1? · · · » y w )

such that /(x) = Q(^Hx),· · · , if/m(x)). In addition / is bounded on VQ lf[ Φ. Let Κ

be the range in R™ of the mapping χ h-> (ι/rHx), · · · » ψτη(χ)) of VQ χ Π Φ· It is easy to

see that Κ and the polynomial Q satisfy the hypotheses of Lemma 2, so that dQ/dy^

= 0. Letting Q(y j , · · . , y j) = Q(0, y 1 ' ' i i > y m _ 1 ) ) we obtain the assertion of the

theorem.

It is possible, of course, to define the action of the group Σ of permutations of

the natural numbers, each of which moves only finitely many integers, in the space /

and c_. Indeed, let χ = (x^, · · · , χ ? · · · ) , and set io~x). - xCT_ Χι^γ cr e Σ. In this case,

just as in the case of H, it is possible to define symmetric functions as constants on

the orbits of points in / (respectively in cQ) under the action of Σ.

Theorem 9. Let /: / -» R be a symmetric polynomial of degree m. Then there ex-

ists a polynomial Q(y,, · · · , y ,) such that fix) = ζ){ψαρ(χ), · • · , if/m{x)), where

a = [p] + 1 for nonintegral p and ap = p for integral p. In particular, on the space cQ of

bounded sequences having a limit every symmetric polynomial is identically a constant.

Theorem 9 has a continuous analog. Let Ά be the group of transformations of the

interval LO, 1J preserving Lebesgue measure. The elements of Ξ act in the spacesL

p[0, lL Ρ = 1, in the following way: o(x){t) = xia'Ht)) for every χ € L [θ, l ] ,

t € [0, l ] , σ e Ξ· It is clear that this definition is correct. Next, let Μ be a H-invari-

ant subset of L\_0, l ] .

Definition. A function / on Μ is called symmetric if f(ax) = f{x) for all σ eH

and χ 6 M.

Natural examples of symmetric polynomials on Μ are the functions

Theorem 9 . Let f: L [0, l] -> R be a symmetric polynomial of degree m. Then

there exists a polynomial Q: Rw -> R, where w = min{m,[p]),such that for any x £ L [0, l]

2 gg A. S. NEMIROVSKII AND S. M. SEMENOV

f(x) = Q[$x(t)dt, ... , \xw{t)dt).

§5. An example of a smooth function not approximate by polynomials

In the infinite-dimensional case the class of functions admitting uniform approxi-

mation by polynomials is very small. It not only does not contain all uniformly con-

tinuous functions, it does not even contain all smooth functions.

Let CîV) be the algebra of uniform limits of polynomials on the ball V of infi-

nite-dimensional (separable) real Hubert space, and let Aîv) be the algebra of uni-

form limits of functions in D°°{V).b

Theorem 10. CjV) / Ajv).

Proof. Let R" be real «-dimensional Euclidean space and V its unit ball. In

V choose a maximal (in cardinal number) system Γ of points such that any two of

them are at distance at least 1/2 from one another. Let R(n) be the number of points

in Γ . Then #(72) > 2n. In fact, if x., · · · , xDt , are the points of Γ and U . , · · · ,η — ι κ (η) Γ η ι

UR, ) are the balls of radius lA with centers at these points, then Vn C U z ^ z by the

maximality of Γ^. Therefore meas Vn < meas Uz·^· = ̂ " ' m e a s U{ = R(n) meas ί/ρ

whence

meas Vn

On the other hand, for a given natural number s let J S be the space of polyno-

mials on Rw of degree not exceeding 5. It is easy to see that dim J s < sns + 1.

Choose n{s) so large that 2 w ( s ) > sW-s)]·5 + 1. Let Τη be the space of functions on

Γ with uniform norm, and let r^: / s , . -» Τ . . be the linear mapping carrying a func-

tion into its restriction to ^n^sy Since dim Tn(s) = R(n(s)) > dim ^^(sy the range of

rs is a proper subspace of Γ Β , . ; therefore there exists <fis £ Τ , . such that \ψ3\ =

1 and | | 0 , - Γ , ρ | | > 1 for all ρ e ? ^ .

Now let VQ 4 be the ball of radius 4 with center at the origin in the infinite-

dimensional real separable Hubert space H, and let {H'.}'*, be balls of radius 1 con-

tained in ô 4 ' w ith distance between centers at least 5/2. We may assume that

V . . is the section of W by a finite-dimensional plane L of dimension n(s) pass-

ing through its center, so that Γ . .Cff . Let Γ = U°° ,Γ , ., and let φ be the func-

tion on Γ defined by φ\γ < , = φ .

It is obvious that ||<£||r = 1. Note that for any polynomial p in Η we have

\\p - 0llp d 1· 1° fact, if deg p = s, then

II Ρ - φ «r > UP - Ψ* Hrn(4) = II x sip \L) - <PS \\Γη($) > ι ·

On the other hand, if Γ = {ΛΓ-IÎJ, then the points x. and x. are at distance at least


1/2 apart (z / /)· If 0(i) is a smooth function on the line, with (9(0) = 1 and 6{t) = 0

for | i | > 1/8, then the functionCO

= 2 φ(^)θ(ΐμ-χ/|Γ)

lies in D°?(H) and obviously ψ\Γ = φ- Therefore for any polynomial p in Η we have

\\P - Ψ\\ν > 1* a n d Ψ i ^οο^νο 4·*· T h e t n e o r e m i s proved.

Remark. With obvious modifications the preceding construction can be realized

not only in Hubert space but also in any infinite-dimensional Banach space E. It shows

that if there is a nontrivial function in Ε of class Du{E) with bounded support, then

there exists a function of the same smoothness not approximable by polynomials. In

particular there are always uniformly continuous functions on Ε not approximable by

polynomials.

§6. Regular functions

In the preceding section we saw that sufficient conditions guaranteeing polyno-

mial approximations for a function on Hubert space Η cannot be reduced to the re-

quirement of uniform continuity in the usual topology of H. Below we shall show that

uniform continuity of the function in another linear topology on H, intermediate between

the strong and weak topology, implies the existence of such approximations. This re-

sult can be considered as an infinite-dimensional variant of the classical Weierstrass

theorem.

We begin with a description of the topology mentioned above.

Definition. Let eO = \e(}\°°_l be an orthogonal basis in Η and let \\X\\-Q =

maxjix, ei ) | . A regular topology in Η is the weakest locally convex topology R in

which all seminorms py(x) = \\UX\\—Q corresponding to bounded linear operators U in

Η ate continuous.

A mapping f: W -* Ε of a subset W of Η into a Banach space Ε is called regular

if it is bounded and uniformly continuous in the topology R on every bounded subset W.

Example. Let /: Η -> Ε be a Frechet differentiable mapping such that in coordi-

nates relative to some orthogonal basis e" we have Σ°?_Λ3/(χ)/θχ.\\ < N(r) for ||x|| < r.

Then for x, y € V we obviously have ||/(x) - f(y)\\ < N{r)\\x - y\\-, and / is regular

on H. In particular, if p: Η -> £ is a polynomial in whose decomposition relative to

coordinates in the basis ~e coordinate powers less than three do not occur, then p is

regular (in fact, by hypothesis dp{x)/dxj\x _Q = d2p(x)/dxj\x _Q = 0, whence2 ^V o > 1 | x j | and Σ ~ J ^ U V ^ - I I < N | 2 | | i > 2 H I V o > | | x | | ) · Thus the

elementary symmetric polynomials φρ(χ) = Σ°° χ? of §4 are regular for p > 2.

We now prove the main result of the section—an approximation theorem for regular

functions.

Theorem 11. Every regular mapping f: W -» Ε can be approximated by polynomials

uniformly on bounded subsets of W.


Proof. It suffices to verify that if W is bounded and / is regular on W, then given

e > 0 there exists a polynomial p( with values in Ε such that \\f — Ρ€\\ψ < €· For defi-

niteness we shall assume that W is contained in the unit ball V of H.

Since / is regular, by the definition of the topology R there exist linear operators

Ul, - · · , Us_ j in Η and a number μ > 0 such that from x, y e W and \\uXx - y)\\~^ <

μ, ι'< s - 1, it follows that ||/(>r) - /(y)|| < f/2. Let //' be the direct sum of 5 copies

of Η and take η > 1 so large that ||ί/.|| < η for i = 1, - · · , s - 1. Define the imbedding

π: Η -> Η' by setting πχ = {xiny/s)' l , (l/.x)( ny/s)~ l , - · - , (U .x^ny/s)-1). Let V' be

the unit ball in //', let W' = n(W) C V' and let / (x) be the function on W' defined by

/ (πχ) = fix). Denote by if the orthogonal basis {e®~, · · - , e°\ in H' (e° is the basis

in each direct summand). Obviously from x, y 6 W and \\x - y\\_ < u. - /z(«\/5)~ lt:

follows that ||/ (x) - f (y)(|< f/2. We shall construct a polynomial q such that

11/ - ΐΛ \ < €· To do this we cover V by "cubes" of the form \x\ \\x - a\\_ < 8\,

on whose intersection with W the function / varies, according to what has been

proved, by no more than e/2 for sufficiently small 8. The "cubes" of the covering are

so well situated that it is possible to construct a polynomial partition of unity on V

"almost subordinate" to this covering; using this partition of unity the desired approx-

imation of / is constructed. The construction presented below depends on three posi-

tive parameters δ, γ and v, which are chosen in a suitable way at the end of the con-

struction.

Let 8 > 0. Consider a partition of the interval [- 1, l] by points a., i = 0, ±1,

. . . , ± m(8), such that -ai = a_i and 3δ/2 < \a{ l - «J < 2δ. For any multi-index (k)

= U j , · · · , &p, 0 < \k{\ < m(8), and (z) = (iv · · - , ij), ϊγ < • · · < z/? let

I

and let IT^.j be the cube with center at α\%\'·

Note that if χ e Π^Α then at least / coordinates of χ are greater than 8/2 in

absolute value, so that ||x|| > Sfi/2.

We call the cube Π{^ admissible if Π ^ Π V ^ 0; in this case (k) is called an

admissible index. It is clear that for an admissible index (k) = (k^, · · · , k^ we have

I < 4δ~ , so that the set /§ of admissible indices is finite. By definition we shall as-

sume that /§ contains the empty index 0 , which corresponds to the cube Il^with cen-

ter at the origin:

n0 = <*e//'||*t<*>·Note that the cubes {U^]^kêl form a covering of V'. We now construct a polynomial

partition of unity of V "almost subordinate" to this covering, and use it to construct

an approximation of / . We need

Lemma 3. Let (k) = (k. ,···, k.) be an admissible index and let a ^ = a 1 " " ' {


be one of the corresponding centers. Given γ > 0, there exists a polynomial F'j? on

H' such that

1) degvP§<N{6, γ);2) vP$>0i«//';

3) ν Ρ $ ( χ ) < Μ ( ό ) 4 . . . 4 for χ£ψ·

for xtV, IJC-aJU ) t<o; *Ρ$(*)<Τ*£ ·.'·**, for

P r o o f . T h e f u n c t i o n s ! | x ' | - c a n b e a p p r o x i m a t e d b y p o l y n o m i a l s o n V . I n f a c t , f o r x e V

«* t < ( | rf") < (ii * if")"2"1 ( s *<2) < ι * f " •so that the functions O-,°°_ x2m)1'2m' which are obviously approximable by polynomials

on V , converge to |1*||— uniformly on V as m -> o°. Therefore there exists a symmet-

ric polynomial Q g(x) such that for χ e V

Now consider the mapping iS •) of V' into the coordinate space R :

,·(*) (v\ ir\ IT v\ ν ν \

l(i) (X) = [ylt, {* (»)•*/» ·*'ι» · · ·» •*///'

Here 1 - T,., is the orthogonal projection onto the subspace generated by the vectors

e . ,' ' ' , e. .z l l l

Note that i$(V') C${k), where £ ( f e ) is a ball in R/ + 1 . Let

Note that these sets do not depend on (z), as can be seen from the preceding construe-

tion.

For χ e n j ^ f l V we have ( ^ ( χ » ! < 0 and \U[^(x))j+l -ak\<8, j= 1,· · · , /.

Taking account of the fact that \a | > 3δ/2 for q ̂ 0, we obtain in that X^ does

not intersect Φ ^ = {£ e R/ + 1I ξ2 ··• ξι + ι = θ\· The set ί>(

2*° i s contained in

{16 R/+11 ξχ > 1} U h € R/+11 max {| ξ7- - α* Μ |} > - f

Thus iDj and i A ' ^° n o t intersect, so that there exists a smooth function <f>/k\ on

®(lfe) such that φα) > 0, Φα)\^ (k)> 1 and 0 ( f c ) = 0 in a neighborhood of 3)2

(fe)(J 35(

3

A).

Letting θ ,^λζ) = φ,^Χζ)/^1! · · · £j + i ' and choosing a polynomial /),,, on R + so

that yh{k)> 0 in R/ + 1 , yh{k) < y in 3)(

2

fe) and yh(k)< M(k){8) in (*°

H ^ ^ i s s o s m a 1 1 t h a t f o r £


yh(k) (I) I t ••· lt+ι > 1,

we set

) = *£ ... xilVhik)(i$lx)).It is easy to verify that yP\j\ is the desired polynomial; in this connection its degree,

by construction, does not exceed some N,kXd, y), and it is possible to set N(8, γ) =

max,^ v f i JV,£.(5, γ) and Μ(δ) = max,M e^Μ,^Χδ). The lemma is proved.

From £>g (x) it is easy to obtain a polynomial yP®hc) such that y Pîpc) > 0

> 1 in Π^Π ^, and yp0< γ for χ e v', \\x\\- > 3S/2.

Now define vectors b\t•) in the following way: if there exists χ € V' such that

II* ~ a[^)h ^ ^ Z 2 ' s e t b\i) = f^' otherwise set b^ = 0. Define the vector b&

analogously.

Consider the series

By assertion 3) of Lemma 3 and the fact that X. . x4 . . . x4 < (X°^,x4)^ <1 in

V , we obtain that our series converges at each point of V and its sum is bounded on

V . By Theorem 3 this series of polynomials of uniformly bounded degrees then con-

verges everywhere on Η , and gP(x) is a polynomial.

By assertion 4) of Lemma 3, y $P(x) > 1 in V', since the cubes I T ^ cover V'.

Let a(8, γ) denote the quantity sup^ /1 sP(x)| . Given ν > 0, there exists a

polynomial Tv(/) on the line such that l/t < Ty{t) < l/t + u for t e [l, α(δ, γ)].

Now let

M.VR(X)= ( Σ ^

Then, using the boundedness of / on If it is possible to show, as above, that

g /? is a polynomial on Η with values in E. Let us estimate ||/ - g v^lli

For'x e W'

(χ) = Σ (fix)- b$) [yP$i (x)] [Y>6P (x)]-i( ) ( * ) /

+ Σ b\n

We estimate the first term. Choose δ so that 3δ < μ^. Then by the choice of μ,, for

each (k) and (z) such that ||x - aj*>||F < 35/2, we have ||/"(x) - b$\\ < e/2, but if

ΙΙ χ- αω11? > 3 δ / 2> t h e n ° < r

p ( t / w < x 4 · · · *?z, r , s p W > ι a n d II/'W2||/|| w . Therefore for any χ € W'

where R(S) is the number of elements in /g. Choosing γ > 0 so that 2y||/)|H//?(5) < f/4,


we obtain that on W' the first term does not exceed 3 e/4 in norm.

For the 8 and y already chosen the norm of the second term can evidently be

bounded by ||/||„,«(§, y), which is arbitrarily small for sufficiently small y.

Thus for suitable δ, y and ν we have ||/ - g ν&\ψΐ •<*· Now it is possible to

set p {χ) = ς R{nx)· The theorem is proved.

Remark. The definition of regular topology and regular mapping can of course be

carried over from the Hubert case to the spaces I 1 < p < oo (as e° we take the

natural basis in / and as ||x||~o the quantity max.|x.|? where χ - \x \ is an element of

/ ). A verbatim analog of Theorem 11 is also valid for the spaces / ; the proof repeats

that given above (with obvious modifications).

For our purposes it is useful to extend the stock of functions for which an approxi-

mation theorem like Theorem 11 is valid. Namely, let us call a mapping f: V -* Ε of

the ball V in Hubert space Η into a Banach space Ε semiregular if it is uniformly

continuous in the usual topology of Η and its restriction to any sphere in V is regular

on this sphere.

Proposition 2. Every semiregular mapping f: V -* Ε can be approximated by poly

nomials uniformly on V.

This proposition, which we need in the next section, can be deduced from Theorem

11 by arguments used at the end of the proof of Theorem 1.

§7. Approximation of symmetric functions

Theorem 10 shows that in the general case a meaningful connection between uni-

form smoothness of a function and the existence of polynomial approximations is lack-

ing. It turns out that there is such a connection for symmetric functions. As a corol-

lary we obtain the assertion promised in §1 on the impossibility of approximating an

arbitrary uniformly continuous function by functions of class D .

We shall show that the problem of uniform approximation of a function having sym-

metry by smooth functions or polynomials is equivalent to approximating them by sym-

metric functions of the same type.

Theorem 12. Let f: V -* R be a symmetric function in an orthonormal basis e~ =

\ej\°°,, and suppose p is a polynomial {or function of class Dn(V)) such that \f ~ p\\v

< €· Then there exists a symmetric polynomial p {respectively a symmetric function of

class Dnu{v)) such that | | / - ? | | v .< e.

Proof. The reasoning is based on a application of symmetrization. We need the

following lemma.

Lemma 4. Let ί/^Ι^, be functions of class Dn{V), where

\\Dmfk{x)\\v^Nm, m = 0 , 1, . . . , n,

and for each m<n the family \Dmfk{x)\r£_l of functions is equicontinuous on V· Let


ω^ίδ) be their common modulus of continuity. Then there exist a sequence {k\°°_, and

a function f e Dn(v) such that

D /t (*) f°r any χ e V,

2) \\Dmf(x)\\v < Nm and ω^ίδ) is the modulus of continuity of Dmf for all m < n.

Proof. Let X = \x.|~ be a countable dense set in V. By Theorem 4, for each /

it is possible to extract from the sequence \(Dmfk(x), h)\c^_l of polynomials in h a

subsequence that converges pointwise. By applying the diagonal process we try to

make all sequences

converge for any /' and any h € H. Since the functions Dmf Ax) are equicontinuous

for each m, these sequences will also converge when x. is replaced by any χ € V;

for χ € V and m < η let

/ (x) = D°f (x) = litn /*. (x), (Dmf (x), h) •= lioit->00 i-*OO

(χ), h).

It is clear that for each m < η (Dmf, h) is a homogeneous form of degree m in h; in

addition < Nm, and ω^ίδ) is the modulus of continuity of Dmf.

Let us verify that f{x) is the desired function. It suffices to verify that Dmf(x) is

the rath differential of /. But for all χ e V and h e Η such that χ + h € V

- h w - Σ τ (D

/=!

By the hypothesis of the lemma, for all k

whence

and consequently Dmfix) - Dmfix) for all m < n. The lemma is proved.

Proof of Theorem 12. Consider the sequence {S p\°° , of functions, where

iSmP)(x) = ~ 2 p(a(x))t

σβΣ-e

and Σ— is the subgroup of Σ— consisting of all permutations of coordinates leaving

fixed the coordinates with indices greater than m. Then the functions S p are uni-

formly bounded polynomials on V of one and the same degree (respectively functions

in D" satisfying the hypotheses of Lemma 4). Applying Theorem 4 (Lemma 4), we ex-

tract a subsequence \Sm p\T_, that converges pointwise on V to a polynomial (to a


function in Dn{v)) p. Since for any σ β Σ - all functions of our sequence beginning from

a certain one are invariant with respect to σ, ρ is symmetric. Since / is symmetric, for

any m

\\f-SmP\\v<\\f-p\\v<E,

whence ||/ — p\\ y < e. The theorem is proved.

We now state a criterion for polynomial approximation of symmetric functions.

Theorem 13. Uniform approximation of a symmetric function f: V -> R by polyno-

mials is possible if and only if f is uniformly continuous on V in the metric

p-(*, y)=\\\x\\-\\y\\\\-m^\{x-y1ei)\.

Proof. Necessity. By Theorems 12 and 8, to prove necessity it suffices to verify

that the standard symmetric polynomials ifjm{x) = Σ ^ ^ χ , e?)m, m > 2, are uniformly

continuous in the metric p—. But for x, y e V and m = 2

while for m > 3oo

I ψ1" (x) - ψ" (y)\ < 2 |(JC, ei)m - {y, et)m | < 2m max \{x - y, et)\,

which was required.

Sufficiency follows immediately from Theorem 11. The theorem is proved.

We now prove the main theorem on polynomial approximation of smooth symmetric

functions.

Theorem 14. Let f e D (V) be a symmetric function. Then f can be approximated

by polynomials uniformly on V.

Proof. By Theorem 11 it suffices to verify that / is uniformly continuous on each

sphere with center at 0 in the metric given by || · || —. To prove this fact we need

Lemma 5. Let f £ D (v) be a symmetric function, and let ω(δ) be the modulus of

continuity of its second derivative. Then we have the decomposition f (χ) = λ(χ)χ +

y(x), where λ{χ) = lim supiôod2f(x)/dx2 is a bounded continuous scalar function and

γ(χ) is a vector such that |y ;(x)| < 4ω(\/2 | x | ) | x | in coordinates relative to the basis ~e.

Proof. We first verify that df{x)/dx.\x__Q = 0. In fact, let χ be a vector such that

xi for a given i, and let σ. be a transposition of the fth and /th coordinates. Then, by the

symmetry of /, df{x)/dx{ = {df/dx)(σ.χ). Since xi = 0, it follows that σ.χ —, χ as

j -• oo, whence df/dx. —. df/dx{ as / — <*>. Since lim-^^d^/dx. = 0, we obtain that

df/dx. = 0.

Now let χ be an arbitrary point of V, and suppose the index i is fixed. By Tay-

lor's formula and what has been proved above


0 =d/(x-*,*,

dxfa, (*)| < ω

Furthermore, if σ. is the same as above, then

Since ||O\JC — ΛΓ |

Thus for all j

whence

. - χ.\

a;

dx\ dxj

d*f(x)

dxf

ax?

| ft, χ, I)·

Combining this equality with the one obtained above from Taylor's formula, we get

(x) \ , R .

— Xi -\- (at {χ) + (jt· (Λ:)) xh

j )>OO dxj ;

which with the estimates for the moduli of x. and β. proves the lemma.

Using this lemma, we prove the uniform continuity of F on SQ with respect to

I · | | - . For definiteness let r = 1. Given e > 0, we show that for sufficiently small δ

we have |/(x) - f(y)\ < e whenever | | x - y\\- < δ with x, y e SQ j .

Using the uniform continuity of / with respect to the Hubert metric, we find α >

0 such that from x, y € SQ j and ||x - y|| < a. follows |/(x) - f{y)\ < e. Now it suf-

fices to verify that for all sufficiently small δ we have |/(x) - f(y)\ < e whenever

II* ~ >ΊΙ ̂ a an<^ II* ~ y\\— £ ^ with x, y £ £„ -. Let χ and y be such that ||x — y|| >

a, ||x— y\\— < δ and x, y € SQ . . It is easy to construct a unit vector ζ orthogonal

to χ and y such that \\ζ\\— < δ. Let

Then x(/) is a curve on SQ j joining χ and y. Taking account of the decomposition of

/ (x) given in Lemma 5 and the orthogonality of x{t) and χ (t) following from the iden-

tity | |*(/)|| = 1, we have

ι ι

f(x)-f (y) = J (f (* (0). *' (0) dt = j (γ (χ (0), χ* (/)) dt.0 0

Furthermore,

In addition


sin-,, Yy

Now let δ(χ) denote the set of indices of the coordinates of the vector χ that are

greater than 8^ in modulus. Using estimates of |y.(x(i))|, the representation of χ (t)

and the relations | | x - y\\ > a, \\x - y | |- < δ and \\ξ\\ — < δ, we obtain, after simple

transformations,

\f{x)-f(y)\ =

i£6(x) o)) *

where C, - 4ω{\/2(ΐ + 2δ)). Since ||x|| = 1, the number of elements in the set δ(χ) does

not exceed δ~ " , so that the first sum is bounded by C . αδ , where C, α does not de-

pend on δ. The second sum is bounded (Cauchy's inequality) by a quantity C^ depend-

ing only on a. Thus

If (*) - 4ω(/2 26])Cl

and the right side is arbitrarily small for small δ, which was required.

The theorem is proved.

Note that the smoothness requirement on / in Theorem 14 cannot be weakened.

This follows from

Proposition 3. There exists a function /, symmetric on H, of smoothness class

DuiH) with a derivative satisfying a Lipschitz condition, that is not approximable by

polynomials on V.

~8mProof. Let ξη = Σ ?

2

= ^ 2~8me., and let Sm be the orbit of ξη under the action of

the group Σ—. Then S is a closed set, invariant with respect to Σ—. It is easy to

verify that sup {|(x, y)\ |x € 5"^, y eS^} < 1/8 for m ^ n. Let Sm be the closed con-

vex hull of Sm; it is also invariant with respect to Σ— and, in addition, for χ € S ,

y e Sn, \\x\\, \\y\\ > 1/2 and m+n we have ||x - y\\ < 1/2. Let Tm = Sm + V 0 > 1 / g . The

set Τ is invariant with respect to Σ—, since the latter is a subgroup of the orthog-

onal group of H.

Let 0{t) be a smooth function on the line, equal to 1 in the neighborhood of the

origin and zero for t > l/l6, and let <j>{t) be another smooth function, equal to zero for

t > 1/4 and 1 for i = 1.

Let gm(x) - φ{\\χ\\2)θ{ρ(χ, Τ )). By Lemma 1 the functions g are uniformly


bounded together with their first derivatives, and their derivatives are equi-Lipschitz.

By construction these functions are symmetric, and their supports are at positive dis-

tances from one another. Furthermore these distances are bounded away from zero. Now

letoo

Then f(x) satisfies the required smoothness and symmetry conditions, but does not

satisfy the necessary condition for polynomial approximation given in Theorem 13. In

fact, the sequence {s^l^.j is fundamental in the metric ρ—, while the sequence

W€m^m = l = K - l ) " ^ . ! is not fundamental. Thus fix) is the desired function.

Note that / not only fails to be approximable by polynomials, but even by func-

tions of class Du(H). Indeed, in the contrary case it could be approximated, according

to Theorem 12, by symmetric functions of class D2(H), and then by Theorem 14 also

by polynomials, which is impossible.

Theorem 14 has a continuous analog.

Theorem 14 . Let f: L [θ, l] -» R be a symmetric function having an a th deriva-

tive,^) uniformly continuous on any ball in L [0, l ] . Then there exists a continuous

function Q: W-p* -* R such that

for any χ e L [θ, l ] .

Corollary. A symmetric function on L [θ, l ] , p > 1, of class D P(L , R) is uni-

formly approximable by polynomials on any bounded subset in L [0, l ] .

Corollary. The Gateaux functional fφ(χ) = /οφ(χ(,ή)dt, where χ € L [θ, l ] , p> 1, and

φ:. R -> R is a continuous function satisfying the condition |<̂ >(AT)| < C|x|^? is uniformly

approximable on the unit ball by functions of class D & if and only if φ is a poly-

nomial and deg φ < [/>].

A complete proof of the theorem and its corollaries can be found in [8].

It is easy to show that if φ is an infinitely smooth function with compact support,

then the functional /^ has, when p is integral, a (p - l)th derivative, uniformly con-

tinuous on any ball, and when p is nonintegral a [/>]th derivative, also uniformly con-

tinuous on any ball. At the same time, by Theorem 14 and the last corollary / . is not

approximable on the unit ball by functions of class D &.

Received 16/OCT/72

(2) Recall that a = ρ or [p] + I according as ρ is integral or not.


BIBLIOGRAPHY

1. V. I. Averbuh and O. G. Smoljanov, Differentiation theory in linear topological spaces,Uspehi Mat. Nauk 22 (1967), no. 6 (138), 201-260 = Russian Math. Surveys 22 (1967), no. 6,201-258. MR 36 #6933.

2. J. Eells, Jr., A setting for global analysis, Bull. Amer. Math. Soc. 72 (1966), 751-807;Russian transl., Uspehi Mat. Nauk 24 (1969), no. 3 (147), 157-210. MR 34 #3590.

3. P. Levy, Problemes concrets d'analyse fonctionnelle, 2nd ed., Gauthier-Villars, Paris,1951; Russian transl., "Nauka", Moscow, 1967. MR 12, 834; 36 #6892.

4. A. S. Nemirovskil, On a certain chain of algebras on a Hilbert sphere, Funkcional. Anal,i Prilozen. 5 (1971), no. 1, 85-86 = Functional Anal. Appl. 5 (1971), 72-73-

5. f Smooth and polynomial approximation of continuous functions on Hilbert space,Dissertation, Moscow State University, Moscow, 1973· (Russian)

6. S. M. Semenov, Polynomials on linear topological spaces, Vestnik Moskov. Univ. Ser. IMat. Meh. 25 (1970), no. 3, 45-49 = Moscow Univ. Math. Bull. 25 (1970), no. 3/4, 35-38. MR 42#8273.

7. _. _ ? Algebras on a Hilbert space which are generated by quadratic forms, Funkcional.Anal, i Prilozen. 5 (1971), no. 2, 89-90= Functional Anal. Appl. 5 (1971), 164-166.

8. , Symmetric functions on L spaces, Dissertation, Moscow State University,Moscow, 1973- (Russian)

Translated by B. SILVER

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ON POLYNOMIAL APPROXIMATION OF FUNCTIONS ON HILBERT SPACE