Reports of the Academy of Sciences of the USSR

1. Volume 143, No. 1

MATHEMATICS

N. A. SAPOGOV

A STRENGTHENING OF THE LOZINSKY–KHARSHILADZE THEOREM ON POLYNOMIAL APPROXIMATIONS

(Presented by Academician S. N. Bernstein on 30 X 1961)

1. Let \(C\) be the space of all \(2\pi\)-periodic complex-valued continuous functions \(f(x)\), defined for \(0 \leq x \leq 2\pi\), with norm
   \[ \|f\|_C=\max_{x\in[0,2\pi]} |f(x)|. \]
   Consider a linear operator \(U(f,x)\) mapping the space \(C\) into its subspace \(\mathscr E_n\), formed by trigonometric polynomials
   \[ E_n=\sum_{|k|\leq n} c_k \exp(ikx) \]
   of order \(\leq n\), and let
   \[ U(\exp(ilx),x)=\sum_{|k|\leq n}\gamma_{l,k}\exp(ikx),\qquad l=0,\pm1,\pm2,\ldots \]
   The equalities \(\gamma_{k,k}=1,\ |k|\leq n;\ \gamma_{l,k}=0,\ l\ne k\), are necessary and sufficient in order that the relations \(U(E_n,x)\equiv E_n\) hold for each of the polynomials \(E_n\) of order \(\leq n\). But we consider arbitrary operators \(U\), imposing no restrictions whatever on the coefficients \(\gamma_{l,k}\).

2. For the given operator \(U(f,x)\) we construct the associated operator \(U^0(f,x)\), setting, by definition,
   \[ U^0(f,x)=\int_0^{2\pi} f(x+t)\Phi(t)\,dt =\sum_{|k|\leq n} c_k(f)\gamma_{k,k}\exp(ikx), \]
   where
   \[ \Phi(t)=(2\pi)^{-1}\sum_{|k|\leq n}\gamma_{k,k}\exp(-ikt) \]
   and \(c_k(f)\) are the Fourier coefficients of the function \(f(x)\). By \(f_t\) we denote \(f(x+t)\), where \(t\) is regarded as a parameter.

Theorem 1. If \(f\in C\) and \(U(f,x)\) is a linear operator mapping \(C\) into \(\mathscr E_n\), then the identity
\[ (2\pi)^{-1}\int_0^{2\pi} U(f_t,x-t)\,dt = \int_0^{2\pi} f(x+t)\Phi(t)\,dt = U^0(f,x). \tag{1} \]
holds.

Proof. Since the closure in the norm \(\|f\|_C\) of the set of all trigonometric polynomials coincides with the space \(C\), it is sufficient to prove identity (1) only for polynomials
\[ f_N(x)=\sum_{|l|\leq N} c_l(f_N)\exp(ilx) \]
of arbitrary order \(N\). We have, putting \(N>n\):
\[ \int_0^{2\pi} U(f_{Nt},x-t)\,dt = \int_0^{2\pi} U\left(\sum_{|l|\leq n} c_l(f_N)\exp(il(x+t)),x-t\right)dt. \tag{2} \]

Equality (2) is equivalent to the content of Lemma 2 from the work \((^1)\). Further:

\[ \begin{aligned} \int_{0}^{2\pi} U\left(\sum_{|l|\leq n} c_l(f_N)\exp(il(x+t)),\,x-t\right)\,dt &= \sum_{|l|\leq n} c_l(f_N)\int_{0}^{2\pi} \left[\exp(ilt)\sum_{|k|\leq n}\gamma_{l,k}\exp(ik(x-t))\right]dt \\ &=2\pi\sum_{|l|\leq n} c_l(f_N)\gamma_{l,l}\exp(ilx). \end{aligned} \tag{3} \]

Equalities (2) and (3) prove Theorem 1. Identity (1) is a generalization of the well-known Marcinkiewicz–Berman identity \((^1)\), which is obtained from (1) under the special assumption that \(\gamma_{l,k}=0,\ k\ne l\). (\(\Phi(t)\) can be identified with any polynomial from \(\mathcal E_n\) by a suitable choice of the diagonal coefficients \(\gamma_{k,k},\ |k|\leq n\).)

1. Theorem 1 leads, in particular, to a lower estimate for the norm of an arbitrary operator \(U\) taking its values in \(\mathcal E_n\):

\[ \|U\|_C \geq \|U^0\|_C. \tag{4} \]

For the proof it suffices in identity (1) to put \(x=0\) and to consider the upper bounds of the moduli of both sides of this identity for all \(f\in C\) satisfying \(\|f\|_C\leq 1\). If, for example, for the operator \(U\) the diagonal coefficients \(\gamma_{k,k}\) satisfy the condition

\[ \gamma_{k,k}=1,\qquad |k|\leq n, \tag{5} \]

then, whatever the other coefficients \(\gamma_{k,l},\ k\ne l\), may be, the estimate
\[ \|U\|_C \geq 4\pi^{-2}\ln n+O(1),\quad n\to\infty, \]
is valid, since under condition (5) we have
\[ \|U^0\|_C=4\pi^{-2}\ln n+O(1),\quad n\to\infty. \]

1. The indicated estimate for the norm of the operator \(U\) makes it possible to strengthen somewhat the well-known theorem of Lozinskii–Kharshiladze \((^{2,3})\) on polynomial approximations of continuous functions.

Theorem 2. Let \(U_n(f,x)\) be linear operators mapping the space \(C\) into its subspaces \(\mathcal E_n\), formed by trigonometric polynomials of order \(\leq n,\ n=1,2,\ldots\). If, for the diagonal coefficients \(\gamma_{k,k}\) of each of the operators \(U_n\), condition (5) is satisfied, then, whatever the remaining coefficients \(\gamma_{l,k},\ l\ne k\), of these same operators may be, the limiting relations
\[ \|U_n(f,x)-f(x)\|_C\to\infty,\quad n\to\infty, \]
cannot hold for all \(f\in C\).

In particular, if \(\gamma_{k,l}=0\) for all \(k\ne l\), then Theorem 2 becomes the theorem of Lozinskii–Kharshiladze, if, of course, one disregards the inessential circumstance that the formulation of this theorem was given by its authors for the case of the space of real continuous functions \(C\).

An analogous result is valid for approximations of continuous functions by algebraic polynomials.

For simplicity we have restricted ourselves to consideration of the classical space \(C\) of continuous periodic functions with norm
\[ \|f\|_C=\max |f(x)|, \]
but Theorem 1 and inequality (4), together with the proofs, carry over almost literally to the case of a number of more general functional spaces, for example, spaces of type \(E,\ \widetilde F_\theta\) from papers \((^{4,5})\). Finally, Theorem 2 can be strengthened a little further if, instead of condition (5), one requires the equalities \(\gamma_{k,k}=1\) to hold for \(|k|\leq \nu_n\), where
\[ \lim_{n\to\infty}\frac{\nu_n}{n}=1. \]
The proof follows from the fact that under this condition the norms \(\|U_n^0\|_C\) increase without bound (see, for example, \((^6)\), p. 148).

1. Of substantial importance is the possibility of generalizing identity (1) to spaces of functions defined on topological groups. Let \(G\) be a bicompact commutative group with invariant measure \(\mu\) \((\mu(G)=1)\), and let \(\chi_k(x)\), \(k=1,2,\ldots\), be the totality of its characters. Let \(C(G)\) denote the normed space of all continuous functions \(f(x)\) on the group \(G\), whose norm is defined by the equality \(\|f\|_C=\sup_{x\in G}|f(x)|\).

Let \(U(f,x)\) be a linear operator mapping \(C(G)\) into its subspace \(X_n(G)\), formed by all polynomials \(\sum_{k=1}^n c_k\chi_k(x)\) of order \(\leq n\) (\(c_k\) are complex numbers). For any \(f\in C(G)\), let \(f_t=f(x+t)\), \(c_k(f)\) be the Fourier coefficient with respect to the character \(\chi_k(x)\), and
\[ U(\chi_l,x)=\sum_{k=1}^n \gamma_{l,k}\chi_k(x). \]

Theorem 3. If \(f\in C(G)\); \(U(f,x)\) is a linear operator mapping \(C(G)\) into the subspace \(X_n(G)\), then
\[ \int_G U(f_t,x-t)\,d\mu(t)=\sum_{k=1}^n c_k(f)\gamma_{k,k}\chi_k(x). \tag{6} \]

Identity (6) is a certain generalization of the identity established by D. L. Berman for bicompact groups \({}^{7}\), where again it is assumed that for the operator \(U\) the coefficients \(\gamma_{l,k}=0\) if \(k\ne l\).

Received
23 X 1961

CITED LITERATURE

\({}^{1}\) D. L. Berman, DAN, 85, No. 1 (1952). \({}^{2}\) S. M. Lozinskii, DAN, 61, No. 2 (1948). \({}^{3}\) I. P. Natanson, Constructive Theory of Functions, 1949. \({}^{4}\) S. M. Lozinskii, DAN, 64, No. 4 (1949). \({}^{5}\) D. L. Berman, DAN, 88, No. 1 (1953). \({}^{6}\) P. P. Korovkin, Linear Operators and Approximation Theory, 1959. \({}^{7}\) D. L. Berman, Izv. Vyssh. uchebn. zaved., Mathematics, No. 4 (17) (1960).