MATHEMATICS

M. M. DRAGILEV

ON THE ORDER OF BEST APPROXIMATION OF FUNCTIONS DEFINED ON A CONTINUUM BY LINEAR AGGREGATES OF ELEMENTS OF AN EXTENDABLE BASIS

(Presented by Academician A. N. Kolmogorov on 18 III 1961)

Let \(K\) be a bounded continuum in the plane \(Z\), nondegenerate and not separating the plane; let \(G\) be a simply connected domain containing \(K\); and let \(\mathfrak A(G)\) and \(\mathfrak A(K)\) be the spaces of functions analytic, respectively, in the domain \(G\) and on \(K\). Suppose that the function \(w=\varphi(z)\) effects a conformal mapping of the doubly connected domain \(G/K\) onto the circular annulus
\[ 1<|w|<R. \]

V. D. Erokhin, generalizing the well-known theorem of S. N. Bernstein on the order of best approximation by polynomials of functions defined on a continuum \((^1)\), showed that for every continuum \(K\) and any domain \(G\) containing it there exists a sequence of analytic functions
\[ f_0(z),\ f_1(z),\ldots,\ f_j(z),\ldots;\qquad f_j(z)\in\mathfrak A(G),\quad j=0,1,\ldots, \tag{1} \]
which has the following property.

Property B. A function \(f(z)\), defined on the continuum \(K\), admits, for every \(\delta>0\) and all natural \(n\), approximations by polynomials
\[ \sum_{j=0}^{n} c_j^{(n)} f_j(z) \]
of the form
\[ \left|\,f(z)-\sum_{j=0}^{n} c_j^{(n)} f_j(z)\,\right| \le C(\delta)\left(\frac{1+\delta}{\rho}\right)^n \qquad (1<\rho<R;\ z\in K), \]
if and only if it is regular in the domain \(G_\rho\), bounded by the preimage of the circle \(|w|=\rho\) under the transformation \(w=\varphi(z)\) \((^2)\).

Such a sequence, according to \((^2)\), is the so-called principal basis of the space \(\mathfrak A(G)\), which in the special case when the function \(\varphi(z)\) maps the exterior of \(K\) onto the exterior of a disk \((\varphi(\infty)=\infty)\), coincides with the sequence of Faber polynomials.

In the present note it is shown that Property B is possessed by every basis of the space \(\mathfrak A(G)\) that is extendable to the continuum (i.e., every common basis of the spaces \(\mathfrak A(G)\) and \(\mathfrak A(K)\)), under the condition that the elements of this basis are numbered in a certain order. The author considers it necessary to emphasize here that the following proof relies essentially on the theorem of V. D. Erokhin stated above.

Let \(\{e_n(z)\}_{n=0}^{\infty}\) be a principal basis, and let the sequence (1) be an arbitrary basis of the space \(\mathfrak A(G)\) extendable to \(K\), and let
\[ f_n(z)=\sum_{i=0}^{\infty} a_{in}e_i(z),\qquad e_n(z)=\sum_{i=0}^{\infty} b_{in}f_i(z)\quad (n=0,1,\ldots). \]

Changing, if necessary, the numbering of the functions \(f_n(z)\), and also multiplying them by suitable constants, we bring the basis to canonical form, ana-

analogously to how this is done in paper \((^3)\). In doing so we shall keep the previous notation for the functions and coefficients.

Lemma 1. The basis (1) satisfies the conditions:

a)
\[ \lim_{j\to\infty}\left[\max_{z\in \overline{G}_\rho}|f_j(z)|\right]^{1/j} =\rho\qquad (1<\rho<R); \]

b)
\[ |a_{ij}|R^{\,i-j}\le \alpha_j;\qquad |b_{ij}|R^{\,i-j}\le \alpha_i;\qquad |a_{ij}|\le \alpha_j;\qquad |b_{ij}|\le \alpha_j \quad (i,j=0,1,\ldots), \]
where \(\alpha_0,\alpha_1,\ldots,\alpha_n,\ldots\) is a certain nondecreasing sequence of positive numbers such that
\[ \lim_{n\to\infty}\alpha_n^{1/n}=1. \]

The first assertion is proved in paper \((^3)\), while the second is equivalent to it, as follows from the matrix criterion for a quasi-power basis established by M. G. Khaplanov \((^4)\).

Lemma 2. For every number \(p>1\) there exists a positive number \(q\) such that
\[ \sum_{i\ge pn+q}\sum_{j=0}^{n}|a_{ij}|\,|b_{ki}|<\frac12 \qquad (k=0,1,\ldots,n;\ n=0,1,\ldots). \]

For the proof it suffices to determine the number \(q>1\) from the condition
\[ \frac12 R^{q/p}\bigl(1-R^{-1/p}\bigr)> \sup_{0\le i<\infty}\bigl[\alpha_i^2 R^{(1/p-1)i}\bigr] \]
and to use the preceding lemma.

We next consider arbitrary polynomials composed of elements of the basis (1),
\[ P_n(z)=\sum_{j=0}^{n}c_j^{(n)}f_j(z);\qquad c_n^{(n)}\ne 0\quad (n=0,1,\ldots). \]
Let \(M_n(K)\) and \(M_n(\rho)\) denote, respectively, the maxima of the modulus of \(P_n(z)\) on the continuum \(K\) and in the domain \(\overline{G}_\rho\) \((1<\rho<R)\).

Lemma 3. The polynomials \(P_n(z)\) satisfy the condition
\[ \overline{\lim}_{n\to\infty}\left[\frac{M_n(\rho)}{M_n(K)}\right]^{1/n} \le \rho \qquad (1<\rho<R). \]

Proof. Expand each polynomial \(P_n(z)\) in the fundamental basis:
\[ P_n(z)=\sum_{j=0}^{n}c_j^{(n)}\sum_{i=0}^{\infty}a_{ij}e_i(z) =\sum_{i=0}^{\infty}\left(\sum_{i=0}^{n}c_j^{(n)}a_{ij}\right)e(z) =\sum_{i=0}^{\infty}\widetilde c_i^{(n)}e_i(z). \]

From the elementary properties of the fundamental basis established in paper \((^2)\) it follows that
\[ |\widetilde c_i^{(n)}|\le M_n(K) \]
for all \(i,\ n=0,1,\ldots\). We use the identity
\[ c_{k^*}^{(n)}=\sum_{j=0}^{n}c_j^{(n)}\sum_{i=0}^{\infty}a_{ij}e_{k^*i} =\sum_{i=0}^{\infty}\sum_{j=0}^{n}c_j^{(n)}a_{ij}b_{k^*i} \qquad (n=0,1,\ldots), \]
in which \(k^*\) denotes the ordinal number of any one of the coefficients \(c_k^{(n)}\) \((k=0,1,\ldots,n)\) having the largest modulus. With the aid of Lemma 2, for a suitable choice of the numbers \(p,q\), we obtain the inequality
\[ |c_{k^*}^{(n)}| \le \sum_{i<pn+q+1}\left|\sum_{j=0}^{n}c_j^{(n)}a_{ij}\right|\,|b_{k^*i}| + \sum_{i\ge n+q}\sum_{j=0}^{n}|c_j^{(n)}|\,|a_{ij}|\,|b_{k^*i}| \le \]
\[ \le \sum_{i<pn+q+1}|\widetilde c_i^{(n)}|\,|b_{k^*i}| + |c_{k^*}^{(n)}| \sum_{i\ge pn+q}\sum_{j=0}^{n}|a_{ij}|\,|b_{k^*i}| \le \]
\[ \le M_n(K)(pn+q+1)\alpha_{pn+q+1} +\frac12|c_{k^*}^{(n)}|. \]

or

\[ |c_{k^*}^{(n)}| \leq 2M_n(K)(pn+q+1)\alpha_{pn+q+1} \qquad (n=0,1,\ldots). \]

Let \(\varepsilon>0\) be arbitrary. From Lemma 1 it follows that the maximum of the modulus of \(f_n(z)\) in the closed domain \(\overline{G}_\rho\) does not exceed \(C(\varepsilon)(\rho+\varepsilon)^n\). But then, obviously,

\[ M_n(\rho)\leq (n+1)|c_{k^*}^{(n)}|C(\varepsilon)(\rho+\varepsilon)^n, \]

whence it follows that

\[ \left[\frac{M_n(\rho)}{M_n(K)}\right]^{1/n} \leq \left[2(n+1)(pn+q+1)\alpha_{pn+q+1}C(\varepsilon)\right]^{1/n} (\rho+\varepsilon), \]

and the assertion of the lemma is obtained by passing successively to the limit first as \(n\to\infty\) and then as \(\varepsilon\to0\).

Theorem. Every basis of the space \(\mathfrak{A}(G)\) extendable to the continuum, after being reduced to canonical form, possesses property \(\mathbf{Б}\) with respect to the given continuum.

The proof is based on Lemmas 1a) and 3 and almost word for word repeats the arguments given in \((^5)\) in the proof of S. N. Bernstein’s theorem.

Rostov-on-Don Institute
of Agricultural Machine Building

Received
18 VII 1961

REFERENCES

1. S. N. Bernstein, Communications of the Kharkov Mathematical Society, ser. 2, 13 (1912).
2. V. D. Erokhin, DAN, 120, No. 4 (1958).
3. M. M. Dzhrbashyan, Mathematical Collection, 59 (95), issue 2 (1961).
4. M. G. Khlaponin, DAN, 80, No. 2 (1951).
5. A. I. Markushevich, Theory of Analytic Functions, 1950.