S. Ya. AL’PER

ON THE BEST APPROXIMATION OF ANALYTIC FUNCTIONS IN THE MEAN OF FIRST DEGREE ON A CIRCLE

(Presented by Academician V. I. Smirnov on 6 V 1963)

1°. We shall consider the problem of determining exact and asymptotic values for the best approximation in the metric \(L\) on the circle \(|z|=1\) of the analytic functions \(\dfrac{1}{z-a}\), \(\ln(z-a)\), \((z-a)^s\), and other functions, by polynomials of degree \(n\). For the best approximation of the indicated functions in the metric \(L\) on the segment \([-1,+1]\) of the real axis for real \(a\), such a question was studied in the works of S. M. Nikol’skii \((^1)\), N. I. Akhiezer \((^2)\), and others. The best approximation in the metric \(L\) of a function \(f(z)\in L\) on the circle \(|z|=r,\ 0<r\leqslant 1\), will be denoted by \(\rho_n^{(1)}[f,r]\):

\[ \rho_n^{(1)}[f,r]=\inf \int_{|z|=r} |f(z)-Q_n(z)|\cdot |dz| \]

over all possible polynomials \(Q_n(z)\) of degree \(n\); \(\rho_n^{(1)}[f,1]=\rho_n^{(1)}[f]\).

We shall use the following criterion:

In order that \(P_n(z)\) be a polynomial of best approximation in the metric \(L_p,\ p\geqslant 1\), of degree \(n\) to the function \(f(z)\in L_p\) on a rectifiable curve \(\Gamma\)*, it is sufficient and (for \(p=1\) and the case where the difference \(f(z)-P_n(z)\) is nonzero almost everywhere on \(\Gamma\)) necessary that, for every polynomial \(Q_n(z)\) of degree \(n\), the equality

\[ \int_{\Gamma} |f(z)-P_n(z)|^{p-1}\operatorname{Re}\{Q_n(z)\operatorname{sign}[\overline{f(z)-P_n(z)}]\}\,dz=0 \]

hold, where \(\operatorname{sign} w=w/|w|\) for \(w\ne 0\) and \(\operatorname{sign} w=0\) for \(w=0\).

In the case when \(\Gamma\) is the segment \([-1,+1]\), \(f(z)\) is a real-valued function, and \(p=1\), this criterion was established by S. M. Nikol’skii \((^3)\).

In the general case this criterion is an analogue of the criterion established by A. F. Timan \((^4)\) with the metric \(L_p\) defined by an integral over a domain. In the case when \(p=1\) and \(\Gamma\) is the circle \(|z|=1\), the polynomial \(P_n(z)\) of degree \(n\) gives the best approximation if the equality

\[ \int_{|z|=1} z^{-m}\frac{f(z)-P_n(z)}{|f(z)-P_n(z)|}\,|dz|=0 \tag{1} \]

holds for all \(m=0,1,2,\ldots,n\).

2°. The starting point is the determination of the quantity \(\rho_n^{(1)}\!\left[\dfrac{1}{z-a}\right]\), \(|a|\ne 1\).

Consider the identity

\[ \frac{1}{z-a}-P_n(z)= M\frac{1+\bar a z+\bar a^{\,2}z^2+\ldots+\bar a^{\,n+1}z^{n+1}}{z-a}, \tag{2} \]

\[ \text{* The condition } f(x)\in L_p \text{ on } \Gamma \text{ means that } \int_{\Gamma}|f(z)|^p\,|dz|<\infty. \]

in which \(P_n(z)\) is a polynomial of degree \(n\), and the number \(M\) is determined from the condition that the residue of the right-hand side at the point \(z=a\) is equal to 1,
\(M=(|a|^2-1)(|a|^{2n+4}-1)^{-1}\). We shall show that \(P_n(z)\) is the polynomial of best approximation in the metric \(L\) on the circle \(|z|=1\) for the function \(\dfrac{1}{z-a}\). Multiplying the numerator and denominator of the right-hand side of (2) by \(\bar z-\bar a\), where \(|z|=1\), we represent this formula in the form

\[ \frac{1}{z-a}-P_n(z) = M\frac{\bar z-\bar a^{\,n+2}z^{n+1}} {1-2|a|\cos(\vartheta-\alpha)+|a|^2}, \]

where \(z=e^{i\vartheta}\), \(a=|a|e^{i\alpha}\). Hence we find

\[ \left|\frac{1}{z-a}-P_n(z)\right| = M \frac{\sqrt{1-2|a|^{n+2}\cos (n+2)(\vartheta-\alpha)+|a|^{2n+4}}} {1-2|a|\cos(\vartheta-\alpha)+|a|^2}. \]

According to condition (1), it is enough to show that the integral

\[ \int_{0}^{2\pi} \frac{\bar z^{\,n+1}-\bar a^{\,n+2}z^{n+1-m}} {\sqrt{1-2|a|^{n+2}\cos (n+2)(\vartheta-\alpha)+|a|^{2n+1}}} \,d\vartheta \tag{3} \]

vanishes for all \(m=0,1,2,\ldots,n\).

We note that for \(k=1,2,\ldots,n+1\) the equality

\[ \int_{0}^{2\pi} \frac{\cos k\varphi} {\sqrt{1-2c\cos (n+2)\varphi+c^2}}\,d\varphi =0, \tag{4} \]

holds, where \(c\) is any number \(>0\) and \(\ne 1\). Indeed, this follows from the Fourier expansion

\[ (1-2c\cos\varphi+c^2)^{-1/2} = \sum_{m=0}^{\infty} A_m\cos m\varphi, \]

in which \(\varphi\) is replaced by the quantity \((n+2)\varphi\). From (4) there follows the vanishing of the integral (3) for \(m=0,1,2,\ldots,n\).

Thus, \(P_n(z)\) is the polynomial of best approximation, and for \(|a|\ne 1\) the formula is valid

\[ \rho_n^{(1)} \left[\frac{1}{z-a}\right] = \frac{|a|^2-1}{|a|^{2n+4}-1} = \int_{0}^{2\pi} \frac{\sqrt{1-2|a|^{n+2}\cos (n+2)\vartheta+|a|^{2n+4}}} {1-2|a|\cos\vartheta+|a|^2} \,d\vartheta . \tag{5} \]

From formula (5), for \(|a|>1\), there follows the asymptotic formula

\[ \rho_n^{(1)} \left[\frac{1}{z-a}\right] \sim \frac{2\pi}{|a|^{\,n+2}}. \tag{6} \]

For \(|a|<1\), from (5) it follows that

\[ \lim_{n\to\infty}\rho_n^{(1)} \left[\frac{1}{z-a}\right] =2\pi. \tag{7} \]

In particular, \(\rho_n^{(1)}[z^{-1}]=2\pi\), and the polynomial of best approximation in mean of the first degree is the identically zero polynomial.

\(3^\circ\). By an analogous method one can find
\(\rho_n^{(1)}\left[\dfrac{z^s}{z^p-a^p}\right]\), where \(p\) is a natural number \(>1\), \(s\) is equal to one of the numbers \(0,1,2,\ldots,p-1\), and \(|a|\ne 1\). For this purpose the identity

\[ \frac{z^s}{z^p-a^p}-Q_n(z) = \frac{|a|^{2p}-1}{|a|^{2p(k+2)}-1}\,H(z), \tag{8} \]

in which \(Q_n(z)\) is some polynomial of degree \(n\); \(n\) is any number of the form \(kp+s\) (\(k\) is an integer \(\geq 0\)) and

\[ H(z)=\frac{z^s\left[1+\bar a^p z^p+\bar a^{2p}z^{2p}+\cdots+\bar a^{(k+1)p}z^{(k+1)p}\right]}{z^p-a^p}. \]

From identity (8) one obtains the formula

\[ \rho_n^{(1)}\left[\frac{z^s}{z^p-a^p}\right] = \frac{|a|^{2p}-1}{|a|^{2p(k+2)}-1} \int_0^{2\pi} \frac{\sqrt{1-2|a|^{p(k+2)}\cos(k+2)p\vartheta+|a|^{2p(k+2)}}} {1-2|a|^p\cos p\vartheta+|a|^{2p}}\,d\vartheta, \tag{9} \]

which is valid for the numbers \(n=kp+s\), and also for \(kp+s+1,\ldots, kp+s+p-1\); \(Q_n(z)\) will be a polynomial of best approximation in the mean of degree \(n,n+1,\ldots,n+p-1\). For \(|a|>1\), (9) gives the asymptotic formula

\[ \rho_n^{(1)}\left[\frac{z^s}{z^p-a^p}\right]\sim \frac{2\pi}{|a|^{p(k+2)}} \]

as \(k\to\infty\) and \(n=kp+s, kp+s+1,\ldots, kp+s+p-1\). For \(|a|<1\) we obtain

\[ \lim_{n\to\infty}\rho_n^{(1)}\left[\frac{z^s}{z^p-a^p}\right]=2\pi. \]

In particular, for \(a=0,\ s=0\) we shall have \(\rho_n^{(1)}[1/z^p]=2\pi\).

\(4^\circ\). Starting from identity (2), with the aid of integration with respect to the parameter \(a\) (first taking \(a\) real, \(a>1\)), one can obtain the asymptotic formula

\[ \rho_n^{(1)}[\ln(z-a)]\sim \frac{2\pi}{n|a|^{n+1}}, \tag{10} \]

valid for any complex \(a\) satisfying \(|a|>1\).

\(5^\circ\). If \(f(z)=(a-z)^s\) for real \(s\) and \(a>1\), then construct a polynomial \(P_n(z)\) of degree \(n\) which coincides with \(f(z)\) at the zeros of the polynomial

\[ R(z)=1+az+a^2z^2+\cdots+a^{n+1}z^{n+1}. \]

Then

\[ (a-z)^s-P_n(z)=\frac{R(z)}{2\pi i} \int_{|\zeta|=r} \frac{(a-\zeta)^s}{(\zeta-z)R(\zeta)}\,d\zeta, \]

where \(1<r<a\). First assuming \(s>-1\) and deforming the contour of integration in the appropriate way, we obtain

\[ (a-z)^s-P_n(z) = -\frac{(1-a^{n+2}z^{n+2})\sin\pi s}{(1-az)\pi} \int_a^\infty \frac{(x-a)^s(1-ax)}{(x-z)(1-a^{n+2}x^{n+2})}\,dx. \]

Estimating the integral with the help of S. N. Bernstein’s asymptotic formula \((5)\) and using the result from \(2^\circ\), we obtain

\[ \rho_n^{(1)}[(z-a)^s]\sim \frac{2\Gamma(s+1)|\sin\pi s|}{n^{s+1}|a|^{\,n-s+1}}. \tag{11} \]

This formula is valid for any complex \(a\), \(|a|>1\), and any real \(s\) different from zero and from a positive integer. Setting \(s=-p\), it can be represented in the form

\[ \rho_n^{(1)}\left[\frac{1}{(z-a)^p}\right]\sim \frac{2\pi n^{p-1}}{\Gamma(p)|a|^{n+p+1}}. \tag{12} \]

If

\[ f(z)=\sum_{m=1}^{k}\frac{A_m}{(z-a)^m}+\varphi(z), \]

where \(|a|>1,\ A_k\ne 0\), and \(\varphi(z)\) is analytic in the disk with center \(z=0\) and radius \(>|a|\), then

\[ \rho_n^{(1)}[f]\sim |A_k|\frac{2\pi n^{k-1}}{(k-1)!\,|a|^{n+k+1}}. \]

Let us note that the passage to the limit as \(a\to 1\) in formula (11) (for \(s>-1\)) or in formula (10) cannot be carried out, since the proof of these formulas does not imply their uniformity with respect to \(a\).

\(6^\circ\). The following analogue holds of S. N. Bernstein’s formula \((^7)\), proved by him for uniform approximation on the interval \([-1,+1]\):

\[ \rho_n^{(1)}[(z-a)^s\Phi(z)]\sim |\Phi(a)|\,\rho_n^{(1)}[(z-a)^s], \tag{13} \]

where \(|a|>1\), \(s\) is not equal to 0 or to a positive integer, and \(\Phi(z)\) is an analytic function in some disk \(|z|<R,\ R>|a|\), under the condition \(\Phi(a)\ne 0\).

\(7^\circ\). Let \(H_1^{(s)}\) denote the class of all functions analytic in the disk \(|z|<1\) for which

\[ \lim_{\rho\to 1}\frac{1}{2\pi}\int_{0}^{2\pi}|f^{(s)}(\rho e^{i\theta})|\,d\theta\le 1, \]

where \(s\) is an integer \(\ge 0\), \(H_1^{(0)}=H_1\). Using the results of K. I. Babenko \((^7)\), one can obtain an exact formula for the best mean approximation of functions of the class \(H_1^{(s)}\).

Theorem. For \(0<r\le 1\), when \(n\ge s\), the formula

\[ \sup_{f\in H_1^{(s)}}\rho_{n-1}^{(2)}[f,r]=2\pi r^{n+1}\alpha_{n,s}, \]

holds, where

\[ \alpha_{n,s}=\frac{1}{n(n-1)\ldots(n-s+1)} \]

for \(s>0\), and \(\alpha_{n,0}=1\).

For \(s=0\) one can obtain an asymptotic formula for the quantity appearing in the theorem, and \(0<r<1\), without relying on K. I. Babenko’s results, but using the Cauchy integral formula and the estimate following from formula (6),

\[ \int_{|z|=r}\left|\frac{1}{\xi-z}-Q_{n-1}(z,\xi)\right|\cdot |dz|<2\pi r^{n+1}(1+\varepsilon_n) \]

for any \(\xi,\ |\xi|=1\), where \(Q_{n-1}(z,\xi)\) is some polynomial of degree \(n-1\) in \(z\), \(\varepsilon_n>0\), and \(\lim \varepsilon_n=0\).

Received
3 V 1963

REFERENCES

1. S. M. Nikol’skii, Izv. AN SSSR, Ser. Mat., 11, 139 (1947).
2. N. I. Akhiezer, Lectures on Approximation Theory, 1947.
3. S. M. Nikol’skii, Izv. AN SSSR, Ser. Mat., 10, 207 (1946).
4. A. F. Timan, RZhMat, No. 4, ref. 2854 (1958).
5. S. N. Bernstein, Extremal Properties of Polynomials, 1937.
6. S. N. Bernstein, Collected Works, 2, 1954, p. 88.
7. K. I. Babenko, Izv. AN SSSR, Ser. Mat., 22, No. 5 (1958).