Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1958. Volume 118, No. 2
MATHEMATICS
SUN YUNG-SHEN
ON THE BEST APPROXIMATION OF CLASSES OF FUNCTIONS REPRESENTABLE IN THE FORM OF A CONVOLUTION
(Presented by Academician N. N. Bogolyubov on 29 VI 1957)
- Let \(H_p\) \((1 \leq p < \infty)\) be the class of real functions \(\varphi(t)\in L^p\) with period \(\omega\), satisfying the condition
\[ \|\varphi\|=\left\{\int_0^\omega |\varphi(t)|^p dt\right\}^{1/p}\leq 1, \]
and let \(H_\infty\) be the class of essentially bounded measurable functions \(\varphi(t)\) with period \(\omega\), for which
\[ \operatorname*{ess\,sup}|\varphi(t)|\leq 1. \]
Suppose that the function \(f(x)\) admits a representation in the form of a convolution:
\[
f(x)=\frac{1}{\omega}\int_0^\omega K(t-x)\varphi(t)\,dt
\quad (\varphi(t)\in H_p),
\tag{1}
\]
where \(K(t)\) is a function with period \(\omega\), and, in the case \(1<p\leq\infty\), \(K(t)\in L^q\)
\[
\left(\frac{1}{p}+\frac{1}{q}=1\right),
\]
while in the case \(p=1\), \(K(t)\) is continuous. The totality of all functions of the form (1) will be denoted by \(C(K,H_p)\).
Here we consider the problem of the best approximation of the class of functions \(f(x)\), representable in the form (1), by polynomials in a given Chebyshev system \((^1)\) of continuous functions with period \(\omega\), \(\{f_k(x)\}\) \((k=1,2,\ldots,n)\); general formulas are established for the quantity
\[
M_n^{(p)}=\sup_{\varphi\in H_p} E_n(f)_C,\qquad 1\leq p\leq\infty,
\tag{2}
\]
where
\[
E_n(f)_C=\min_{\alpha_k}\max_{0\leq x\leq\omega}
\left|f(x)-\sum_{k=1}^n \alpha_k f_k(x)\right|
\]
(\(\alpha_k\) are real numbers) is the best approximation of the function \(f(x)\) by polynomials in the system \(\{f_k(x)\}\); the properties of extremal functions \(f_0(x)\), for which \(E_n(f_0)_C=M_n^{(p)}\), are investigated, and the relations between the quantity \(M_n^{(p)}\) and the best approximation of the corresponding kernel \(K(t)\) in the metric \(L^q\) are studied.
For the case \(p=\infty\), this problem was studied earlier by Favard \((^2)\), Akhiezer and Krein \((^3)\), Nadem \((^4)\), Nikol’skii \((^5)\), and others.
- Let
\[ 0\leq x_1<x_2<\cdots<x_{n+1}<\omega. \]
Put \((^6)\)
\[ M_i= \begin{vmatrix} f_1(x_1) & f_1(x_2) & \cdots & f_1(x_{i-1}) & f_1(x_{i+1}) & \cdots & f_1(x_{n+1})\\ f_2(x_1) & f_2(x_2) & \cdots & f_2(x_{i-1}) & f_2(x_{i+1}) & \cdots & f_2(x_{n+1})\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ f_n(x_1) & f_n(x_2) & \cdots & f_n(x_{i-1}) & f_n(x_{i+1}) & \cdots & f_n(x_{n+1}) \end{vmatrix}, \]
\[ M=\sum_{i=1}^{n+1}M_i,\qquad a_i=(-1)^i\frac{M_i}{M}\quad (i=1,2,\ldots,n+1). \]
By \(a_i^{(0)}\) we shall denote the value of \(a_i\) for \(x_j=x_j^{(0)}\) \((j=1,2,\ldots,n+1)\), and by \(\{x_i^{(0)}\}\) a system of points
\[
0\le x_1^{(0)}<x_2^{(0)}<\cdots<x_{n+1}^{(0)}<\omega .
\]
In what follows, by a Chebyshev alternant of the function \(f_0(x)\) we shall mean a system of points \(\{x_i^{(0)}\}\) at which the difference between the given function and its polynomial of best approximation assumes the maximal values \(E_n(f_0)_c\) with alternating signs.
3. The case \(1<p\le \infty\).
Theorem 1. Let \(K(t)\in L^q\left(\dfrac1p+\dfrac1q=1\right)\).
Then:
\[ 1^\circ.\quad M_n^{(p)} = \frac1\omega \max_{0<x_1<\cdots<x_{n+1}<\omega} \left\{ \int_0^\omega \left| \sum_{i=1}^{n+1} a_i K(t-x_i) \right|^q dt \right\}^{1/q}. \tag{3} \]
\[ 2^\circ.\quad \text{There exists a function } f_0(x)\in C(K,H_p) \text{ for which } E_n(f_0)_c=M_n^{(p)}. \]
\[ 3^\circ.\quad \text{The maximum in formula (3) is attained for those and only those systems of points } \{x_i^{(0)}\} \text{ which are Chebyshev alternants for some extremal function } f_0(x). \]
Generally speaking, the extremal function \(f_0(x)\) is not unique. In the case \(1<p<\infty\), each extremal function \(f_0(x)\) is determined by a unique function \(\varphi_0(t)\in H_p\). In order that, for \(M_n^{(p)}>0\), the function \(\varphi_0(t)\in H_p\) determine an extremal function \(f_0(t)\), it is necessary and sufficient that there exist a system of points \(\{x_i^{(0)}\}\) such that
\[ M_n^{(p)} = \frac1\omega \left\{ \int_0^\omega \left| \sum_{i=1}^{n+1} a_i^{(0)}K(t-x_i^{(0)}) \right|^q dt \right\}^{1/q} \]
and, almost everywhere,
\[ \varphi_0(t) = \left\{ \frac1{\omega M_n^{(p)}} \right\}^{q-1} \left| \sum_{i=1}^{n+1} a_i^{(0)}K(t-x_i^{(0)}) \right|^{q-1} \operatorname{sign} \left\{ \sum_{i=1}^{n+1} a_i^{(0)}K(t-x_i^{(0)}) \right\} \]
on the interval \(0\le t\le \omega\).
The extremal function \(f_0(x)\) is unique if and only if all the functions
\[ \left| \sum_{i=1}^{n+1} a_i^{(0)}K(t-x_i^{(0)}) \right| \operatorname{sign} \left\{ \sum_{i=1}^{n+1} a_i^{(0)}K(t-x_i^{(0)}) \right\} \]
coincide almost everywhere on \([0,\omega)\) for all systems of points \(\{x_i^{(0)}\}\) for which the maximum in formula (3) is attained.
In the case \(p=\infty\), one and the same extremal function \(f_0(x)\) may be determined by different functions \(\varphi_0(t)\in H_\infty\). In order that the function \(\varphi_0(t)\in H_\infty\) determine an extremal function \(f_0(x)\), it is necessary and sufficient that there exist a system of points \(\{x_i^{(0)}\}\) such that
\[ M_n^{(\infty)} = \frac1\omega \int_0^\omega \left| \sum_{i=1}^{n+1} a_i^{(0)}K(t-x_i^{(0)}) \right| dt, \]
\[ \varphi_0(t) = \operatorname{sign} \left\{ \sum_{i=1}^{n+1} a_i^{(0)}K(t-x_i^{(0)}) \right\} \]
almost everywhere on the set
\[ \mathcal E \left\{ t:\sum_{i=1}^{n+1} a_i^{(0)}K(t-x_i^{(0)})\ne 0 \right\}, \qquad \operatorname{mes}\mathcal E>0. \]
For there to exist a unique function \(\varphi_0(t)\in H_\infty\) determining the extremal function \(f_0(x)\), it is necessary and sufficient that all the functions
\[ \operatorname{sign}\left\{\sum_{i=1}^{n+1} a_i^{(0)}K\left(t-x_i^{(0)}\right)\right\} \]
coincide almost everywhere on \([0,\omega)\) for all systems of points \(\{x_i^{(0)}\}\) for which the maximum in formula (3) is attained.
In the following two theorems we shall assume that \(K(t)\in L^q[0,2\pi)\) \((1\le q<\infty)\); \(\varphi(t)\in H_p[0,2\pi)\) \(\left(\frac1p+\frac1q=1\right)\); \(T_n(t)\) is a trigonometric polynomial of degree not exceeding \(n\), and
\[
E_n(f)_c=\min_{T_{n-1}}\|f(x)-T_{n-1}(x)\|_c,
\]
where \(f(x)\) is an arbitrary continuous function with period \(2\pi\).
Theorem 2. Let \(K(t)\in L^q[0,2\pi)\) \((1\le q<\infty)\) and
\[ f(x)=\frac1\pi\int_0^{2\pi} K(t-x)\varphi(t)\,dt, \tag{4} \]
where \(\varphi(t)\in H_p[0,2\pi)\). Then
\[ \sup_{\varphi\in H_p}E_n(f)_c\le \frac1\pi\min_{T_{n-1}} \left\{\int_0^{2\pi}\left|K(t)-T_{n-1}(t)\right|^q\,dt\right\}^{1/q}, \tag{5} \]
where
\[
\frac1p+\frac1q=1.
\]
If \(1<p<\infty\), then in order that inequality (5) become an equality, it is necessary and sufficient that there exist a system of points
\[
0\le x_1^{(0)}<\cdots<x_{2n}^{(0)}<2\pi
\]
such that:
\[ 1^\circ.\qquad \sup_{\varphi\in H_p}E_n(f)_c = \frac1\pi \left\{ \int_0^{2\pi} \left| \sum_{i=1}^{2n}\widetilde a_i^{(0)} K\left(t-x_i^{(0)}\right) \right|^q \,dt \right\}^{1/q}, \tag{6} \]
where
\[ \widetilde a_i^{(0)}=(-1)^i\frac{Q_i^{(0)}}{Q^{(0)}},\qquad Q^{(0)}=\sum_{i=1}^{2n}Q_i^{(0)},\qquad Q_i^{(0)}=\prod_{p,q\ne i}\sin\frac{x_p^{(0)}-x_q^{(0)}}2, \tag{7} \]
\[ 2n\ge p>q\ge1,\qquad p,q\ne i,\qquad i=1,2,\ldots,2n. \]
\[ 2^\circ.\qquad \operatorname{sign}\{K_n(t-x_i^{(0)})\}\cdot \operatorname{sign}\{K_n(t-x_{i+1}^{(0)})\}\le0 \]
almost everywhere on the interval \(0\le t\le2\pi\), \(i=1,2,\ldots,2n-1\).
\(3^\circ.\) All the functions \(\left|K_n(t-x_i^{(0)})\right|\) are proportional on the interval \(0\le t\le2\pi\), \(i=1,2,\ldots,2n\), where
\[
K_n(t)=K(t)-T_{n-1}^*(t),
\]
and \(T_{n-1}^*(t)\) is the polynomial of best approximation to the function \(K(t)\) of degree \(n-1\) in \(L^q\).
If, however, \(p=\infty\), then for (5) to become an equality it is necessary that there exist a system of points
\[
0\le x_1^{(0)}<x_2^{(0)}<\cdots<\cdots<x_{2n}^{(0)}<2\pi
\]
such that:
\[ 1^\circ.\qquad \sup_{\varphi\in H_\infty}E_n(f)_c = \frac1\pi \int_0^{2\pi} \left| \sum_{i=1}^{2n}\widetilde a_i^{(0)} K\left(t-x_i^{(0)}\right) \right|\,dt. \tag{8} \]
\[ 2^\circ.\qquad \operatorname{sign}\{K_n(t-x_i^{(0)})\}\cdot \operatorname{sign}\{K_n(t-x_{i+1}^{(0)})\}\le0 \]
almost everywhere on the interval \(0\leq t\leq 2\pi\), \(i=1,2,\ldots,2n-1\), where \(K_n(t)=K(t)-T^*_{n-1}(t)\), and \(T^*_{n-1}(t)\) is any polynomial of best approximation in the mean to the function \(K(t)\) of order \(n-1\).
If these conditions are satisfied for some trigonometric polynomial \(T^*_{n-1}(t)\) of best approximation in the mean to the function \(K(t)\) of order \(n-1\), then (5) turns into the equality (5′).
Theorem 3. If \(K(t)\in L\), \(K(t+2\pi)=K(t)\), \(\varphi(t)\) is any continuous function with period \(2\pi\),
\[ f(x)=\frac{1}{\pi}\int_{0}^{2\pi}K(t-x)\varphi(t)\,dt, \tag{9} \]
then
\[ E_n(f)_C \leq \frac{1}{\pi}\left\{ \max_{0<x_1<\cdots<x_{2n}<2\pi} \int_{0}^{2\pi}\left|\sum_{i=1}^{2n}\widetilde a_i K(t-x_i)\right|\,dt \right\}E_n(\varphi)_C, \tag{10} \]
where \(\widetilde a_i\) is a function of the parameters \(x_j\) \((1\leq j\leq 2n,\ j\ne i)\), defined by (7).
The inequality (10) cannot be improved.
Corollary. Let \(f(x)\) be a periodic function with period \(2\pi\), having a continuous \(r\)-th derivative \(f^{(r)}(x)\); then
\[ E_n(f)_C \leq \frac{K_r}{n^r}E_n\bigl(f^{(r)}\bigr)_C,\qquad n=1,2,\ldots \tag{11} \]
If, moreover, the conjugate function \(\widetilde f(x)\) has a continuous \(r\)-th derivative \(\widetilde f^{(r)}(x)\), then
\[ E_n(f)_C \leq \frac{\widetilde K_r}{n^r}E_n\bigl(\widetilde f^{(r)}\bigr)_C,\qquad n=1,2,\ldots \tag{12} \]
Here \(K_r,\ \widetilde K_r\) are the Favard constants \((^3,^7)\). The constants cannot be improved.
- We now consider the case \(p=1\), \(K(t)\in C[0,\omega]\). Let \(H_V\) be the class of functions \(g(t)\) of bounded variation on \([0,\omega]\) with norm \(\|g\|_V=\int_{0}^{\omega}|dg|\leq 1\). Put
\[ F(x)=\frac{1}{\omega}\int_{0}^{\omega}K(t-x)\,dg(t). \tag{13} \]
Theorem 4. If \(K(t)\in C[0,\omega]\), then
\[ M_n^{(1)}=\max_{g\in H_V}E_n(F)_C=\frac{1}{\omega}E_n(K)_C. \tag{14} \]
This work was carried out under the supervision of S. B. Stechkin, to whom I express my deep gratitude.
Moscow State University
named after M. V. Lomonosov
Received
24 VI 1957
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