Abstract
Full Text
UDC 517.512.6
MATHEMATICS
M.-B. A. BABAEV
ON THE APPROXIMATION OF POLYNOMIALS OF TWO VARIABLES BY FUNCTIONS OF THE FORM \(\varphi(x)+\psi(y)\)
(Presented by Academician I. N. Vekua, 30 I 1970)
In the present work, for a certain class of functions of two variables, a simple method is proposed for computing the values of the best approximation by functions of the form \(\varphi(x)+\psi(y)\); a method is established for finding the best approximating function for polynomials of two variables, by means of which the best approximating function is found.
Let \(f(x,y)\) be a function continuous on the rectangle \(Q=[a,b;c,d]\), and
\[ E[f,\varphi+\psi;Q]_C=E[f]=\inf_{\varphi+\psi}\|f-(\varphi+\varphi)\|,\qquad \|f\|=\sup_{x,y\in Q}|f|. \]
Denote by \(\Pi\) the class of functions \(f(x,y)\) having the following property: for arbitrary \(x''\ge x'\), \(y''\ge y'\) from \(Q\), the inequality
\[ f(x'',y'')+f(x',y')\ge f(x'',y')+f(x',y'') \]
holds.
Let \(\Pi_{-}\) be the class of functions \(f(x,y)\) for which \((-f)\in\Pi\). It is not difficult to understand that if \(f_{xy}\ge0\), then \(f\in\Pi\). Indeed, for arbitrary \(x''\ge x'\), \(y''\ge y'\) we have
\[ 0\le \int_{x'}^{x''}\int_{y'}^{y''} f_{xy}\,dx\,dy = f(x'',y'')+f(x',y')-f(x'',y')-f(x',y''). \]
The converse, in general, is not true, since the function \(f(x,y)\) need not be differentiable.
Similarly, if \(f_{xy}\le0\), then \(f\in\Pi_{-}\).
Theorem 1. For an arbitrary continuous function \(f(x,y)\in\Pi\),
\[ E[f,\varphi+\psi;Q]_C=\frac14[f(b,d)+f(a,c)-f(a,d)-f(b,c)]. \]
This result, which is a strengthening of a theorem of T. J. Rivlin and R. J. Sibner \((^1)\) (the case \(f_{xy}\ge0,\ Q=[0,1;0,1]\)), also differs in the method of proof.
We also note the relatively complicated methods \((^{2-4})\) for determining the value of the best approximation in the general case.
Corollary 1. For an arbitrary continuous function \(f(x,y)\in\Pi_{-}\),
\[ E[f,\varphi+\psi;Q]_C=-\frac14[f(b,d)+f(a,c)-f(a,d)-f(b,c)]. \]
Corollary 2. If the mixed derivative \(f_{xy}\) preserves its sign on the rectangle \(Q\), then
\[ E[f,\varphi+\psi;Q]_C=\frac14|f(b,d)+f(a,c)-f(a,d)-f(b,c)|. \]
Corollary 3. Let \(f=x^\alpha y^\beta g(x,y)\), where \(\alpha,\beta\ge0\) are real numbers and \(Q=[0,b;0,d]\).
Then:
a) if \(f\in \Pi\), then \(E[f,\varphi+\psi,Q]_C=\frac14 f(b,d)\);
b) if \(f\in \Pi_{-}\), then \(E[f,\varphi+\psi,Q]_C=-\frac14 f(b,d)\).
Corollary 4. Let \(S=[0,1;\,0,1]\). Then for arbitrary real \(\alpha,\beta\ge 0\),
\[ E[x^\alpha y^\beta,\varphi+\psi;\,S]_C = E[xy,\varphi+\psi;\,S]_C = \frac14 . \]
This follows immediately from Corollary 3 and from the fact that \(x^\alpha y^\beta\in \Pi\). Consider the approximation of the polynomial in two variables
\[ z=\sum_{p=0}^{m}\sum_{q=0}^{n} A_{pq}x^p y^q \]
on the rectangle \(Q\) by all possible functions of the form \(\varphi(x)+\psi(y)\) in the metric of the space \(C\). Taking into account
\[ \sum_{p=0}^{m}\sum_{q=0}^{n} A_{pq}x^p y^q = \sum_{p=1}^{m}\sum_{q=1}^{n} A_{pq}x^p y^q + \sum_{p=0}^{m} A_{p0}x^p + \sum_{q=1}^{n} A_{0q}y^q, \]
we conclude that
\[ E\left[\sum_{p=0}^{m}\sum_{q=0}^{n} A_{pq}x^p y^q,\varphi+\psi\right] = E\left[\sum_{p=1}^{m}\sum_{q=1}^{n} A_{pq}x^p y^q,\varphi+\psi\right], \]
and if \(\varphi_0+\psi_0\) is the best approximant to the polynomial \(\displaystyle \sum_{0}^{m}\sum_{0}^{n} A_{pq}x^p y^q\), then
\[ \varphi_0+\psi_0- \left( \sum_{p=0}^{m} A_{p0}x^p + \sum_{q=1}^{n} A_{0q}y^q \right) \]
is the best approximant to the polynomial \(\displaystyle \sum_{1}^{m}\sum_{1}^{n} A_{pq}x^p y^q\).
Therefore, in what follows we shall assume
\[ z=\sum_{1}^{m}\sum_{1}^{n} A_{pq}x^p y^q. \]
We propose the following method \(\alpha\) for determining the best approximating function:
\[ z=\sum_{1}^{m}\sum_{1}^{n} A_{pq}x^p y^q, \]
\[ z_1=z,\qquad z_{2n}=z_{2n-1}-g_n,\qquad z_{2n+1}=z_{2n}+h_n, \]
where
\[ g_n=g_n(x)=\frac12\left[\max_{c\le y\le d}^{\wedge} z_{2n-1}+\min_{c\le y\le d}^{\vee} z_{2n-1}\right], \]
\[ h_n=h_n(y)=\frac12\left[\max_{a\le x\le b}^{\wedge} z_{2n}+\min_{a\le x\le b}^{\vee} z_{2n}\right], \]
\[ \max^{\wedge} z=\sum_{1}^{m}\sum_{1}^{n} \max A_{pq}x^p y^q, \]
\[ \min^{\vee} z=\sum_{1}^{m}\sum_{1}^{n} \min A_{pq}x^p y^q; \]
\(\widehat{\max} z\) denotes the sum of the maxima of all terms of the polynomial, and \(\check{\min} z\) the sum of the minima. Obviously,
\[ \widehat{\max} z \geq \max z; \qquad \check{\min} z \leq \min z . \]
Denote
\[ \lim_{n\to\infty} g_n(x)=g_0(x)=g_0,\qquad \lim_{n\to\infty} h_n(y)=h_0(y)=h_0 . \]
We assert that
\[ \|z-(g_0-h_0)\|=E[z], \]
i.e. \(g_0(x)+h_0(y)\) is a best approximating function.
We shall need some auxiliary facts.
Lemma 1. If, for nonnegative numbers \(c,d,y\), for some fixed \(i\),
\[ y^i \geq \frac12(c^i+d^i), \qquad y^{i+1}<\frac12(c^{i+1}+d^{i+1}), \]
then
\[ y^{i+k}<\frac12(c^{i+k}+d^{i+k}),\qquad k=2,3,\ldots \]
Lemma 2. If, for nonnegative numbers \(c,d\), and \(y\),
\[ y<\frac12(c+d), \]
then
\[ y^i<\frac12(c^i+d^i),\qquad i=2,3,\ldots \]
Corollary. If \(y^n \geq \frac12(c^n+d^n)\), then
\[ y^i \geq \frac12(c^i+d^i),\qquad i=1,2,\ldots,n-1. \]
Indeed, the existence of at least one inequality
\[
y^{i_0}<\frac12(c^{i_0}+d^{i_0}),\qquad i_0\leq n-1,
\]
would imply, by one of Lemmas 1 and 2, that
\[
y^n<\frac12(c^n+d^n).
\]
Theorem 2. Method \(\alpha\) makes it possible to obtain, for the polynomial
\[ z=\sum_{p=1}^{m}\sum_{q=1}^{n} A_{pq}x^p y^q,\qquad A_{pq}\geq 0, \]
the best approximating function of the form \(\varphi(x)+\psi(y)\) on the rectangle \([a,b;c,d]\), \(a,c\geq 0\), and this function is the polynomial
\[ \frac12\sum_{p=1}^{m}\sum_{q=1}^{n} A_{pq} \left[x^p(c^q+d^q)+y^q(a^p+b^p)-\frac12(a^p+b^p)(c^q+d^q)\right]. \]
The author expresses gratitude to Acad. of the Academy of Sciences of the Azerb. SSR I. I. Ibragimov for his attention to the work and discussion of the results.
Institute of Mathematics and Mechanics
Academy of Sciences of the Azerb. SSR
Received
14 I 1970
CITED LITERATURE
- T. J. Rivlin, R. J. Sibner, Am. Math. Monthly, 72, No. 10, 1101 (1965).
- S. P. Diliberto, E. G. Straus, Pacific J. Math., 1, 195 (1951).
- M.-B. A. Babaev, Dokl. AN AzerbSSR, 23, No. 1, 3 (1967).
- M.-B. A. Babaev, Dokl. AN AzerbSSR, 23, No. 2, 3 (1967).