UDC 517.512.6

MATHEMATICS

R. A. RAITSIN

ON BEST MEAN APPROXIMATION BY POLYNOMIALS OF FUNCTIONS HAVING A REAL SINGULAR POINT

(Presented by Academician S. N. Bernstein on May 6, 1966)

S. N. Bernstein \((^{1-3})\) first observed that the sequence of best uniform approximations of the function \(|x-c|^p\) \((-1<c<1,\ p>0)\) on the interval \([-1,1]\) by algebraic polynomials of degree \(n\) is asymptotically equal to \((\sqrt{1-c^2})^p\mu(p)n^{-p}\) \((n\to\infty)\), where \(\mu(p)\) is a certain constant, and indicated a method for finding the value of \(\mu(p)\). These works play a fundamental role in the study of the properties of best approximation by polynomials of other functions with singularities of the same type.

Subsequently I. I. Ibragimov \((^4)\) applied S. N. Bernstein’s method to the estimation of best uniform approximation of functions of the form \(x^r|x|^\alpha \ln^m|x|\) \((r+\alpha>0,\ m\ge 0,\ r\) and \(m\) are integers, \(\alpha\) is a real number) on the interval \([-1,1]\) by means of a polynomial of degree \(n\); he also considered the case of the function \((a-x)^{r+\alpha}\ln^m(a-x)\) \((a\ge 1)\).

S. M. Nikol’skii \((^5)\) investigated the asymptotic properties of the best approximation of the function \(|a-x|^s\) by polynomials in the metric of the space \(L\) on the interval \([-1,1]\), and obtained the corresponding asymptotic equalities.

The present note is devoted to best approximation by polynomials in the metric of the space \(L_q(-1,1)\) \((1\le q\le\infty)\) of the function \((x-a)^r\times |x-a|^\alpha\ln^m|x-a|\), where \(r\) and \(m\) are integers, and \(\alpha\) and \(a\) are real numbers. In these investigations the main role is played by a general limit theorem of S. N. Bernstein (see \((^6)\), Theorem VII bis), and also Theorem 1, generalizing the well-known method of S. N. Bernstein \((^3)\) (this theorem is analogous to S. N. Bernstein’s theorems not only in formulation, but also in the method of proof).

Theorem 1 (generalization of Bernstein’s inequality). Suppose that for the best approximation of a function \(f(x)\) \((f(x)\in L_q(-2,2))\) the following conditions are satisfied

\[ \text{1)}\quad E_{n-1}(f;a,b)_{L_q} < \left(1+\frac{B\ln n}{\alpha_n}\right) E_n(f;a,b)_{L_q} \quad (-2<a<0<b<2) \tag{1} \]

where \(B>0\) is some constant; \(\alpha_n\) is a monotonically increasing sequence \((1\le \alpha_n\le n)\) such that, for any constant \(\gamma\) \((0\le \gamma<1)\),

\[ \lim_{n\to\infty}\frac{\ln n}{\alpha_n^{\,1-\gamma}}=0 \quad\text{and}\quad \lim_{n\to\infty}\frac{\alpha_{n+o(n)}}{\alpha_n}=1. \]

\[ \text{2)}\quad E_n(x^{2p\nu}f(x);a,b)_{L_q} < \frac{C^{2\nu}(2\nu)^{2\nu}}{n^{2\nu}}\, E_n(f;a,b)_{L_q}, \tag{2} \]

where \(p\) and \(\nu\) are natural numbers; \(C>0\) is some constant independent of \(\nu\) and \(n\).

3) There exists a natural number \(j\) such that

\[ \lim_{n\to\infty} n^j E_n(f;a,b)_{L_q}=\infty . \tag{3} \]

Then

\[ E_n\left(\sum_{k=1}^{l} B_k f(x-a_k); -1,1\right)_{L_q} = \]

\[ = [1+o(1)]\left\{\sum_{k=1}^{l} |B_k|^q E_n^q(f(x-a_k); -1,1)_{L_q}\right\}^{1/q}, \tag{4} \]

where \(B_k\) are constants, \(-1<a_k<1,\ a_k\ne a_i\ (k\ne i),\ l<\infty\).

Consider the best weighted approximation

\[ E_{n,r(x)}(f;a,b)_{L_q} = \inf_{c_k}\left\{\int_a^b \left|f(x)-\sum_{k=1}^{n} c_k x^k\right|^q [r(x)]^q\,dx\right\}^{1/q}, \tag{5} \]

where the weight \(r(x)\) satisfies the following conditions:

1) \(r(x)\ge \beta>0\);

2) \[ \left\{\int_a^b [r(x)]^q\,dx\right\}^{1/q}\le M<\infty \quad (-2<a<0<b<2); \]

3) the function \(r(x)\) is continuous at the point \(x=0\).

Corollary. If the function \(f(x)\) is continuous outside every neighborhood of zero and satisfies the conditions of Theorem 1, then the following asymptotic equality holds \((n\to\infty)\):

\[ E_{n,r(x)}(f;a,b)_{L_q} = [1+o(1)]r(0)E_n(f;a,b)_{L_q}. \tag{6} \]

Lemma. For any \(0<\delta_n<1\), for all natural \(n>1\), on intervals \([a,b]\), where \(a<0<b\), the inequality holds \((r+\alpha>-1/q,\ r\text{ and }m\ge 0\text{ are integers})\)

\[ E_{n-1}(x^r|x|^\alpha \ln^m|x|;a,b)_{L_q} \le \frac{(1+\delta_n)^{n+1/q}+1}{(1+\delta_n)^{n-r-\alpha}-1} \times \]

\[ \times \left( 1+ \frac{C(1+\delta_n)^{n-r-\alpha}\ln(1+\delta_n)} {[(1+\delta_n)^{n+1/q}+1]\ln n} \right) E_n(x^r|x|^\alpha \ln^m|x|;a,b)_{L_q}, \tag{7} \]

where \(C>0\) is some constant.

If in this inequality we put \((1+\delta_n)^{\,n-r-\alpha}=2n\), then we obtain (see \((^3,^7)\))

\[ E_{n-1}(f;a,b)_{L_q} < \left(1+\frac{D\ln n}{n}\right)E_n(f;a,b)_{L_q}, \tag{8} \]

where \(D>0\) is some constant.

Thus, for the function \(x^r|x|^\alpha\ln^m|x|\) condition (1) of Theorem 1 is satisfied. It is not difficult to verify that for this function conditions (2) with \(p=1\) and condition (3) are satisfied. Thanks to the general limit theorem of S. N. Bernstein \((^6)\), this makes it possible to prove the following theorem.

Theorem 2. If \(r+\alpha>-1/q,\ m\ge 0\), and \(|a_k|<1\) (\(r\) and \(m\) are integers, \(\alpha\) and \(a_k\) are real numbers), then the equality holds \((n\to\infty)\)

\[ E_n\left(\sum_{k=1}^{l} B_k (x-a_k)^r |x-a_k|^\alpha \ln^m|x-a_k|; -1,1\right)_{L_q} = \]

\[ = [1+o(1)] \left\{ \sum_{k=1}^{l} \left|B_k\left(\sqrt{1-a_k^2}\right)^{r+\alpha+1/q}\right|^q \right\}^{1/q} E_n(x^r|x|^\alpha \ln^m|x|; -1,1)_{L_q} \tag{9} \]

(\(B_k\) are constants, \(l<\infty\)).

If \(\alpha\) is not an even integer, then

\[ \lim_{n\to\infty}\frac{n^{r+\alpha+1/q}}{(\ln n)^m} E_n\bigl(x^r|x|^\alpha \ln^m |x|;\,-1,1\bigr)_{L_q} = A_1\bigl(x^r|x|^\alpha\bigr)_{L_q}<\infty; \tag{10} \]

if \(\alpha\) is an even integer, then

\[ \lim_{n\to\infty}\frac{n^{r+\alpha+1/q}}{(\ln n)^{m-1}} E_n\bigl(x^{r+\alpha}\ln^m |x|;\,-1,1\bigr)_{L_q} = m A_1\bigl(x^{r+\alpha}\ln |x|\bigr)_{L_q}<\infty, \tag{11} \]

where \(A_1[f(x)]_{L_q}\) is the best approximation of the function \(f(x)\) by entire functions of degree \(\le 1\) in the metric of the space \(L_q(-\infty,\infty)\).

Relations (9) and (10), when \(r\) is even and \(m=0\), for \(q=\infty\) constitute the theorem of S. N. Bernstein \((^3)\), and for \(q=1\) the result of S. M. Nikol’skii \((^5)\). For arbitrary natural \(r\) and \(m=0\), for \(q=\infty\) this asymptotic equality was proved in the book of A. F. Timan \((^7)\), and for arbitrary \(q\) it was given in a note \((^8)\). Relations (10) and (11) were obtained by I. I. Ibragimov \((^4)\) (\(q=\infty\)) (see also \((^9)\)). In the case \(l=m=q=1,\ r=\alpha=0\), equalities (9) and (11) were obtained by another method by V. I. Gukevich \((^{10})\).

Using the corollary of Theorem 1, we obtain the theorem:

Theorem 3. Whatever the noninteger \(s>-1/2q\) and the integer \(m\ge 0\) may be, for the best approximation of the function \((1-x)^s\ln^m(1-x)\) in the metric of the space \(L_q\) \((1\le q<\infty)\) with weight \((1-x^2)^{-1/2q}\) by algebraic polynomials on \([-1,1]\) there exists the limit

\[ \lim_{n\to\infty} \frac{n^{2s+1/q}}{(\ln n)^m} E_{n,(1-x^2)^{-1/2q}} \bigl[(1-x)^s\ln^m(1-x);\,-1,1\bigr]_{L_q} = 2^{m-s-1/q} A_1\bigl(|x|^{2s}\bigr)_{L_q}, \tag{12} \]

and in the case when \(s\ge 0\) and \(m>0\) are integers,

\[ \lim_{n\to\infty} \frac{n^{2s+1/q}}{(\ln n)^{m-1}} E_{n,(1-x^2)^{-1/2q}} \bigl[(1-x)^s\ln^m(1-x);\,-1,1\bigr]_{L_q} = \]

\[ = m2^{m-s-1/q} A_1\bigl(x^{2s}\ln|x|\bigr)_{L_q}. \tag{13} \]

For the best approximation of the function \((a-x)^s\ln^m(a-x)\) \((a>1)\) in the metric of the space \(L_q(-1,1)\), it is not difficult to obtain the inequalities

\[ \frac{ 2^{(q+1)/q}(a^2-1)^{(s+1)/2}(\ln n)^m(1-\varepsilon_n) }{ |\Gamma(-s)|\,n^{s+1}(a+\sqrt{a^2-1})^{n+2} } \le E_n\bigl[(a-x)^s\ln^m(a-x);\,-1,1\bigr]_{L_q} \le \]

\[ \le \frac{ 2^{1/q}(a^2-1)^{(s-1)/2}(\ln n)^m(1+\varepsilon_n) }{ |\Gamma(-s)|\,n^{s+1}(a+\sqrt{a^2-1})^n }, \tag{14} \]

where \(\varepsilon_n\to 0\) as \(n\to\infty\), \(s\) is not a nonnegative integer, and

\[ \frac{ 2^{(q+1)/q}(a^2-1)^{(s+1)/2}s!\,m(\ln n)^{m-1}(1-\varepsilon_n) }{ n^{s+1}(a+\sqrt{a^2-1})^{n+2} } \le E_n\bigl[(a-x)^s\ln^m(a-x);\,-1,1\bigr]_{L_q} \le \]

\[ \le \frac{ 2^{1/q}(a^2-1)^{(s-1)/2}s!\,m(\ln n)^{m-1}(1+\varepsilon_n) }{ n^{s+1}(a+\sqrt{a^2-1})^n }, \tag{15} \]

if \(s\ge 0\) is an integer.

I express my deep gratitude to Prof. A. F. Timan for posing the problem and for his attention to the present work.

Dnepropetrovsk Chemical-Technological Institute
named after F. E. Dzerzhinsky

Received
9 IV 1966

CITED LITERATURE

\(^1\) S. N. Bernstein, Extremal properties of polynomials, Moscow–Leningrad, 1937, pp. 58–102.
\(^2\) S. N. Bernstein, Izv. AN SSSR, OMEN, 169 (1938).
\(^3\) S. N. Bernstein, DAN, 18, 379 (1938).
\(^4\) I. I. Ibragimov, Izv. AN SSSR, ser. matem., 10, 429 (1947).
\(^5\) S. M. Nikol’skii, Izv. AN SSSR, ser. matem., 11, 139 (1947).
\(^6\) S. N. Bernstein, DAN, 58, 525 (1947).
\(^7\) A. F. Timan, Theory of approximation of functions of a real variable, Moscow, 1960, pp. 426–450.
\(^8\) R. A. Raipin, DAN, 164, 51 (1965).
\(^9\) S. N. Bernstein, DAN, 54, 667 (1946).
\(^ {10}\) V. I. Gukevich, DAN, 77, 785 (1951).