Abstract Generated abstract
The paper studies best approximation by algebraic polynomials in the spaces \(L_q(-1,1)\), \(1 \le q \le \infty\), for functions with real singularities of the form \((x-a)^r |x-a|^\alpha \ln^m |x-a|\). Using a general limit theorem of S. N. Bernstein, it proves a generalized Bernstein-type inequality and derives asymptotic formulas for sums of such singular terms at finitely many interior points. The results include weighted approximation consequences and limiting constants expressed through best approximation by entire functions of degree at most one on the real line. The paper also gives asymptotic results for endpoint singularities with Jacobi-type weight and estimates for functions \((a-x)^s \ln^m(a-x)\) with a singularity outside the interval.
Full Text
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).