Full Text
UDC 517.512
MATHEMATICS
G. V. ZHIDKOV
A CONSTRUCTIVE CHARACTERIZATION OF ONE CLASS OF NONPERIODIC FUNCTIONS
(Presented by Academician S. N. Bernstein on 30 XI 1965)
Let \(f(x) \in L_2[-1,1]\); \(E_n^{(2)}[f]\) is the best approximation to \(f(x)\) on \([-1,1]\) by algebraic polynomials of degree not exceeding \(n\) in the metric \(L_2\).
Denote
\[
f_h(x)=\frac{1}{\pi}\int_0^\pi f\bigl(x\cos h+\sqrt{1-x^2}\sin h\cos\theta\bigr)\,d\theta .
\]
The article considers the question of the structural properties of the class of functions satisfying the condition
\[
E_n^{(2)}[f]\leq M/n^{s+\gamma},\qquad n>s,\qquad n'=1,2,\ldots,\qquad 0<\gamma<1,
\]
where \(M\) is a constant independent of \(n\).
Theorem 1. In order that the inequality
\[
E_n^{(2)}[f]\leq M/n^{s+\gamma}
\]
hold, it is necessary and sufficient that
\[
\left\{\int_{-1}^{1}\left[\frac{d^s f_h(x)}{dx^s}-\frac{d^s f(x)}{dx^s}\right]^2(1-x^2)^s\,dx\right\}^{1/2}\leq c h^\gamma,
\]
where \(c\) is a constant independent of \(h>0,\ n>s,\ 0<\gamma<1\).
It should be noted that the question of approximation of functions in the metric \(L_p\) by algebraic polynomials with a certain fractional-rational weight was considered by M. K. Potapov and G. K. Lebedev \((^4)\).
In proving the lemmas and theorems, the article uses the classical method of \(2^n\)-summation of S. N. Bernstein \((^1,^2)\).
Lemma 1. For the Legendre polynomial \(P_k(\cos h)\), for any \(h\) and \(k\) the inequality
\[
1-P_k(\cos h)\leq k^2h^2/2
\tag{1}
\]
holds, and respectively
\[
1-P_k(\cos h)\geq 4k^2h^2/3\pi^3,\qquad 0\leq kh\leq \pi .
\tag{2}
\]
Proof. We use the expression for the Legendre polynomial \((^3)\)
\[
P_k(\cos h)=
2\frac{(2k-1)!!}{(2k)!!}\cos kh
+
2\frac{(2k-3)!!}{(2k-2)!!}\frac{1}{2}\cos (k-2)h+\cdots
\]
\[
\cdots+
2\frac{(2k-2m-1)!!}{(2k-2m)!!}\frac{(2m-1)!!}{(2m)!!}\cos (k-2m)h+\cdots
\]
Since \(1=P_k(1)\), it follows that
\[
1-P_k(\cos h)=
4\frac{(2k-1)!!}{(2k)!!}\sin^2 k\frac{h}{2}
+
4\frac{(2k-3)!!}{(2k-2)!!}\frac{1}{2}\sin^2 (k-2)\frac{h}{2}
+\cdots
\]
\[
\cdots+
4\frac{(2k-2m-1)!!}{(2k-2m)!!}\frac{(2m-1)!!}{(2m)!!}\sin^2 (k-2m)\frac{h}{2}
+\cdots
\tag{3}
\]
Hence, from the equality \(1=P_k(1)\), taking into account the inequality \(\sin^2 x\leq x^2\), we obtain (1).
On the basis of the inequality
\[ \left[\frac{(2m)!!}{(2m-1)!!}\right]^2\frac{1}{2m+1}<\frac{\pi}{2}< \left[\frac{(2m)!!}{(2m-1)!!}\right]^2\frac{1}{2m} \tag{4} \]
we have
\[ \frac{(2k-2m-1)!!}{(2k-2m)!!}\frac{(2m-1)!!}{(2m)!!}> \frac{2}{\pi}\frac{1}{\sqrt{(2k-2m+1)(2m+1)}}> \frac{2}{\pi(k+1)}. \tag{5} \]
Since \(0\leq kh\leq\pi\), it follows from (3), (5), and the inequality \(\sin x\geq 2x/\pi\), where \(0\leq x\leq \pi/2\), that
\[ 1-P_k(\cos h)\geq \frac{8h^2}{\pi^3(k+1)} \left[k^2+(k-2)^2+\cdots+(k-2m)^2+\cdots\right] \geq \frac{4k^2h^2}{3\pi^3}. \tag{6} \]
Let \(f(\cos\beta)\sqrt{\sin\beta}\) be a square-summable function on \([0,\pi]\). Denote
\[ b_k=\frac{2k+1}{2}\int_0^\pi f(\cos\beta)P_k(\cos\beta)\sin\beta\,d\beta, \qquad a_k^2=\frac{2b_k^2}{2k+1}. \tag{7} \]
Under these assumptions, the following assertion is valid.
Lemma 2. In order that the inequality
\[ \left(\sum_{k=n}^{\infty}a_k^2\right)^{1/2}\leq M/n^{s+\gamma} \tag{8} \]
hold, it is necessary and sufficient that
\[ \left\{\int_0^\pi \left[ \frac{d^s}{d\cos\beta^s}\frac{1}{\pi}\int_0^\pi f(\cos R)\,d\theta - \frac{d^s}{d\cos\beta^s}f(\cos\beta) \right]^2 \sin^{2s+1}\beta\,d\beta \right\}^{1/2} \leq ch^\gamma, \tag{9} \]
where \(\cos R=\cos\beta\cos h+\sin\beta\sin h\cos\theta,\quad n>s,\quad 0<\gamma<1.\)
Proof. Suppose that (9) holds. From the relation
\[ f(\cos\beta)\sim \sum_{k=0}^{\infty} b_kP_k(\cos\beta), \]
applying the addition theorem for Legendre polynomials
\[ P_k(\cos R)=P_k(\cos\beta)P_k(\cos h)+2\sum_{m=1}^{k} \frac{(k-m)!}{(k+m)!}P_k^m(\cos\beta)P_k^m(\cos h)\cos m\theta, \]
we obtain
\[ \left[ \frac{d^s}{d\cos\beta^s}\frac{1}{\pi}\int_0^\pi f(\cos R)\,d\theta - \frac{d^s}{d\cos\beta^s}f(\cos\beta) \right]\sin^s\beta \sim \]
\[ \sim \sum_{k=s}^{\infty}[P_k(\cos h)-1]P_k^s(\cos\beta)b_k. \]
By Parseval’s equality, taking into account (7) and the fact that
\[ \int_0^\pi \{P_k^s(\cos\beta)\}^2\sin\beta\,d\beta = \frac{2}{2k+1}\frac{(k+s)!}{(k-s)!}, \]
we shall have
\[ \int_{0}^{\pi}\left[\frac{d^{s}}{d\cos\beta^{s}}\,\frac{1}{\pi}\int_{0}^{\pi} f(\cos R)\,d\theta-\frac{d^{s}}{d\cos\beta^{s}}f(\cos\beta)\right]^{2}\sin^{2s+1}\beta\,d\beta = \sum_{k=s}^{\infty}[P_k(\cos h)-1]^2\frac{(k+s)!}{(k-s)!}\,a_k^2 . \tag{10} \]
For any \(n>s\), from condition (9) we have
\[ \frac{(n+s)!}{(n-s)!}\sum_{k=n}^{2n-1}[P_k(\cos h)-1]^2a_k^2 \leq \sum_{k=s}^{\infty}[P_k(\cos h)-1]^2\frac{(k+s)!}{(k-s)!}\,a_k^2 \leq c^2h^{2\gamma}. \]
Putting \(h=1/n\) and taking into account here that \(1\leq kh<2\), on the basis of inequality (2) we obtain
\[ \frac{(n+s)!}{(n-s)!}\sum_{k=n}^{2n-1}a_k^2 \leq \frac{(n+s)!}{(n-s)!}\sum_{k=n}^{2n-1}k^4h^4a_k^2 \leq \frac{c_1}{n^{2\gamma}},\qquad c_1=\frac{9}{16}\pi^6c^2 . \tag{11} \]
Since \(n>s\), for \(i=1,2,\ldots,s-1\) we have \(n-i>\frac{s-i}{s}n\). Hence, also from the inequality
\((n+1)(n+2)\cdots(n+s)>n^s\), it follows that
\[ \frac{(n-s)!}{(n+s)!}\leq \frac{s^s}{s!}\frac{1}{n^{2s}} . \]
The last inequality and (11) give
\[ \sum_{k=n}^{2n-1}a_k^2\leq \frac{c_2}{n^{2s+2\gamma}},\qquad c_2=c_1\frac{s^s}{s!}. \tag{12} \]
Therefore
\[ \sum_{k=n}^{\infty}a_k^2 = \sum_{j=0}^{\infty}\sum_{k=2^jn}^{2^{j+1}n-1}a_k^2 \leq \frac{c_2}{n^{2s+2\gamma}}\sum_{j=0}^{\infty}2^{-2j(s+\gamma)} <\infty . \]
Taking into account the convergence of the series on the right, we obtain (8).
Let us show that (9) follows from (8). The inequality holds
\[ \sum_{k=s}^{\infty}[P_k(\cos h)-1]^2\frac{(k+s)!}{(k-s)!}a_k^2 \leq 2^s\sum_{k=s}^{\infty}[P_k(\cos h)-1]^2k^{2s}a_k^2 . \]
Represent the following sum by two terms:
\[ \sum_{k=s}^{\infty}[P_k(\cos h)-1]^2k^{2s}a_k^2 = \sum_{k=s}^{n-1}+\sum_{k=n}^{\infty},\qquad n=\left[\frac{1}{h}\right]. \]
Put
\[ \gamma_k=\sum_{m=k}^{\infty}m^{2s}a_m^2\leq M_2/k^{2\gamma}. \tag{13} \]
Inequality (13) follows from (12) and is equivalent to (8). Taking inequalities (1) and (13) into account, we shall have
\[ \sum_{k=s}^{n-1}[P_k(\cos h)-1]^2k^{2s}a_k^2 \leq \sum_{k=1}^{n}k^4h^4k^{2s}a_k^2 = h^4\sum_{k=1}^{n}k^4(\gamma_k-\gamma_{k-1}) \leq \]
\[ \leq h^4\{\gamma_1+(2^4-1^4)\gamma_2+(3^4-2^4)\gamma_2+\cdots+[n^4-(n-1)^4]\gamma_n\} \leq \]
\[ \leq 6h^4\sum_{k=1}^{n}k^3\gamma_k \leq 6M_2h^4\sum_{k=1}^{n}k^{3-2\gamma} \leq M_3h^{2\gamma}. \tag{14} \]
Since \(|P_k(\cos h)| \leqslant 1\), we have
\[ \sum_{k=n}^{\infty} [P_k(\cos h)-1]^2 k^{2s} a_k^2 \leqslant 4 \sum_{k=n}^{\infty} k^{2s} a_k^2 \leqslant 4M_2 h^{2\gamma}. \tag{15} \]
Combining the estimates (14) and (15), we obtain (9).
Theorem 1 follows from Lemma 2, if we put \(x=\cos\beta\) and take into account that
\[ \left\{\int_{-1}^{1}\left[f(x)-\sum_{k=0}^{n-1} b_k P_k(x)\right]^2 dx\right\}^{1/2} = \left(\sum_{k=n}^{\infty} a_k^2\right)^{1/2} = E_n^{(2)}[f]. \]
Next we shall prove a theorem on the absolute convergence of the series in Legendre polynomials, which is an analogue of S. N. Bernstein’s theorem on the absolute convergence of the trigonometric Fourier series (1).
Theorem 2. If the function \(f(x)\) satisfies the condition \(|f_h(x)-f(x)| \leqslant c h^\gamma\), \(-1 \leqslant x \leqslant 1\), \(h>0\), \(\gamma>1/2\), then the series \(\sum_{k=2}^{\infty} |a_k|\) converges.
Proof. For \(s=0\), from (10) we obtain
\[ \int_0^\pi \left[\frac{1}{\pi}\int_0^\pi f(\cos R)\,d\theta - f(\cos\beta)\right]^2 \sin\beta\,d\beta = \sum_{k=0}^{\infty} [P_k(\cos h)-1]^2 a_k^2. \]
Putting \(x=\cos\beta\), from the condition of Theorem 2 we shall have
\[ \sum_{k=0}^{\infty} [P_k(\cos h)-1]^2 a_k^2 \leqslant 2c^2 h^{2\gamma}. \]
Choose an arbitrary natural number \(N\) and put \(h=1/N\). Then
\[ \sum_{k>N/2}^{N} [P_k(\cos h)-1]^2 a_k^2 \leqslant 2c^2 N^{-2\gamma}. \]
Hence, and from inequality (2),
\[ \sum_{k>N/2}^{N} a_k^2 \leqslant 2c_1 N^{-2\gamma}. \]
In particular, if we put \(N=2^\nu\), \(\nu=1,2,\ldots\), then
\[ \sum_{k=2^{\nu-1}+1}^{2^\nu} a_k^2 \leqslant 2c_1 2^{-2\gamma\nu}. \]
Hence, and from Bunyakovsky’s inequality,
\[ \sum_{k=2^{\nu-1}+1}^{2^\nu} |a_k| \leqslant \left(\sum_{k=2^{\nu-1}+1}^{2^\nu} a_k^2\right)^{1/2} \left(\sum_{k=2^{\nu-1}+1}^{2^\nu} 1\right)^{1/2} \leqslant \sqrt{c_1}\,2^{\nu(1/2-\gamma)}. \]
Consequently,
\[ \sum_{k=2}^{\infty} |a_k| = \sum_{\nu=1}^{\infty}\sum_{k=2^{\nu-1}+1}^{2^\nu} |a_k| \leqslant \sqrt{c_1}\sum_{\nu=1}^{\infty} 2^{(1/2-\gamma)\nu} <\infty. \]
Peoples’ Friendship University
named after P. Lumumba
Received
27 XI 1965
REFERENCES
- S. N. Bernstein, Collected Works, 1, article No. 16, Publishing House of the Academy of Sciences of the USSR, 1952.
- S. N. Bernstein, Collected Works, 2, article No. 82, Publishing House of the Academy of Sciences of the USSR, 1954.
- E. V. Gobson, The Theory of Spherical and Ellipsoidal Functions, IL, 1952.
- M. K. Potapov, Collection: Investigations on Contemporary Problems of the Constructive Theory of Functions, Moscow, 1961.