Abstract Generated abstract
This paper studies the approximation of periodic functions in \(L_p\) by linear summation processes applied to their Fourier series, with errors compared to best trigonometric approximation. It proves general estimates for arbitrary triangular summability matrices, including analogues of Lebesgue-type inequalities for \(p=1,\infty\) and sharper bounds for \(1<p<\infty\). The results are specialized to Zygmund means, Fejér sums, Bernstein-Rogozinskii sums, and Jackson-Vallée-Poussin sums, yielding order estimates in terms of sequences of best approximation errors. Further theorems give improved \(L_p\) bounds for \(1<p<\infty\) and extend the same types of estimates to corresponding trigonometric interpolation sums.
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MATHEMATICS
M. F. TIMAN
SOME LINEAR PROCESSES OF SUMMATION OF FOURIER SERIES AND BEST APPROXIMATION
(Presented by Academician V. I. Smirnov, March 15, 1962)
- Consider the space \(L_p\) \((1 \leq p \leq \infty)\) of all measurable periodic functions of period \(2\pi\), for which, when \(1 \leq p < \infty\),
\[ \|f(x)\|_{L_p} = \left\{\int_0^{2\pi} |f(x)|^p\,dx\right\}^{1/p} <\infty, \]
and, when \(p=\infty\),
\[ \|f(x)\|_{L_\infty} = \operatorname{vrai\,sup}_{0\leq x\leq 2\pi}|f(x)|<\infty. \]
Let \(\{\lambda_k^{(n)}\}\) \((k=0,1,2,\ldots,n;\ n=1,2,\ldots;\ \lambda_0^{(n)}=1;\ \lambda_{n+1}^{(n)}=0)\) be an arbitrary triangular matrix of numbers, and
\[ U_n(f;x;\lambda) = \frac{a_0}{2} + \sum_{k=1}^{n}\lambda_k^{(n)} \bigl(a_k\cos kx+b_k\sin kx\bigr), \]
where \(a_k, b_k\) are the Fourier coefficients of the function \(f(x)\). Denote
\[ R_n(f;\lambda)_{L_p} = \|f(x)-U_n(f;x;\lambda)\|_{L_p}. \]
Theorem 1. If \(f(x)\in L_p\) \((1\leq p\leq \infty)\), then for any matrix \(\{\lambda_k^{(n)}\}\) the inequality
\[
R_n(f;\lambda)_{L_p}
\leq
C\left\{
\sum_{k=0}^{n}
\left|\lambda_k^{(n)}-2\lambda_{k+1}^{(n)}+\lambda_{k+2}^{(n)}\right|
(n-k+1)E_k(f)_{L_p}
\times
\right.
\]
\[
\left.
{}\times
\sum_{\nu=n-k}^{n}\frac{1}{\nu+1}
+
\left|1-\lambda_1^{(n)}\right|
\sum_{\nu=0}^{n}E_\nu(f)_{L_p}
\right\},
\tag{1}
\]
where
\[ E_n(f)_{L_p} = \inf_{\alpha_k,\beta_k} \left\| f(x)- \sum_{k=0}^{n}(\alpha_k\cos kx+\beta_k\sin kx) \right\|_{L_p} \qquad (1\leq p\leq \infty), \]
and \(C\) is an absolute constant. Moreover, in the case \(1<p<\infty\),
\[ R_n(f;\lambda)_{L_p} \leq C_p \sum_{k=0}^{n} \left|\lambda_k^{(n)}-\lambda_{k+1}^{(n)}\right| E_k(f)_{L_p}. \tag{2} \]
Inequality (1) for \(p=\infty\) and \(p=1\) may be regarded as a generalization of Lebesgue’s inequality
\[ \|f(x)-S_n(f;x)\|_{L_p} \leq C E_n(f)_{L_p}\ln n \qquad (p=1,\ p=\infty;\ n>1), \]
which corresponds to the case \(\lambda_k^{(n)}=1\) \((k=0,1,\ldots,n)\), while inequality (2) is a generalization of the well-known inequality
\[ \|f(x)-S_n(f;x)\|_{L_p} \leq C_p E_n(f)_{L_p} \qquad (1<p<\infty), \]
where \(S_n(f;x)\) is the partial sum of the Fourier series of the function \(f(x)\).
- Inequality (1) contains various estimates for particular methods of summation of Fourier series. We shall consider some of them here.
Normal Zygmund means. Let
\[ \lambda_k^{(n)}=1-\left(\frac{k}{n+1}\right)^r \quad (k=0,1,2,\ldots,n;\ r\geqslant 1). \tag{3} \]
Theorem 2. For the matrix (3), for any \(r\geqslant 1\), the inequality
\[ R_n(f;\lambda)_{L_p}\leqslant \frac{C}{n^r} \sum_{\nu=1}^{n}\nu^{r-1}E_{\nu-1}(f)_{L_p} \quad (1\leqslant p\leqslant \infty), \tag{4} \]
holds, where \(C\) is an absolute constant\(^*\).
In particular, for \(r=1\), for Fejér sums the inequality
\[ R_n(f;\lambda)_{L_p}\leqslant \frac{C}{n} \sum_{k=0}^{n}E_k(f)_{L_p} \quad (1\leqslant p\leqslant \infty). \tag{5} \]
holds.
The last estimate can be obtained directly from the integral representation for Fejér sums\(^ {**}\). Indeed, by monotonicity of
\[ \omega_2(f;t)_{L_p}=\sup_{|h|\leq t}\|f(x+h)-2f(x)+f(x-h)\|_{L_p}, \]
\[ \left\|f(x)-\frac{1}{n}\sum_{\nu=0}^{n-1}S_\nu(f;x)\right\|_{L_p} \leqslant C_1\left\{\omega_2\left(f;\frac{1}{n}\right)_{L_p} +\frac{1}{n}\int_{\pi/n}^{\pi}\frac{\omega_2(f;t)_{L_p}}{t^2}\,dt\right\} \leqslant \]
\[ \leqslant C_2\frac{1}{n}\int_{\pi/n}^{\pi}\frac{\omega_2(f;t)_{L_p}}{t^2}\,dt = \frac{C_2}{n}\sum_{\nu=1}^{n-1} \int_{\pi/(\nu+1)}^{\pi/\nu} \frac{\omega_2(f;t)_{L_p}}{t^2}\,dt. \]
It remains to take into account that (see (3), p. 344)
\[ \omega_2\left(f;\frac{1}{n}\right)_{L_p} \leqslant \frac{M}{n^2}\sum_{k=1}^{n}kE_{k-1}(f)_{L_p} \quad (1\leqslant p\leqslant \infty). \]
Bernstein–Rogozinskii sums and Jackson–Vallée-Poussin sums. Let
\[ \lambda_k^{(n)}=\cos \frac{k\pi}{2n+1} \quad (k=0,1,\ldots,n); \tag{6} \]
\[ \lambda_k^{(2n)}= \begin{cases} 1-\dfrac{3}{2}\left(\dfrac{k}{n}\right)^2+\dfrac{3}{4}\left(\dfrac{k}{n}\right)^3, & (0\leqslant k\leqslant n),\\[6pt] \dfrac{1}{4}\left(2-\dfrac{k}{n}\right)^3, & (n\leqslant k\leqslant 2n),\\[6pt] 0, & (k>2n). \end{cases} \tag{7} \]
Theorem 3. For the matrices (6) and (7) the inequality
\[ R_n(f;\lambda)_{L_p}\leqslant \frac{C}{n^2}\sum_{\nu=1}^{n}\nu E_{\nu-1}(f)_{L_p} \quad (1\leqslant p\leqslant \infty), \tag{8} \]
holds, where \(C\) is an absolute constant.
\(\ ^*\) Theorem 2 was reported by the author on 12 X 1961 at the seminar of the Department of Higher Mathematics of the Dnepropetrovsk Agricultural Institute. We note that the means appearing on the right-hand side of inequality (4) play an important role in inverse theorems of constructive function theory (see (3), Chs. 6 and 7) and, as far as we know, were first introduced in paper (1) (see (1), Theorem 4).
\(\ ^{**}\) Estimate (5) was obtained by another method by S. B. Stechkin (5).
- There exist examples showing that, in the general case, for \(p=\infty\) and \(p=1\) the estimates given in Theorems 2 and 3 cannot be improved in order.
At the same time, using the methods applied by the author earlier in papers \((^{2,4})\), one can show that in a number of cases these estimates are crude even in order, if \(1<p<\infty\). In particular, if \(f(x)\in L_p\), \((1<p<\infty)\), and \(E_n(f)_{L_p}=O\left(\dfrac1n\right)\), then, for example, for Fejér sums we have the inequality
\[ \left\|f(x)-\frac1n\sum_{\nu=0}^{n-1}S_\nu(f;x)\right\|_{L_p} \le C_p \begin{cases} \dfrac1n(\ln n)^{1/p}, & (1<p\le 2),\\[6pt] \dfrac1n(\ln n)^{1/2}, & (2\le p<\infty), \end{cases} \]
which gives a sharper estimate than the known \(O(\ln n/n)\).
Theorem 4. If \(f(x)\in L_p\), \((1<p<\infty)\), then for the matrix (3) the inequality
\[ R_n(f;\lambda)_{L_p} \le \frac{C_p}{n^r} \left\{\sum_{\nu=1}^{n}\nu^{\gamma r-1}E_{\nu-1}^{\gamma}(f)_{L_p}\right\}^{1/\gamma}, \tag{9} \]
holds, where \(\gamma=p\) for \(1<p\le 2\), and \(\gamma=2\) for \(2\le p<\infty\).
Theorem 5. If \(f(x)\in L_p\), \((1<p<\infty)\), then for the matrices (6) and (7) the inequality
\[ R_n(f;\lambda)_{L_p} \le \frac{C_p}{n^2} \left\{\sum_{\nu=1}^{n}\nu^{2\gamma-1}E_{\nu-1}^{\gamma}(f)_{L_p}\right\}^{1/\gamma}, \tag{10} \]
holds, where \(\gamma=p\) for \(1<p\le 2\), and \(\gamma=2\) for \(2\le p<\infty\).
There are examples showing that, in this form, these inequalities, in order, can no longer be improved in the general case.
- The same estimates as in the theorems stated above* are valid if, in the left-hand sides of the corresponding inequalities, instead of the sums \(U_n(f;x;\lambda)\), one considers the interpolation sums
\[ U_n^*(f;x;\lambda) = \frac{a_0^{(n)}}{2} + \sum_{\nu=1}^{n}\lambda_\nu^{(n)} \left(a_\nu^{(n)}\cos \nu x+b_\nu^{(n)}\sin \nu x\right), \]
where
\[ a_\nu^{(n)} = \frac{2}{2n+1} \sum_{k=0}^{2n} f\left(\frac{2k\pi}{2n+1}\right) \cos \nu\frac{2k\pi}{2n+1}, \]
\[ b_\nu^{(n)} = \frac{2}{2n+1} \sum_{k=0}^{2n} f\left(\frac{2k\pi}{2n+1}\right) \sin \nu\frac{2k\pi}{2n+1}. \]
Dnepropetrovsk Agricultural Institute
Received
7 III 1962
CITED LITERATURE
- A. F. Timan, M. F. Timan, DAN, 71, No. 1 (1950).
- M. F. Timan, Matem. sborn., 46 (88), No. 1, 125 (1958).
- A. F. Timan, Theory of Approximation of Functions of a Real Variable, Moscow, 1960.
- M. F. Timan, Izv. Vyssh. ucheb. zaved., ser. matem., No. 6, 109 (1961).
- S. B. Stechkin, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 62 (1961).
* For continuous functions \((p=\infty)\), these results are obtained with the aid of a relation between the operators \(U_n^*\) and \(U_n\), established by V. F. Vlasov.