Abstract
Full Text
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)|.
]
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}^{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}^{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
\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}^{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)\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)
\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}^{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}^{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,}|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)
+\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}^{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}^{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}^{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