Sharp Estimates for the Norms of Differences in Terms of the Norms of the Derivatives and the Best Majorants of Moduli of Continuity of Periodic Functions期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Sharp Estimates for the Norms of Differences in Terms of the Norms of the Derivatives and the Best Majorants of Moduli of Continuity of Periodic Functions

Authors:

O L Vinogradov V V Zhuk

Abstract:

Let C be the space of 2 pgr

-periodic continuous functions with uniform norm, let ohgr

_r(f,h) be a modulus of continuity of order r of a function f, and let

$Hr = \{ f \in C:{\omega }_r (f,t) \leqslant t^r for all t > 0\} .$

Then

$\phi _r (h) = \mathop {sup}\limits_{f \in H_r } { \omega }_r (f,h) = \frac{{2^{r + 2} }}{\pi }\sum\limits_{l = 0}^\infty {( - 1)} \frac{{\sin ^r (l + 1/2)h}}{{(2l + 1)^{r + 1} }};$

for $h \in (0,{\pi ]}$ .An explicit formula for the sum of the series on the right-hand side is derived. Analogs of phiv

_r(h) also obtained for other spaces, in particular, for the space L. Sharp estimates for a series of convolution operators are obtained in terms of the norm of the second-order derivative of a function, in particular, sharp estimates for the norm of deviation of the Steklov function of order r are derived in terms of the norm of the second-order derivative. Bibliography: 10 titles.

Keywords:

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