Absolutely convergent Fourier series and generalized Zygmund classes of functions

Authors

We investigate the order of magnitude of the modulus of smoothness of a function

$f$ with absolutely convergent Fourier series. We give sufficient conditions in terms of the Fourier coefficients in order that

$f$ belongs to one of the generalized Zygmund classes

$(\mathrm{Zyg}(\alpha, L)$ and

$(\mathrm{Zyg} (\alpha, 1/L)$ , where

$0\le \alpha\le 2$ and

$L= L(x)$ is a positive, nondecreasing, slowly varying function and such that

$L(x) \to \infty$ as

$x\to \infty$ . A continuous periodic function

$f$ with period

$2\pi$ is said to belong to the class

$(\mathrm{Zyg} (\alpha, L)$ if 26740 |f(x+h) - 2f(x) + f(x-h)| \le C h^\alpha L\left(\frac{1}{h}\right)\qquad \text{for all

$x\in \mathsf T$ and

$h>0$ }, 26740 where the constant

$C$ does not depend on

$x$ and

$h$ ; and the class

$(\mathrm{Zyg} (\alpha, 1/L)$ is defined analogously. The above sufficient conditions are also necessary in case the Fourier coefficients of

$f$ are all nonnegative.