Absolutely convergent Fourier series and generalized Zygmund classes of functions - Móricz, Ferenc (MATHEMATICA SCANDINAVICA, 2009)
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 $(Zyg(α, L)$ and $(Zyg (α, 1/L)$, where $0\le α\le 2$ and $L= L(x)$ is a positive, nondecreasing, slowly varying function and such that $L(x) → ∞$ as $x→ ∞$. A continuous periodic function $f$ with period $2π$ is said to belong to the class $(Zyg (α, L)$ if 26740 |f(x+h) - 2f(x) + f(x-h)| \le C h^α 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 $(Zyg (α, 1/L)$ is defined analogously. The above sufficient conditions are also necessary in case the Fourier coefficients of $f$ are all nonnegative.