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 0α2 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)| C h^α L(1h)for all xT 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.