Absolutely convergent Fourier series and generalized Zygmund classes of functions

• Autores: Ferenc Móricz
• Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 104, Nº 1, 2009, págs. 124-131
• Idioma: inglés
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• Resumen

  • 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≤α≤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^\alpha L\left(\frac{1}{h}\right)\qquad \text{for all x∈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.