Almost everywhere summability of Fourier series with indication of the set of convergence

详细

In this paper, the following problem is studied. For what multipliers {λ_k,n} do the linear means of the Fourier series of functions f ∈ L₁[−π, π],

\(\begin{array}{*{20}c}
{\sum\limits_{k = - \infty }^\infty {\lambda _{k,n} \widehat{f_k }e^{ikx} ,} } & {where \widehat{f_k } is the kth Fourier coefficient, } \\

\end{array} \)

, converge as n→∞ at all points at which the derivative of the function ∫₀^xf exists? In the case λ_k,n = (1 − |k|/(n + 1)), a criterion of the convergence of the (C, 1)-means and, in the general case λ_k,n = ϕ(k/(n + 1)), a sufficient condition of the convergence at all such points (i.e., almost everywhere) are obtained. In the general case, the answer is given in terms of whether ϕ(x) and xϕ′(x) belong to the Wiener algebra of absolutely convergent Fourier integrals. New examples are given.