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

2015/06/20 by R. M. Trigub, Trigub, R. M.
Mathematics · #42A45 #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Mathematical Analysis and Transform Methods #Primary 42A24 #Secondary 42A38

paper · pdf · doi:10.48550/arxiv.1506.06243

openalex publication_date 2015/06/20 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

The following problem is studied in this paper: Which multipliers \λ_{k, n}\ ensure the convergence, as n→ ∞, of the linear means of the Fourier series of functions f∈ L₁[-π, π] ∑_k=-∞^∞ λ_{k, n}f_k e^ikx, where \widehatf_k is the k-th Fourier coefficient, at a point at which the derivative of the function ∫₀^x f exists. A criterion for the convergence of the (C, 1)-means (λ_{k, n}=(1-\frac |k|n+1)₊) is found, while in the general case λ_{k, n}=ϕ(\frac kn+1) a sufficient condition is derived for the convergence at all such points (that is, almost everywhere). The answer is given in terms of the belonging of ϕ(x) and xϕ'(x) to the Wiener algebra of absolutely convergent Fourier integrals. The obtained results are supplemented by some examples.

Citations

• Recherches sur la convergence des s�ries de fourier

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