Almost everywhere summability of Fourier series with indication of the set of convergence
- Авторлар: Trigub R.1
-
Мекемелер:
- Sumy State University
- Шығарылым: Том 100, № 1-2 (2016)
- Беттер: 139-153
- Бөлім: Article
- URL: https://journals.rcsi.science/0001-4346/article/view/149586
- DOI: https://doi.org/10.1134/S0001434616070130
- ID: 149586
Дәйексөз келтіру
Аннотация
In this paper, the following problem is studied. For what multipliers {λk,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 ∫0xf exists? In the case λk,n = (1 − |k|/(n + 1)), a criterion of the convergence of the (C, 1)-means and, in the general case λk,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.{\sum\limits_{k = - \infty }^\infty {\lambda _{k,n} \widehat{f_k }e^{ikx} ,} } & {where \widehat{f_k } is the kth Fourier coefficient, } \\
\end{array} \)
Авторлар туралы
R. Trigub
Sumy State University
Хат алмасуға жауапты Автор.
Email: roald.trigub@gmail.com
Украина, Sumy