On everywhere divergence of trigonometric Fourier series
Abstract
The following theorem is established.Theorem. Let a function \varphi\colon \lbrack 0,+\infty)\to[0,+\infty) and a sequence \{\psi(m)\} satisfy the following condition: the function \varphi(u)/u is non-decreasing on (0,+\infty), \psi(m)\geqslant 1 (m=1,2,\dots) and \varphi(m)\psi(m)=o(m\sqrt{\ln m}/\sqrt{\ln\ln m}\,) as m\to\infty. Then there is a function f\in L \lbrack -\pi,\pi \rbrack such that \displaystyle \int _{-\pi}^\pi\varphi(\vert f(x)\vert)\,dx<\infty and \limsup_{m\to\infty}S_m(f,x)/\psi(m)=\infty for all x\in \lbrack -\pi,\pi \rbrack here S_m(f) is the m-th partial sum of the trigonometric Fourier series of f.
- Publication:
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Sbornik: Mathematics
- Pub Date:
- February 2000
- DOI:
- 10.1070/SM2000v191n01ABEH000449
- Bibcode:
- 2000SbMat.191...97K