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Mat. Zametki, 1968, Volume 3, Issue 5, Pages 597–603 (Mi mz6718)  

Remarks on Fourier series

R. M. Trigub

Sumy branch of Khar'kov Polytechical Institute named after V. I. Lenin

Abstract: We prove the following propositions. An even integrable function whose Fourier coefficients form a convex sequence is absolutely continuous if and only if its Fourier series converges absolutely. If the function $f(t)$ is convex on $[0, \pi]$, $f(t)=f(\pi-t)$, then for odd $n$ $b_n=\frac2\pi\int_0^\pi f(t)\sin nt dt=\frac4\pi\frac{f(\pi/n)}n+\gamma_n$, $\sum_{n>1}|\gamma_n|<10\lceil f(\pi/2)\rceil$ while for even $n$, $b_n=0$.

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English version:
Mathematical Notes, 1968, 3:5, 380–383

Bibliographic databases:

UDC: 517.5
Received: 28.06.1967

Citation: R. M. Trigub, “Remarks on Fourier series”, Mat. Zametki, 3:5 (1968), 597–603; Math. Notes, 3:5 (1968), 380–383

Citation in format AMSBIB
\Bibitem{Tri68}
\by R.~M.~Trigub
\paper Remarks on Fourier series
\jour Mat. Zametki
\yr 1968
\vol 3
\issue 5
\pages 597--603
\mathnet{http://mi.mathnet.ru/mz6718}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=227685}
\zmath{https://zbmath.org/?q=an:0212.41702}
\transl
\jour Math. Notes
\yr 1968
\vol 3
\issue 5
\pages 380--383
\crossref{https://doi.org/10.1007/BF01150993}


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