On the uniqueness of trigonometric series

G. G. Gevorkyan

Sbornik. Mathematics, Tome 68 (1991) no. 2, pp. 325-338

Résumé

It is proved that \begin{equation} \frac {a_0}2+\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx)=\sum_{n=0}^\infty A_n(x) \end{equation} is the Fourier series of an integrable function $f(x)$ if and only if 1) $\lim\limits_{h\to0}S(x,h)=f(x)$ almost everywhere, and 2) $\lim\limits_{\lambda\to\infty}\inf\lambda\mu\{x\in[0,2\pi]\colon S^\ast(x)>\lambda\}=0$, where $S(x,h)=\sum\limits_{n=0}^\infty A_n(x)\biggl(\dfrac{\sin nh}{nh}\biggr)^2$ and $S^\ast(x)=\sup\limits_{h>0}|S(x,h)|$. Bibliography: 6 titles.

Bibliographie

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[3] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1976 | MR

[4] Zigmund A., Trigonometricheskie ryady, t. 1, Mir, M., 1965 | MR

[5] Gevorkyan G. G., “O edinstvennosti ryadov po sisteme Franklina”, Matem. zametki, 46:2 (1989), 51–58 | MR

[6] Aleksandrov A. B., “Ob $A$-integriruemosti granichnykh znachenii garmonicheskikh funktsii”, Matem. zametki, 30:1 (1981), 59–72 | MR | Zbl