An extremal problem for periodic functions with small support. (English. Russian original) Zbl 1062.42002

Gorbachev, D. V.

Math. Notes 73, No. 5, 724-729 (2003); translation from Mat. Zametki 73, No. 5, 773-778 (2003).

N. N. Andreev, S. V. Konyagin and A. Yu. Popov [Math. Notes 60, No. 3, 241-247 (1996); translation from Mat. Zametki 60, No. 3, 323–332 (1996; Zbl 0911.42001)] raised the following extremum problem: Suppose that \(0 < h \leq 1/2\) and \(K(h)\) is the class of 1-periodic continuous even real functions \(f\) satisfying the following conditions:
(i) \(f(x) = \sum^\infty_{n=0}a_n\cos(2\pi nx)\),
(ii) \(\sum^\infty_{n=0} | a_n|=1\),
(iii) \(f(x)=0\) for \(h\leq | x| \leq 1/2\).
It is required to evaluate the quantity \[ B(h) := \sup_{f\in K(h)} a_0=\sup_{f\in K(h)}\int^h_{-h} f(x)dx. \] In the paper under review, \(B(h)\) is evaluated for \(h = 1/2, 1/3, 1/5\).

Reviewer: Ferenc Móricz (Szeged)

MSC:

42A16

Fourier coefficients, Fourier series of functions with special properties, special Fourier series

Keywords:

extremum problem; periodic function; Fourier series