zbMATH — the first resource for mathematics

An extremal problem for periodic functions with small support. (English. Russian original) Zbl 1062.42002
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 $$h = 1/2, 1/3, 1/5$$.

MSC:
 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
Full Text: