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Extremal problems for functions with small support. (English. Russian original) Zbl 0911.42001
Math. Notes 60, No. 3, 241-247 (1996); translation from Mat. Zametki 60, No. 3, 323-332 (1996).
Given $$0< h\leq 1/2$$, define the class $$K_2(h)$$ of functions by the following properties:
(i) $$f(x)= \sum^\infty_{n=0} a_n\cos(2\pi nx)$$,
(ii) $$\sum| a_n|= 1$$,
(iii) $$f(x)= 0$$ for $$h\leq| x|\leq 1/2$$.
The problem the authors attack is to find an estimate of $$a_0= \int^h_{-h} f(x)dx$$. Setting $A_2(h):= \sup_{f\in K_2(h)} \int^h_{-h} f(x)dx,$ the following theorems are proved:
(1) $$\lim_{h\to 0} A_2(h)/n= L$$ exists,
(2) $$A_2(1/4)= 2/(\pi+ 4)$$,
(3) $$L> 1.16$$,
(4) $$L\leq{2\over \pi} \int^\pi_0 {\sin t\over t} dt= 1.179\dots$$ .
Reviewer: F.Móricz (Szeged)

##### MSC:
 42A05 Trigonometric polynomials, inequalities, extremal problems 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series 42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
##### Keywords:
trigonometric series; extremal problems
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##### References:
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