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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)

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.)
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