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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.)
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