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On an extremal problem for periodic functions with small support. (English. Russian original) Zbl 1205.42001
Math. Notes 80, No. 6, 881-884 (2006); translation from Mat. Zametki 80, No. 6, 940-942 (2006).
From the text: Suppose that $$0<h\leq 1/2$$ and $$K(h)$$ is the class of continuous periodic even real functions
$f(x)= \sum_{n=0}^\infty a_n\cos(2\pi nx)$
for which $$\sum_{n=0}^\infty |a_n|=1$$ and $$f(x)\equiv 0$$ for $$h\leq |x|\leq 1/2$$. We consider the extremal problem
$B(h)= \sup_{f\in K(h)} a_0= \sup_{f\in K(h)} \int_{-h}^h f(x)\,dx.$
Here we consider the case $$h=1/6$$.
##### 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.)
Full Text:
##### References:
 [1] S. Konyagin and I. Shparlinski, Character Sums with Exponential Functions and their Applications, Cambridge Univ. Press, Cambridge, 1999. · Zbl 0933.11001 [2] N. N. Andreev, S. V. Konyagin, and A. Yu. Popov Mat. Zametki [Math. Notes], 60 (1996), no. 3, 323–332. [3] D. V. Gorbachev, Selected Problems in the Theory of Functions and Approximation theory and Their Applications [in Russian], Izd. TulGU, Tula, 2004. [4] H. Bateman and A. Erdélyi, Higher Transcendental Functions, vol. 1, McGraw-Hill, New York-Toronto-London, 1953; Russian transl.: Nauka, Moscow, 1965. · Zbl 0143.29202
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