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Some trigonometric extremal problems and duality. (English) Zbl 0776.42002

Summary: In this paper we present a minimax theorem of infinite dimension. The result contains several earlier duality results for various trigonometrical extremal problems including a problem of Fejér. Also the present duality theorem plays a crucial role in the determination of the exact number of zeros of certain Beurling zeta functions, and hence leads to a considerable generalization of the classical Beurling’s Prime Number Theorem. The proof uses functional analysis.

MSC:

42A05 Trigonometric polynomials, inequalities, extremal problems
49J35 Existence of solutions for minimax problems
46B25 Classical Banach spaces in the general theory
11M41 Other Dirichlet series and zeta functions
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