Turán Extremal Problem for Periodic Functions with Small Support and Its Applications

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• 作者：Gorbachev ; D. V. ; Manoshina ; A. S.
• 关键词：Tur&aacute ; n extremal problem ; periodic function ; linear programming ; Fourier series ; Fejé ; r polynomial ; van der Corput set
• 刊名：Mathematical Notes
• 出版年：2004
• 出版时间：2004
• 年：2004
• 卷：76
• 期：5/6
• 页码：640-652
• 全文大小：173 KB

文摘

We study a Tur&aacute;n extremal problem on the largest mean value of a <math>1math>-periodic even function with nonnegative Fourier coefficients, fixed value at zero, and support on a closed interval <math>[-h,h]math>, <math>0<h\le1/2math>. We show how the solution of this extremal problem for rational numbers <math>h=p/qmath> is related to the solution of two finite-dimensional problems of linear programming. The solution of the Tur&aacute;n problem for rational numbers <math>hmath> of the form <math>2/qmath>, <math>3/qmath>, <math>4/qmath>, <math>p/(2p+ \nomathbreak 1)math> is obtained. Applications of the Tur&aacute;n problem to analytic number theory are given.