V.I. Ivanov. Pointwise Tur$\acute{\mathrm{a}}$n problem for periodic positive definite functions ... P. 156-175

Full text (in Russian)

We study the pointwise Tur$\acute{\mathrm{a}}$n problem on the largest value at an arbitrary point $x$ of a $1$-periodic positive definite function supported on the interval $[-h, h]$ and equal to $1$ at zero. For rational values of $x$ and $h$, the problem reduces to a discrete version of the Fej$\acute{\mathrm{e}}$r problem on the largest value of the $\nu$th coefficient of an even trigonometric polynomial of order $p-1$ that has zero coefficient 1 and is nonnegative on a uniform grid $k/q$, $k=0,\dots,q-1$. The discrete Fej$\acute{\mathrm{e}}$r problem is solved for a number of values of the parameters $\nu$, $p$, and $q$. In all the cases, we construct extremal polynomials and quadrature formulas, which yield an estimate for the largest coefficient.

Keywords: Fourier transform and series, periodic positive definite function, pointwise Tur$\acute{\mathrm{a}}$n problem, quadrature formula, extremal polynomial

Received August 29, 2018

Revised November 09, 2018

Accepted November 12, 2018

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 16-01-00308).

Valerii Ivanovich Ivanov, Dr. Phys.-Math. Sci., Prof., Tula State University, 300012 Tula,
e-mail: ivaleryi@mail.ru

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