V.P. Zastavnyi. On extremal trigonometric polynomials ... P. 70-91

Let $\mathscr{F}_n$ be the set of all trigonometric polynomials of order $\le n$, $n\in\mathbb{N}$. For multipliers $H:\mathscr{F}_n\to\mathscr{F}_n$, we prove an interpolation formula $H(f)(t)=\sum_{k=0}^{2n-1}\Lambda_k f\left(t-\tau+{k\pi}/{n}\right),$ which is used to obtain the following inequalities and criteria for an extremal polynomial in them (Theorem 4):
$$\int_{\mathbb{T}}J\left(|H(f)(t)|\right)\,dt \le \int_{\mathbb{T}}J\left(\varkappa|f(t)|\right)\,dt\,;\;\; \| H(f)\|_p\leqslant \varkappa\|f\|_p,\,1\le p\le\infty,\;\varkappa=|\Lambda_0|+\ldots+|\Lambda_{2n-1}|>0. $$ Here the function $J$ is convex and nondecreasing on $[0,+\infty)$. The main goal of this work is to describe all extremal polynomials in the above inequalities. Theorem 5 proves that  if the function $J$ is convex and strictly increasing on $[0,+\infty)$ and two conditions are satisfied: $(1)$ $\exists s\in\mathbb{Z}:\,\overline{\Lambda_{s}} \Lambda_{s+1}<0$ and $(2)$ $\exists \varepsilon\in\mathbb{C}$, $|\varepsilon|=1:$ $\varepsilon \Lambda_k (-1)^k\ge0$, $k\in\mathbb{Z}$, then only polynomials of the form $f(t)=\mu e^{int}+\nu e^{-int}$, $\mu,\nu\in\mathbb{C}$ are extremal in these inequalities. The main cases in this theorem are the cases $p=\infty$ and $p=1$. Theorem 6 proves that if the function $J$ is convex and strictly increasing on $[0,+\infty)$ and the operator $H$ satisfies the Szegö condition (the nonnegativity of a special trigonometric polynomial), then, in all cases different from one exceptional case, only polynomials of the form $f(t)=\mu e^{int}+\nu e^{-int}$, $\mu,\nu\in\mathbb{C}$, are extremal in these inequalities. In the exceptional case, there are other extremal polynomials. In this paper we give general examples of operators $H$ that satisfy the conditions of Theorem 6 (Example 1, Theorems 7 and 8). In particular, S.T. Zavalishchin's operator (Example 2) and the fractional derivative operator $H(f)(t)=f^{(r,\beta)}(t)$, $\beta\in\mathbb{R}$, $r\ge1$, $\varkappa=n^r$ (Corollary 3), satisfy these conditions. In this paper we also describe extremal polynomials in the Trigub and Boas inequalities (for some values of the parameters, not only polynomials of the form $\mu e^{int}+\nu e^{-int}$ are extremal).

Keywords: extremal trigonometric polynomial, Bernstein condition, Szegö condition, Weil-Nagy derivative, Bernstein-Szegö inequality, positive definite function, Boas-Civin method

Received June 28, 2023

Revised August 9, 2023

Accepted September 11, 2023

Funding Agency: The research was carried out under a state assignment (code FRRE-2023-0015 in the Unified State Information System for Recording Research, Development, and Technological Work for Civil Purposes).

Viktor Petrovych Zastavnyi, Dr. Phys.-Math. Sci., Donetsk State University, Donetsk, 283001 Russia, e-mail: zastavn@rambler.ru

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Cite this article as: V.P. Zastavnyi. On extremal trigonometric polynomials. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 4, pp. 70–91.