ACTA issues

Minimization of maxima of nonnegative and positive definite cosine polynomials with prescribed first coefficients

Szilárd Gy. Révész

Acta Sci. Math. (Szeged) 60:3-4(1995), 589-608

Abstract. Let ${{\msbm T}}={{\msbm R}}/2\pi{{\msbm Z}}$ be the circle group, ${\bf C}({{\msbm T}})$ be the set of continuous functions on ${\msbm T}$, and ${\cal T}$ be the set of the trigonometrical polynomials. Let $f\in{\cal T}$ be nonnegative, even and positive (semi)definite. In number theory and analysis itself various extremal problems are related to the determination of the least or largest possible value of $f(0)$ under various conditions on the degree, spectrum set, or the value of some prescribed coefficients. We define $$\eqalign{{\cal F}(a)=\{f\in{\bf C}({{\msbm T}}):f(x)\sim1+a\cos x+\sum_{k=2}^\infty a_k &\cos kx\geq0 (\forall x),\cr &a_k\geq0 (k\in{{\msbm N}})\}}$$ and the extremal quantity $$\alpha(a)=\inf\lbrace f(0): f\in{\cal F}(a)\rbrace.$$ The aim of the paper is to collect as much information about $\alpha(a)$ as possible. Although the motivation of studying $\alpha(a)$ stems from the application of $\alpha(a)$ to extremal problems, e.g. one posed by Landau, we do not describe these connections here. Let us only mention that the new results of the paper, in particular concerning the behavior of $\alpha$ at $a\to2$, provide essential help in those questions. However, the general approach here, and in particular the precise answer to the problem of estimation of Fourier coefficients of a nonnegative Fourier series from information regarding the first coefficient, as formulated in Theorem 2.1, seems to be of independent interest. Special emphasis is given to the description of $\alpha(a)$ at $a\to2$. This analysis leads to considerable improvements upon earlier results of French and Steckin. However, our results settle only the order of magnitude of $\alpha(a)$, and a precise asymptotic description remained an open question.

AMS Subject Classification (1991): 42A05, 42A82, 41A15, 46A55

Received October 27, 1994. (Registered under 5653/2009.)