A Class of Extremal Functions for the Fourier Transform

Abstract

Abstract.We determine a class of real valued, integrable functions fix) and corresponding functions MA[x) such that fix) < MA[x) for all x, the Fourier transform MA[t) is zero when |/| > 1, and the value of MA[0) is minimized. Several applications of these functions to number theory and analysis are given.

Application of Tauberian Theorem to the Exponential Decay of the Tail Probability of a Random Variable

“…As mentioned previously, because the axis of convergence is not compact, it is difficult to check whether ϕ(s) satisfies the theorem assumption or not. The following extension by Graham-Vaaler [5] solves this difficulty by relaxing the limit ( 13) of e −t S(t). This theorem is called a finite form of Ikehara's theorem because λ is restricted to some range of values.…”

The Gaussian core model in high dimensions

“…[3] for spherical codes, and as in that paper we prove a corresponding optimality result under certain conditions (Proposition 6.3). Our construction of the auxiliary function for the linear programming bound is also analogous to the solution of the Beurling-Selberg extremal problem in analytic number theory [5,6,7,8,19,20,37], which arose independently in Selberg's work on sieve theory and Beurling's work on analytic functions (see [31, p. 226] for historical comments). In particular, the relevant case for our work is Gaussian subordination [6,5].…”

A survey on Beurling-Selberg majorants and some consequences of the Riemann hypothesis

“…Over the recent years considerable progress was accomplished in terms of understanding the Beurling-Selberg extremal problem (1.1), both via "hard analysis" techniques, that solve the problem for families of functions by using suitable integral representations, and via "soft analysis" techniques, in the sense that once one has the problem solved for a family of functions with a free parameter, one can integrate this parameter and produce the solution for new classes of functions. This general layout was traced by Graham and Vaaler in [16] and developed with more generality in [5,6,7,8,9].…”

“…The problem (1.1) is hard in the sense that there is no general known way to produce a solution given any f : R → R. Besides the original examples f (x) = sgn(x) of Beurling and f (x) = χ [a,b] (x) of Selberg, the solution for the exponential family f (x) = e −λ|x| , λ > 0, was discovered by Graham and Vaaler in [16], with a first glimpse of the technique of integration on the free parameter λ to produce solutions for a family of even and odd functions. Later, the problem for f (x) = x n sgn(x) and f (x) = (x + ) n , where n is a positive integer, was considered by Littmann in [23,24,25].…”

“…Other classical applications of the solutions of these problems to analytic number theory include Hilbert-type inequalities [8,16,24,29,35], Erdös-Turán discrepancy inequalities [8,21,35], optimal approximations of periodic functions by trigonometric polynomials [2,8,9,35] and Tauberian theorems [16]. The extremal problem in higher dimensions, with applications, is considered in [1,17].…”