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.

Exponential decay of measures and Tauberian theorems

“…If the abscissa of convergence σ 0 of ϕ satsfies σ 0 P p´8, 0s and ϕ has a pole at σ 0 , then it follows from Corollary 1.4 that lim xÑ8 1 x log PpX ą xq " σ 0 . This is the main result in [Nak05] , where complex methods were used (as in the proof of Graham-Vaaler's theorem with minorant and majorant functions, see [GV81]). Unlike approach in these articles, we use methods from real analysis.…”

Entire Approximations to the Truncated Powers

“…Further examples of one-sided approximations with this property can be found in an article of S. W. Graham and Vaaler [5]. For a general connection between interpolants and best approximations see, e.g., Pinkus [16] or Timan [23].…”

“…These problems include the large sieve inequality (extremal majorants are used in a proof of Selberg [20], see also the survey [24] by Vaaler), a multi-dimensional version of the large sieve (Holt and Vaaler [7]), a quantitative version of the Wiener-Ikehara Tauberian Theorem (Graham and Vaaler [5]), and proofs for Hilbert-type inequalities ( [14], Selberg [20], Vaaler [24]). …”

The Wiener–Ikehara theorem by complex analysis

“…[10] or [11]. See also the sharp "finite form" of the Wiener-Ikehara theorem in [5]. Can one modify the complex method to do as well?…”