Gaussian Subordination for the Beurling-selberg Extremal Problem

We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function e−πλx 2 by entire functions of exponential type. The combination of the Gaussian and a general distribution approach provides the solution of the extremal problem for a wide class of even functions that includes most of the previously known examples (for instance [3], [4], [10] and [17]), plus a variety of new interesting functions such as |x|α for −1 < α; log ` (x2 + α2)/(x2 + β2) ́ , for 0 ≤ α < β; log ` x2+α2 ́ ; and x2n log x2 , for n ∈ N. Further applications to number theory include optimal approximations of theta functions by trigonometric polynomials and optimal bounds for certain Hilbert-type inequalities related to the discrete Hardy-Littlewood-Sobolev inequality in dimension one. Introduction We recall that an entire function F : C → C is of exponential type at most 2πδ if for every > 0 there exists a positive constant C, such that the inequality |F (z)| ≤ Ce )|z| holds for all z ∈ C. For a given function f : R→ R, the Beurling-Selberg extremal problem consists of finding an entire function F (z) of exponential type at most 2πδ, such that the integral ∫ ∞ −∞ |F (x)− f(x)|dx (0.1) is minimized. An important variant of this problem, useful in many applications to number theory and analysis, occurs when we impose the additional condition that F (z) is real valued on R and satisfies F (x) ≥ f(x) for all x ∈ R. In this case a function F (z) that minimizes the integral (0.1) is called an extreme majorant of f(x). Extreme minorants are defined in an analogous manner. This extremal problem was solved in unpublished work of A. Beurling in the late 1930’s for the function f(x) = sgn(x). Later A. Selberg used translates of Beurling’s extremal function to majorize and minorize the characteristic function of an interval. Selberg made use of this construction to obtain a sharp form of the large sieve inequality. Further applications in analytic number theory are discussed in [23] and [24]. An outline of the early development of this theory, including simple Date: 789, January 31, 2010. 2000 Mathematics Subject Classification. Primary 41A30, 41A52. Secondary 41A05, 41A44, 42A82.