Gaussian subordination for the Beurling-Selberg extremal problem

Abstract

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 (x 2 + α 2 )/(x 2 + β 2 ) , for 0 ≤ α < β; log x 2 +α 2 ; and x 2n log x 2 , 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.

One-sided band-limited approximations of some radial functions

“…See §4 for the statements of the theorems. These theorems generalize the so-called distribution and Gaussian subordination methods of [12]. The main idea behind these methods goes back to the paper of Graham and Vaaler [24].…”

“…This theorem is essentially a corollary of Theorem 3 of [12]. The proof simply uses the product structure and positivity of G λ (x ) in conjunction with Theorem 3 of [12].…”

“…Such problems are considered in [5,6,11,12,13,14,34,35,36,51]. Some work has also been done in the several variables setting.…”

“…It would be interesting to determine the analogue of the above theorem for minorants of G λ (x ) (i.e. the high dimensional analogue of Theorem 2 of [12]) where the extremal functions cannot be obtained by a tensor product of lower dimensional extremal functions.…”

Interpolation formulas with derivatives in de Branges spaces II

“…These are functions F of prescribed exponential type that minimize the L 1 (R, µ)-distance (µ is some non-negative measure) from a given function g and such that F lies below or above g on R. These functions have the special property that they interpolate the target function g (an its derivative) at a certain sequence of real points and have several special properties that are very useful in applications to analytic number theory, being the key to provide sharp (or improved) estimates. For instance, in connection to: large sieve inequalities [24,28], Erdös-Turán inequalities [15,28], Hilbert-type inequalities [12,14,15,22,28], Tauberian theorems [22] and bounds in the theory of the Riemann zeta-function and general L-functions [7,8,9,11,16,18,19]. Further constructions and applications can also be found in [6,10,13,21,25].…”