Extremal problems in de Branges spaces: the case of truncated and odd functions

Abstract

In this paper we find extremal one-sided approximations of exponential type for a class of truncated and odd functions with a certain exponential subordination. These approximations optimize the L 1 (R, |E(x)| −2 dx)-error, where E is an arbitrary Hermite-Biehler entire function of bounded type in the upper half-plane. This extends the work of Holt and Vaaler (Duke Math J 83:203-247, 1996) for the signum function. We also provide periodic analogues of these results, finding optimal one-sided approximations by trigonometric polynomials of a given degree to a class of periodic functions with exponential subordination. These extremal trigonometric polynomials optimize the L 1 (R/Z, dϑ)-error, where ϑ is an arbitrary nontrivial measure on R/Z. The periodic results extend the work of Li and Vaaler (Indiana Univ Math J 48(1):183-236, 1999), who considered this problem for the sawtooth function with respect to Jacobi measures. Our techniques are based on the theory of reproducing kernel Hilbert spaces (of entire functions and of polynomials) and on the construction of suitable interpolations with nodes at the zeros of Laguerre-Pólya functions.

Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function

“…As in the Paley-Wiener case, there are three distinct qualitative regimes for the solution, and these depend on the roots of A and B (observe that if E(z) = e −iπz , then A(z) = cos πz and B(z) = sin πz, which have roots exactly at β ∈ 1 2 N). Similar extremal problems in de Branges and Euclidean spaces were considered in [6,8,27,30,33].…”

Extremal functions in de Branges and Euclidean spaces, II

“…We expect that the extension of this framework to a multidimensional setting presented here might reveal other interesting applications. 8 2. Entire functions with prescribed nodes 2.1.…”

Interpolation formulas with derivatives in de Branges spaces II

“…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]. …”