Extremal functions in de Branges and Euclidean spaces

Abstract

Abstract. In this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on R N . These extremal functions minimize the L 1 (R N , |x| 2ν+2−N dx)-distance to the original function, where ν > −1 is a free parameter. To achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function F λ (x) = e −λ|x| , where λ > 0, in the general framework of de Branges spaces of entire functions. We then specialize the construction to a particular family of homogeneous de Branges spaces to approach the multidimensional Euclidean case. Finally, we extend the result from the exponential function to a class of subordinated radial functions via integration on the parameter λ > 0 against suitable measures. Applications of the results presented here include multidimensional versions of Hilbert-type inequalities, extremal one-sided approximations by trigonometric polynomials for a class of even periodic functions and extremal one-sided approximations by polynomials for a class of functions on the sphere S N−1 with an axis of symmetry.

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

“…For general even measures ϑ, the case of even periodic functions with exponential subordination was considered in [1,8]. Li and Vaaler [26] solved this extremal problem for the sawtooth function…”

“…They solved this problem (in fact, for a more general class of measures) by establishing an interesting connection with the theory of de Branges spaces of entire functions [13]. This idea was further developed in [8] for a class of even functions f with exponential subordination and in [5,29] for characteristic functions of intervals, both with respect to general de Branges measures. In particular, the optimal construction in [5] was used to improve the existing bounds for the pair correlation of zeros of the Riemann zeta-function, under the Riemann hypothesis, extending a classical result of Gallagher [15].…”

“…The purpose of this paper is to complete the framework initiated in [8], where the case of even functions was treated. Here we develop an analogous extremal theory for a wide class of truncated and odd functions with exponential subordination, with respect to general de Branges measures (these are described below).…”

Interpolation formulas with derivatives in de Branges spaces II

“…Also, the assumption that ϕ ′ (t) ≥ δ for all t ∈ T B together with formula (2.6) implies that 12) where the first inequality is Hölder's inequality, the second one due to (6.9) and the separation of T B , the third one due to (6.11) and the closure under differentiation of H p (E ν ).…”

“…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].…”