A problem on completeness of exponentials

Abstract

Let µ be a finite positive measure on the real line. For a > 0, denote by Ea the family of exponential functionsThe exponential type of µ is the infimum of all numbers a such that the finite linear combinations of the exponentials from Ea are dense in L 2 (µ). If the set of such a is empty, the exponential type of µ is defined as infinity. The well-known type problem asks to find the exponential type of µ in terms of µ. In this note we present a solution to the type problem and discuss its relations with known results.

Two-spectra theorem with uncertainty

“…Denote by η the counting measure of Λ. It follows from the Type Theorem of [40,41], and in fact from much earlier results on the type problem, that T η = 1. Moreover, as follows for instance from the results of [35], L 2 (η) = P W 1 (as sets).…”

Spectrality of the planar Sierpinski family

“…For example, in 1998 Jorgensen and Pedersen [16] found the first singular, non-atomic, self-similar spectral measure supported on 1 4 -Cantor set. Nowadays, there is a large literature on this topic [1][2][3][4][5][6][7][8]10,[13][14][15][16]19,20,26,28,29,31]. Most of this literature deals with the issue in one dimension.…”

“…It relates analysis, geometry and topology (see, e.g. [7,14,16,[26][27][28]30] and references therein), in which the good functions are complex exponentials. The best approximation appears when L 2 (μ) has a basis consisting of complex exponentials (Fourier basis).…”

Heisenberg uniqueness pairs for some algebraic curves and surfaces

“…Heisenberg uniqueness pair has a close relation with the long-standing problem of determining the exponential types for finite measure which is eventually about exploring the density of the set {e iλt : λ ∈ Λ} in L 2 (µ). For more details, we refer to Poltoratski [15,16]. In particular, the question of HUP can be thought as a dual problem of gap problem (see [15]).…”