The spectral function of shift-invariant spaces

Abstract

The shift-invariant spaces are closed subspaces of L2(Rn) that are invariant under all shifts (i.e., integer translations). The theory of shift-invariant subspaces of L2(Rn) plays an important role in many areas, most notably in the theory of wavelets, spline systems, Gabor systems, and approximation theory [BMM; BDR1; BDR2; BL; Bo1; HLPS; Ji; RS1; RS2; Rz2]. The study of analogous spaces for L2(T,H) with values in a separable Hilbert space H in terms of range function, often called doubly invariant spaces, is quite classical and goes back to Helson [He1]. The general structure of shift-invariant (SI) spaces was revealed in the work of de Boor, DeVore, and Ron [BDR1] with the use of fiberization techniques based on range function. In particular, conditions under which a finitely generated SI space has a generating set satisfying some desirable properties (e.g., stability, orthogonality or quasi-orthogonality) were given. This has been further developed in the work of Ron and Shen [RS1] with the introduction of the technique of Gramians and dual Gramians. The general properties of SI spaces and shift-preserving operators have also been studied by the first author [Bo1]. The contribution of this paper is a systematic study of yet another tool in SI spaces, apparently overlooked in the previous research, which we call the spectral function. This function was introduced by the second author in his Ph.D. thesis. It was motivated by [BDR1] and is similar to the multiplicity function studied by Baggett, Medina, and Merrill [BMM]. More precisely, to every SI subspace of L2(Rn) we associate a function on R that contains much useful information about that space. Although [BDR1] and Helson’s range function [He1] is the origin of this approach, it is thanks to Weiss (see [WW]) that the spectral function has a very elementary definition. Namely, for every SI space V ⊂ L2(Rn), there exists a countable family of functions8 whose integer shifts form a tight frame with constant 1 for the space V, and the spectral function of V is defined as the sum of the squares of the Fourier transforms of the elements of 8 (see Lemma 2.3). It can be shown that such a function is well-defined, additive on orthogonal sums, and bounded by 1. Moreover, it behaves nicely under dilations and modulations, which makes it useful in the study of wavelet and Gabor systems. For example, it