Analytic functions in shift-invariant spaces and analytic limits of level dependent subdivision

TL;DR

This paper characterizes all analytic subspaces within finitely generated shift-invariant spaces and describes the analytic functions produced by level-dependent subdivision schemes, confirming exponential polynomials as the exclusive analytic functions in this context.

Contribution

It provides a complete characterization of analytic subspaces in shift-invariant spaces and explicitly describes the analytic functions generated by non-stationary subdivision schemes.

Findings

Exponential polynomials are the only analytic functions generated by subdivision schemes with finitely supported masks.

Explicit descriptions of elements in analytic subspaces are provided.

Analytic subspaces of shift-invariant spaces are fully characterized.

Abstract

The structure of exponential subspaces of finitely generated shift-invariant spaces is well understood and the role of such subspaces for the approximation power of refinable function vectors and related multi-wavelets is well studied. In this paper, in the univariate setting, we characterize all analytic subspaces of finitely generated shift-invariant spaces and provide explicit descriptions of elements of such subspaces. Consequently, we depict the analytic functions generated by level dependent (non-stationary) subdivision schemes with masks of unbounded support. And we confirm the belief that the exponential polynomials are indeed the only analytic functions generated by such subdivision schemes with finitely supported masks.