Frames for weighted shift-invariant spaces

TL;DR

This paper establishes the equivalence between the frame property and closedness in weighted shift-invariant spaces, and constructs specific sequences and spaces with advantageous sampling and stability properties.

Contribution

It proves the equivalence of frame and closedness properties and introduces sequences and spaces with useful sampling and stability features.

Findings

Frame property is equivalent to closedness in weighted shift-invariant spaces.

Constructed sequences $\

V^p_ots

Abstract

We prove the equivalence of the frame property and the closedness for a weighted shift-invariant space. We also construct a sequence $\Phi^{2k+1}$ and the sequence of spaces $V^p_\mu(\Phi^{2k+1})$, $k\in{\mathbb{N}}$, on $\mathbb{R},$ with the useful properties in sampling, approximations and stability.