Construction of frames for shift-invariant spaces

TL;DR

This paper develops methods to construct frames for shift-invariant spaces in weighted L^p spaces, including Riesz bases, using functions with compactly supported Fourier transforms or compact support.

Contribution

It introduces two novel constructions of frames for shift-invariant spaces, one using functions with compactly supported Fourier transforms and the other with compact support, leading to Riesz bases.

Findings

Constructed p-frames for weighted shift-invariant spaces.

Provided methods to generate Riesz bases with compact support functions.

Ensured the generated spaces are closed and shift-invariant.

Abstract

We construct a sequence ${\phi_i(\cdot-j)\mid j\in{\ZZ}, i=1,...,r}$ which constitutes a $p$-frame for the weighted shift-invariant space [V^p_{\mu}(\Phi)=\Big{\sum\limits_{i=1}^r\sum\limits_{j\in{\mathbb{Z}}}c_i(j)\phi_i(\cdot-j) \Big| {c_i(j)}_{j\in{\mathbb{Z}}}\in\ell^p_{\mu}, i=1,...,r\Big}, p\in[1,\infty],] and generates a closed shift-invariant subspace of $L^p_\mu(\mathbb{R})$. The first construction is obtained by choosing functions $\phi_i$, $i=1,...,r$, with compactly supported Fourier transforms $\hat{\phi}_i$, $i=1,...,r$. The second construction, with compactly supported $\phi_i,i=1,...,r,$ gives the Riesz basis.