Stockwell-Like Frames for Sobolev Spaces

TL;DR

This paper introduces a new family of frames tailored for Sobolev spaces, inspired by the Stockwell transform, providing a hybrid of Gabor and Wavelet frames with explicit constructions.

Contribution

It constructs explicit, computable frames that characterize Sobolev norms and seminorms, bridging Gabor and Wavelet frame approaches.

Findings

Frames accurately describe Sobolev norms

Explicit examples of frames are provided

Frames are computationally feasible

Abstract

We construct a family of frames describing Sobolev norm and Sobolev seminorm of the space $H^s(\mathbb{R}^d)$. Our work is inspired by the Discrete Orthonormal Stockwell Transform introduced by R.G. Stockwell, which provides a time-frequency localized version of Fourier basis of $L^2([0,1])$. This approach is a hybrid between Gabor and Wavelet frames. We construct explicit and computable examples of these frames, discussing their properties.