Stockwell-Like Frames for Sobolev Spaces

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.

28 pages, 8 figures