Stockwell-like frames for Sobolev spaces

We construct a family of frames describing the norm and seminorm of the space $H^s(\R^d)$. We also characterise Besov spaces modeled on $L^2(\R^d)$. Our work is inspired by the Discrete Orthonormal Stockwell Transform introduced by R.G. Stockwell, which provides a time-frequency localised version of the 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 and comparing them with the existing literature.