Dilation-and-modulation Parseval frames in the half space

Abstract

Due to [Formula: see text] not being a group under addition, [Formula: see text] admits no traditional wavelet or Gabor frame. This paper addresses a class of dilation-and-modulation ([Formula: see text]) Parseval frames for [Formula: see text]. We present a parametric expression of all [Formula: see text]-Parseval frames (frame sequences) and [Formula: see text]-orthonormal bases (systems); and develop an approach to obtain all [Formula: see text]-Parseval frames (frame sequences) and [Formula: see text]-orthonormal bases (systems) starting with one [Formula: see text]-Parseval frame for [Formula: see text].