Wavelets on Irregular Grids with Arbitrary Dilation Matrices, and Frames Atoms for L^2(R^d)

Abstract

In this article, we develop a general method for constructing wavelets {|det A_j|^{1/2} g(A_jx-x_{j,k}): j in J, k in K}, on irregular lattices of the form X={x_{j,k} in R^d: j in J, k in K}, and with an arbitrary countable family of invertible dxd matrices {A_j in GL_d(R): j in J} that do not necessarily have a group structure. This wavelet construction is a particular case of general atomic frame decompositions of L^2(R^d) developed in this article, that allow other time frequency decompositions such as non-harmonic Gabor frames with non-uniform covering of the Euclidean space R^d. Possible applications include image and video compression, speech coding, image and digital data transmission, image analysis, estimations and detection, and seismology.