Wavelets on Irregular Grids with Arbitrary Dilation Matrices, and Frame Atoms for L2(r)

In this article, we develop a general method for constructing wavelets {| detAj |ψ(Ajx − xj,k) : j ∈ J, k ∈ K} on irregular lattices of the form X = {xj,k ∈ R : j ∈ J, k ∈ K}, and with an arbitrary countable family of invertible d × d matrices {Aj ∈ GLd(R) : j ∈ J} that do not necessarily have a group structure. This wavelet construction is a particular case of general atomic frame decompositions of L2(R) 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. Possible applications include image and video compression, speech coding, image and digital data transmission, image analysis, estimations and detection, and seismology.