Frames in two dimensions arising from wavelet transforms

Abstract

A mathematical scheme for generating functions spanning the Hilbert spacedτ ds s 2 is advanced. The completeness condition is guaranteed by the fact that the proposed functions constitute a frame for L 2 (R 2 , (dτ ds/s 2 )). Such functions are built up out of the following ingredients: (i) the wavelet transform, and (ii) a frame for L 2 (R, dt).

From cardinal spline wavelet bases to highly coherent dictionaries

“…Their equivalent in the context of signal processing were first introduced by Gabor and nowadays frequently appear in some contexts under the name of Gabor frames or Weyl-Heisenberg frames [12,13]. On the other hand, the particular class of coherent states arising from the affine group on the real line, called wavelets [6,[11][12][13][14][15], have also been broadly applied in physics [16][17][18][19], as well as in signal processing. The applications of wavelets notably increased when fast techniques, arising from the multi-resolution analysis scheme, became available.…”

On the truncation of the harmonic oscillator wavepacket

“…Such a theory, developed in 1952 by Duffin and Shaffer in the context of harmonic analysis [1] has been applied, for over fifteen years, to construct coherent states. More specifically, we should mention affine coherent states, also called wavelets, and Weyl-Heisenberg coherent states, also known as Gabor frames [2][3][4][5][6][7][8][9][10][11]. We recall in the next paragraph the general definition of frames and a few properties, which is all what we need for the purpose of the present effort.…”