4-element Numerical Function Wavelet Transform And Multiscale Analysis

Let P be the affine group of the real line R, and let H be the set of all quaternions. Thus L2(R, H; dx) denotes the space of all square integrable quaternion-valued functions. Prom the viewpoint of square integrable group representations we study the theory of continuous wavelet transforms on L2(R, H;dx) associated with the group P, and give the Calderon reproducing formula. Noticing that complex matrix-valued function space L2d(R, Cnxn)can be identified with L2(R, H;dx) by an isometric operator, we study the multiresolution analysis on L2(R, H; dx) by drawing on the theory of multires-olution analysis on matrix-valued function space. Furthermore, we give the construction of scaling function and wavelets.