Algebraic Frames for Commutative Hyperharmonic Analysis of Signals and Images

Abstract

Integral transforms and the signal representations associated with them are important tools in applied mathematics and signal theory. The Fourier transform and the Laplace transform are certainly the best known and most commonly used integral transforms. However, the Fourier transform is just one of many ways of signal representation and there are many other transforms of interest. In the past 20 years, other analytical methods have been proposed and applied, for example, wavelet, Walsh, Legendre, Hermite, Gabor, fractional Fourier analysis, etc. Regardless of their particular merits they are not as useful as the classical Fourier representation that is closely connected to such powerful concepts of signal theory as linear and nonlinear convolutions, classical and high–order correlations, invariance with respect to shift, ambiguity and Wigner distributions, etc. To obtain the general properties and important tools of the classical Fourier transform for an arbitrary orthogonal transform we associate to it generalized shift operators and develop the theory of abstract harmonic analysis of signals and linear and nonlinear systems that are invariant with respect to these generalized shift operators.