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This paper presents a fundamentally new algebraic approach to the analysis and synthesis of discrete orthogonal basis functions. It provides the theoretical background to unify Fourier, Gabor and discrete orthogonal polynomial moments. For the first time, a set of objective tests are proposed to measure the quality of basis functions. It consists of two main sections: the theoretical background on the generation and orthogonalization of basis functions together with a new solution for the computation of spectra from incomplete data, as well as the implementation of interpolation for all orthogonal basis functions; a new approach to discrete orthogonal polynomials, proving that there is one and only one unitary discrete polynomial basis. Furthermore, the concept of anisotropic moments is introduced and applied to 2D seismic data, which is an image processing problem. The new polynomial basis is numerically better conditioned than the discrete cosine transform. This opens the door to new image compression algorithms, offering a higher compression ratio than the well known JPEG method, for the same numerical effort.