This paper is about generalized Fourier descriptors, and their application to the research of invariants under group actions. A general methodology is developed, crucially related to Pontryagin's, Tannaka's, Chu's and Tatsuuma's dualities, from abstract harmonic analysis. Application to motion groups provides a general methodology for pattern recognition. This methodology generalizes the classical basic method of Fourier-invariants of contours of objects. In the paper, we use the results of this theory, inside a Support-Vector-Machine context, for 3D objects-recognition. As usual in practice, we classify 3D objects starting from 2D information. However our method is rather general and could be applied directly to 3D data, in other contexts. Our applications and comparisons with other methods are about human-face recognition, but also we provide tests and comparisons based upon standard data-bases such as the COIL data-base. Our methodology looks extremely efficient, and effective computations are rather simple and low cost. The paper is divided in two parts: first, the part relative to applications and computations, in a SVM environment. The second part is devoted to the development of the general theory of generalized Fourier-descriptors, with several new results, about their completeness in particular. These results lead to simple formulas for motion-invariants of images, that are "complete" in a certain sense, and that are used in the first part of the paper. The computation of these invariants requires only standard FFT estimations, and one dimensional integration.