Construction of group transforms subject to several performance criteria

A transform, L, is defined in this paper and studied subject to the two opposing performance criteria, computational effectiveness and accuracy of approximation of a given unitary transform measured in distance related to the Hilbert-Sehmidt norm. The matrices which represent the L-transform constitute a special class of group matrices, which were introduced by Frobenius in 1895. These (n × n)-matrices depend upon certain free parameters whose choice determines their behavior with respect to the two criteria. Hence, part of these parameters can be chosen to affect the computational performance of L whereas the other part is used to minimize its distance from a given unitary transform. The computational complexity of L is eased by making L sparse rather than a product of sparse matrices as in the FFT algorithm. It is shown that if only one of these two criteria is used then either all the parameters can be used to reduce the computational complexity of L to that of the fast Haar transform (in nontrivial cases) or else all of these parameters can be used to achieve (asymptotically) distance zero from a given unitary transform. If both criteria are used, then the more parameters that are used for one of them, the less parameters we have to deal with for the other one and vice versa-an inverse proportional dependence. Various examples are given.