Optimal bi-orthonormal approximation of signals

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Abstract
Signal representation using nonorthogonal bases in general, and Gabor expansion specifically, is required for systems description and modeling in physics, biology, and engineering. In this paper the problem of signal approximation by partial sets of a given nonorthogonal basis is addressed, motivated by the essentially practical requirement of signal representation in infinite-dimensional spaces. Utilizing the bi-orthonormal approach, a general theorem for optimal vector approximation in Hilbert spaces is suggested, based on distinction between two bi-orthonormal sets related to a partial basis. A sufficient and necessary condition interrelating these sets is given, and a general systematic method for deriving finite bi-orthonormal sets is presented. This method employs an algebraic approach and thus obviates, in the case of function spaces, the need for solving integral equations. It is concluded that in cases of significant nonorthogonality, there is a major advantage in utilizing the optimal approximation approach regarding the resultant accuracy and calculation efficiency, from both the theoretical and numerical viewpoints.

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