Optimal bi-orthonormal approximation of signals

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 biorthonormal approach, a general theorem for optimal vector approximation in Hilbert spaces is suggested, based on distinction between two biorthonormal sets related to a partial basis. A sufficient and necessary condition interrelating these sets is given, and a general systematic method for deriving finite biorthonormal sets is presented. This method uses 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, the optimal approximation approach has greater accuracy and calculation efficiency, from both the theoretical and numerical viewpoints