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This volume containing the Proceedings of the NATO Advanced Study Institute embodies as such survey articles and a wealth of recent results from the theory of splines, spline-wavelets, multivariate wavelet decomposition, interpolation theory, polynomial approximation, near-minimax approximations, rational interpolants, and Padé approximants in one and several variables, radial basis approximation, optimization theory, orthogonal polynomials, and more. The volume will therefore be of interest to researchers and graduate students in mathematics and engineering for whom the latest developments in approximation theory are of interest.