Approximation by linear combinations of translates of a single function
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We study approximation by arbitrary linear combinations of n translates of a single function of periodic functions. We construct some linear methods of this approximation for univariate functions in the class induced by the convolution with a single function, and prove upper bounds of the L^p-approximation convergence rate by these methods, when n →∞, for 1 ≤ p ≤∞. We also generalize these results to classes of multivariate functions defined the convolution with the tensor product of a single function. In the case p=2, for this class, we also prove a lower bound of the quantity characterizing best approximation of by arbitrary linear combinations of n translates of arbitrary function.
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