We present a method for the fast reconstruction of high-dimensional sparse algebraic polynomials in Chebyshev form and for the fast approximation of multivariate non-periodic functions from samples, when the frequency locations belonging to the non-zero or largest Chebyshev coefficients are unknown. We only assume that we have given a generally very large index set of possible frequencies, e.g. a d-dimensional full grid. We determine the frequency locations in a dimension-incremental way from samples along reconstructing rank-1 Chebyshev lattices. We demonstrate the high performance of the proposed method in numerical examples in up to 15 dimensions.
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