Efficient multivariate approximation on the cube
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For the approximation of multivariate non-periodic functions h on the high-dimensional cube [-1/2,1/2]^d we combine a periodization strategy for weighted L_2-integrands with efficient approximation methods. We prove sufficient conditions on d-variate torus-to-cube transformations ψ:[-1/2,1/2]^d→[-1/2,1/2]^d and on the non-negative weight function ω such that the composition of a possibly non-periodic function with a transformation ψ yields a smooth function in the Sobolev space H_mix^m(T^d). In this framework we adapt certain L_∞(T^d)- and L_2(T^d)-approximation error estimates for single rank-1 lattice approximation methods as well as algorithms for the evaluation and reconstruction of multivariate trigonometric polynomials on the torus to the non-periodic setting. Various numerical tests in up to dimension d=5 confirm the obtained theoretical results for the transformed approximation methods.
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