Sharp inverse estimates for radial basis function interpolation: One-to-one correspondence between smoothness and approximation rates
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While direct statemets for kernel based approximation on domains Ω⊂ℝ^d using radial basis functions are well researched, there still seems to be a gap in comparison with corresponding inverse statements. In this paper we sharpen inverse statements and by this close a “gap of d/2" which was raised in the literature. In particular we show that for a large class of finitely smooth radial basis function kernels such as Matérn or Wendland kernels, there exists a one-to-one correspondence between the smoothness of a function and its L^2(Ω) approximation rate via kernel interpolation: If a function can be approximation with a given rate, it has a corresponding smoothness and vice versa
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