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While direct statemets for kernel based approximation on domains $\Omega\subset \mathbb{R}^d$ using radial basis functions are well researched, therestill seems to be a gap in comparison with corresponding inverse statements. Inthis 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 largeclass of finitely smooth radial basis function kernels such as Mat\'ern orWendland kernels, there exists a one-to-one correspondence between thesmoothness of a function and its $L^2(\Omega)$ approximation rate via kernelinterpolation: If a function can be approximation with a given rate, it has acorresponding smoothness and vice versa

Sharp inverse estimates for radial basis function interpolation: One-to-one correspondence between smoothness and approximation rates