On best approximation of classes by radial functions

We investigate the radial manifolds Rn, generated by a linear combination of n radial functions on Rd. We consider the best approximation of function classes by the manifold Rn. In particular, we prove that the deviation of the manifold Rn from the Sobolev class W2r,d in the Hilbert space L2 behaves asymptotically as n-r/d-1. We show the connection between the manifold Rn and the space of algebraic polynomials Pd,s of degree s. Namely, we prove there exist constants c1 and c2 such that the space Pd,s is either contained or not in Rn as n ≥ c1sd-1 or n ≤ c2sd-1, respectively.

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