New explicit-in-dimension estimates for the cardinality of high-dimensional hyperbolic crosses and approximation of functions having mixed smoothness

Abstract

We are aiming at sharp and explicit-in-dimension estimations of the cardinality of s-dimensional hyperbolic crosses where s may be large, and applications in high-dimensional approximations of functions having mixed smoothness. In particular, we provide new tight and explicit-in-dimension upper and lower bounds for the cardinality of hyperbolic crosses. We apply them to obtain explicit upper and lower bounds for ε-dimensions–the inverses of the well known Kolmogorov N-widths–in the space L2(Ts) of modified Korobov classes Ur,a(Ts) on the s-torus Ts:=[−π,π]s. The functions in this class have mixed smoothness of order r and depend on an additional parameter a which is responsible for the shape of the hyperbolic cross and controls the bound of the smoothness component of the unit ball of Kr,a(Ts) as a subset in L2(Ts). We give also a classification of tractability for the problem of ε-dimensions of Ur,a(Ts). This theory is extended to high-dimensional approximations of non-periodic functions in the weighted space L2([−1,1]s,w) with the tensor product Jacobi weight w by tensor products of Jacobi polynomials with powers in hyperbolic crosses.