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

We are aiming at sharp and explicit-in-dimension estimations of the cardinality of ss-dimensional hyperbolic crosses where ss 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 εε-dimensions–the inverses of the well known Kolmogorov NN-widths–in the space L2(Ts)L2(Ts) of modified Korobov classes Ur,a(Ts)Ur,a(Ts) on the ss-torus Ts:=[−π,π]sTs:=[−π,π]s. The functions in this class have mixed smoothness of order rr and depend on an additional parameter aa 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)Kr,a(Ts) as a subset in L2(Ts)L2(Ts). We give also a classification of tractability for the problem of εε-dimensions of Ur,a(Ts)Ur,a(Ts). This theory is extended to high-dimensional approximations of non-periodic functions in the weighted space L2([−1,1]s,w)L2([−1,1]s,w) with the tensor product Jacobi weight ww by tensor products of Jacobi polynomials with powers in hyperbolic crosses.