2013/09/20 by Chernov, Alexey, Dung, Dinh
#FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1309.5170
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 Kolmogorov N-widths and ε-dimensions of a modified Korobov class parametrized by positive a of s-variate periodic functions having mixed smoothness r, as a function of three variables N,s,a and ε, s,a, respectively, when N,s may be large, ε may be small and a may range from 0 to infinity. Based on these results we describe a complete classification of tractability for the problem of ε-dimensions of the modified Korobov class. In particular, we prove the introduced exponential tractability of this problem for a>1. All of these methods and results are also extended to high-dimensional approximations of non-periodic functions by Jacobi polynomials with powers in hyperbolic crosses.