The N-Widths of Hardy-Sobolev Spaces of Several Complex Variables

Abstract

Let B denote the unit ball in C n with boundary S and let σ( v) be the standard normalized measure on S( B). For fixed 1 ≤ p ≤ ∞, R≥ 1 let BH p ( B R ) ( BA p ( B R )) denote the unit ball of the Hardy space H p (resp. the Bergman space A p ) in B R ≔ RB and for l ∈ N let H R ( l, p, n) (resp. A R ( l, p, n)) denote the class of those functions which have the lth radial derivative belonging to BH p ( B R ) ( BA p ( B R )) for l = 0, let H R (0, p, n) ≔ BH p ( B R ) ( A R (0, p, n) ≔ BA p ( B R )). The values of Kolmogorov, Gel′fand, and Bernstein and linear N-widths of classes H R ( l, p, n) and A R ( l, p, n) in the metrics L p (σ) and L p ( v) (except for A R ( l, p, n) in L p (σ)) are found. For all 1≤ p, q ≤ ∞, R > 1 the asymptotic estimates of N-widths for classes H R ( l, p, n) and A R ( l, p, n) in the spaces L q (σ) and L q ( v) are also obtained.