Estimates for $n$-widths of sets of smooth functions on complex spheres

Abstract

In this work we investigate n-widths of multiplier operators Λ * and Λ, defined for functions on the complex sphere Ω d of C d , associated with sequences of multipliers of the type, respectively, for a bounded function λ defined on [0, ∞). If the operators Λ * and Λ are bounded from L p (Ω d ) into L q (Ω d ), 1 ≤ p, q ≤ ∞, and U p is the closed unit ball of L p (Ω d ), we study lower and upper estimates for the n-widths of Kolmogorov, linear, of Gelfand and of Bernstein, of the sets Λ * U p and ΛU p in L q (Ω d ). As application we obtain, in particular, estimates for the Kolmogorov n-width of classes of Sobolev, of finitely differentiable, infinitely differentiable and analytic functions on the complex sphere, in L q (Ω d ), which are order sharp in various important situations.