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 { λ m , n ∗ } m , n ∈ N , λ m , n ∗ = λ ( m + n ) and { λ m , n } m , n ∈ N , λ m , n = λ ( max { m , n } ) , 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 Λ U 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.