In this paper, using an equivalent characterization of the Besov space by its wavelet coefcients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bs+t q0(Lp0(Ω)) → Bs q1(Lp1(Ω)), t > max{d(1/p0 1/p1), 0}, 1 ≤ p0, p1, q0, q1≤∞, where Bs+t q0(Lp0(Ω)) is a Besov space defned on the bounded Lipschitz domain Ω Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.
Let β > 0 and Sβ := {z ∈ C : |Imz| < β} be a strip in the complex plane. For an integer r ≥ 0, let H∞r,β denote those real-valued functions f on R, which are analytic in Sβ and satisfy the restriction |f(r)(z)| ≤ 1, z ∈ Sβ. For σ > 0, denote by Bσ the class of functions f which have spectra in (2πσ,2πσ). And let Bσ⊥ be the class of functions f which have no spectrum in (2πσ,2πσ). We prove an inequality of Bohr type f ∞≤√πλΛσr∞ k=0 (1)k(r+1) (2k + 1)r sinh((2k + 1)2σβ) , f ∈ H∞r,β∩ Bσ⊥ , where λ∈ (0,1), Λ and Λ are the complete elliptic integrals of the first kind for the moduli λ and λ = √1 λ2, respectively, and λ satisfies 4ΛβπΛ = σ1. The constant in the above inequality is exact.
