SOME INEQUALITIES BETWEEN THE BEST SIMULTANEOUS APPROXIMATION AND MODULUS OF CONTINUITY IN THE WEIGHTED BERGMAN SPACE - Muqim S. Saidusainov (Ural Mathematical Journal)
Some inequalities between the best simultaneous approximation of functions and their intermediate derivatives, and the modulus of continuity in a weighted Bergman space are obtained. When the weight function is \(γ(ρ)=ρ^α,\) \(α>0\), some sharp inequalities between the best simultaneous approximation and an \(m\)th order modulus of continuity averaged with the given weight are proved. For a specific class of functions, the upper bound of the best simultaneous approximation in the space \(B_{2,γ_{1}},\) \(γ_{1}(ρ)=ρ^{α},\) \(α>0\), is found. Exact values of several \(n\)-widths are calculated for the classes of functions \(W_{p}^{(r)}(ω_{m},q)\).