Abstract
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 \(\gamma(\rho)=\rho^\alpha,\) \(\alpha>0\), 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 \(B_{2,\gamma_{1}},\) \(\gamma_{1}(\rho)=\rho^{\alpha},\) \(\alpha>0\), is found. Exact values of several \(n\)-widths are calculated for the classes of functions \(W_{p}^{(r)}(\omega_{m},q)\).
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