SOME INEQUALITIES BETWEEN THE BEST SIMULTANEOUS APPROXIMATION AND MODULUS OF CONTINUITY IN THE WEIGHTED BERGMAN SPACE

Muqim S. Saidusainov     (University of Central Asia, 55 Qimatsho Imatshoev, Khorog, GBAO, Tajikistan)

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 \(W_{p}^{(r)}(\omega_{m},q)\).


Keywords


The best simultaneous approximation, Modulus of continuity, Upper bound, \(n\)-widths

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References


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DOI: http://dx.doi.org/10.15826/umj.2023.2.014

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