ON WIDTHS OF SOME CLASSES OF ANALYTIC FUNCTIONS IN A CIRCLE

Mirgand Sh. Shabozov     (Tajik National University, 17 Rudaky Ave., Dushanbe, 734025, Tajikistan)
Muqim S. Saidusainov     (University of Central Asia, 155 Qimatsho Imatshoev, Khorog, GBAO, Tajikistan)

Abstract


We calculate exact values of some $n$-widths of the class \(W_{q}^{(r)}(\Phi),\) \(r\in\mathbb{Z}_{+},\) in the Banach spaces \(\mathscr{L}_{q,\gamma}\) and \(B_{q,\gamma},\) \(1\leq q\leq\infty,\) with a weight \(\gamma\). These classes consist of functions \(f\) analytic in the unit circle, their \(r\)th order derivatives \(f^{(r)}\) belong to the Hardy space \(H_{q},\) \(1\leq q\leq\infty,\) and the averaged moduli of smoothness of boundary values of \(f^{(r)}\) are bounded by a given majorant \(\Phi\) at the system of points \(\{\pi/(2k)\}_{k\in\mathbb{N}}\); more precisely,
$$ \frac{k}{\pi-2}\int_{0}^{\pi/(2k)}\omega_{2}(f^{(r)},2t)_{H_{q,\rho}}dt\leq \Phi\left(\frac{\pi}{2k}\right) $$
for all \(k\in\mathbb{N}\), \(k>r.\)


Keywords


Modulus of smoothness, The best approximation, \(n\)-widths, The best linear method of approximation

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References


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DOI: http://dx.doi.org/10.15826/umj.2024.2.011

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