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arXiv · 1709.00705

On moduli of smoothness with Jacobi weights

Kirill A. Kopotun·Dany Leviatan·Igor A. Shevchuk
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Abstract

The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights $(1-x)^α(1+x)^β$ for functions in the Jacobi weighted $L_p[-1,1]$, $0<p\le \infty$, spaces. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted $L_p$ spaces. If $1\le p\le\infty$, then these moduli are equivalent to certain weighted $K$-functionals (and so they are equivalent to certain weighted Ditzian-Totik moduli of smoothness for these $p$), while for $0<p<1$ they are equivalent to certain "Realization functionals".

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Kirill A. Kopotun, Dany Leviatan, Igor A. Shevchuk. 2018-11-09. On moduli of smoothness with Jacobi weights. https://arxiv.org/abs/1709.00705

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