On the Moduli of Smoothness with Jacobi Weights

K. A. Kopotun; D. Leviatan; I. A. Shevchuk

doi:10.1007/s11253-018-1509-9

On the Moduli of Smoothness with Jacobi Weights

K. A. Kopotun^*, D. Leviatan, I. A. Shevchuk

^*Corresponding author for this work

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

We introduce the moduli of smoothness with Jacobi weights (1 − x)^𝛼(1 + x)^β for functions in the Jacobi weighted spaces L_p[−1, 1], 0 < p ≤ ∞. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted spaces L_p. If 1 ≤ p ≤ 1, then these moduli are equivalent to certain weighted K-functionals (and, hence, 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.”.

Original language	English
Pages (from-to)	437-466
Number of pages	30
Journal	Ukrainian Mathematical Journal
Volume	70
Issue number	3
DOIs	https://doi.org/10.1007/s11253-018-1509-9
State	Published - 1 Aug 2018

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Natural Sciences and Engineering Research Council of Canada

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abstract = "We introduce the moduli of smoothness with Jacobi weights (1 − x)𝛼(1 + x)β for functions in the Jacobi weighted spaces Lp[−1, 1], 0 < p ≤ ∞. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted spaces Lp. If 1 ≤ p ≤ 1, then these moduli are equivalent to certain weighted K-functionals (and, hence, 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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AB - We introduce the moduli of smoothness with Jacobi weights (1 − x)𝛼(1 + x)β for functions in the Jacobi weighted spaces Lp[−1, 1], 0 < p ≤ ∞. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted spaces Lp. If 1 ≤ p ≤ 1, then these moduli are equivalent to certain weighted K-functionals (and, hence, 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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On the Moduli of Smoothness with Jacobi Weights

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