Geodesic
Generalized one-dimensional Dunkl transform in direct problems of approximation theory
Matematičeskie zametki, Tome 116 (2024) no. 2, pp. 245-260

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On the real line, we study the generalized Dunkl harmonic analysis depending on a parameter $r\in\mathbb{N}$. The case of $r=0$ corresponds to the usual Dunkl harmonic analysis. All constructions depend on the parameter $r\geqslant 1$. The differences and the moduli of smoothness are defined using a generalized translation operator. The Sobolev space and the $K$-functional are defined using a differential-difference operator. An approximate Jackson-type inequality is proved. The equivalence of the $K$-functional and the modulus of smoothness is established.
Keywords: generalized Dunkl transform, generalized translation operator, convolution, $K$-functional, modulus of smoothness, Jackson inequality. 
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     author = {V. I. Ivanov},
     title = {Generalized one-dimensional {Dunkl} transform in direct problems of approximation theory},
     journal = {Matemati\v{c}eskie zametki},
     pages = {245--260},
     publisher = {mathdoc},
     volume = {116},
     number = {2},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2024_116_2_a6/}
}
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V. I. Ivanov. Generalized one-dimensional Dunkl transform in direct problems of approximation theory. Matematičeskie zametki, Tome 116 (2024) no. 2, pp. 245-260. http://geodesic.mathdoc.fr/item/MZM_2024_116_2_a6/