Abstract
In a Hilbert space L 2,α := L 2(ℝ, |x|2α+1 dx), α > − 1/2, we study the generalized Dunkl translations constructed by the Dunkl differential-difference operator. Using the generalized Dunkl translations, we define generalized modulus of smoothness in the space L 2,α . Based on the Dunkl operator we define Sobolev-type spaces and K-functionals. The main result of the paper is the proof of the equivalence theorem for a K-functional and a modulus of smoothness.
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Original Russian Text © E.S. Belkina and S.S. Platonov, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 8, pp. 3–15.
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Belkina, E.S., Platonov, S.S. Equivalence of K-functionals and modulus of smoothness constructed by generalized Dunkl translations. Russ Math. 52, 1–11 (2008). https://doi.org/10.3103/S1066369X0808001X
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DOI: https://doi.org/10.3103/S1066369X0808001X