Geodesic
Equivalence of $K$-functionals and moduli of smoothness constructed by generalized Dunkl translations
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2008), pp. 3-15

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In a Hilbert space $L_{2,\alpha}:=L_2(\mathbb{R},|x|^{2\alpha+1}dx)$, $\alpha>-1/2$, 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 $L_{2,\alpha}$. On the base of 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.
Keywords: Dunkl operator, generalized Dunkl translation, $K$-functional, modulus of smoothness. 
@article{IVM_2008_8_a0,
     author = {S. S. Platonov and E. S. Belkina},
     title = {Equivalence of $K$-functionals and moduli of smoothness constructed by generalized {Dunkl} translations},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {3--15},
     publisher = {mathdoc},
     number = {8},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2008_8_a0/}
}
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S. S. Platonov; E. S. Belkina. Equivalence of $K$-functionals and moduli of smoothness constructed by generalized Dunkl translations. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2008), pp. 3-15. http://geodesic.mathdoc.fr/item/IVM_2008_8_a0/