Estimates of functionals by generalized moduli of continuity generated by the dunkl operators

Estimates of functionals by deviations of Steklov type averages generated by Dunkl type operators

“…As is known, the operator T possesses the following properties. For the Dunkl operators these properties can be found in [15,16,2]. For Dunkl type operators these properties are established in a similar way; some of them can be found in [9].…”

“…The first assertion is proved in [21, p. 655], and the second one is proved in [2]. If, in addition, the function G is even, then…”

“…In Section 7, we discuss how to construct linear combinations of Steklov type averages so that to obtain the best order of their deviations. The results of this paper in the case of the Dunkl operator were obtained in [2].…”

“…We define the second order Steklov kernel with step h > 0 by the equality loc (R, A)), then S h f ∈ W (2) p (R, A) (W (2) p,loc (R, A)), and…”

“…Furthermore, it is reasonable to consider T c instead of the mapping T , which can lead to more delicate estimates as in [2], for m ∈ N we set…”