Positive $$L^p$$Lp-Bounded Dunkl-Type Generalized Translation Operator and Its Applications

By Dmitry V. Gorbachev, V. Ivanov, S. Tikhonov. Constructive approximation, 2017

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We prove that the spherical mean value of the Dunkl-type generalized translation operator $$\tau ^y$$τy is a positive $$L^p$$Lp-bounded generalized translation operator $$T^t$$Tt. As applications, we prove the Young inequality for a convolution defined by $$T^t$$Tt, the $$L^p$$Lp-boundedness of $$\tau ^y$$τy on radial functions for $$p>2$$p>2, the $$L^p$$Lp-boundedness of the Riesz potential for the Dunkl transform, and direct and inverse theorems of approximation theory in $$L^p$$Lp-spaces with the Dunkl weight.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/be41421a73168e7e45b53087c99be7826d82bf4b