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

D. V. Gorbachev; V. I. Ivanov; S. Yu Tikhonov

doi:10.1007/s00365-018-9435-5

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

D. V. Gorbachev, V. I. Ivanov, S. Yu Tikhonov

Research output: Contribution to journal › Article › Research

19 Citations (Scopus)

Abstract

© 2018, Springer Science+Business Media, LLC, part of Springer Nature. We prove that the spherical mean value of the Dunkl-type generalized translation operator τ y is a positive L p -bounded generalized translation operator T t . As applications, we prove the Young inequality for a convolution defined by T t , the L p -boundedness of τ y on radial functions for p> 2 , the L p -boundedness of the Riesz potential for the Dunkl transform, and direct and inverse theorems of approximation theory in L p -spaces with the Dunkl weight.

Original language	English
Pages (from-to)	555-605
Journal	Constructive Approximation
Volume	49
DOIs	https://doi.org/10.1007/s00365-018-9435-5
Publication status	Published - 15 Jun 2019

Keywords

Convolution
Dunkl transform
Generalized translation operator
Riesz potential

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title = "Positive L p -Bounded Dunkl-Type Generalized Translation Operator and Its Applications",

abstract = "{\textcopyright} 2018, Springer Science+Business Media, LLC, part of Springer Nature. We prove that the spherical mean value of the Dunkl-type generalized translation operator τ y is a positive L p -bounded generalized translation operator T t . As applications, we prove the Young inequality for a convolution defined by T t , the L p -boundedness of τ y on radial functions for p> 2 , the L p -boundedness of the Riesz potential for the Dunkl transform, and direct and inverse theorems of approximation theory in L p -spaces with the Dunkl weight.",

keywords = "Convolution, Dunkl transform, Generalized translation operator, Riesz potential",

author = "Gorbachev, {D. V.} and Ivanov, {V. I.} and Tikhonov, {S. Yu}",

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doi = "10.1007/s00365-018-9435-5",

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TY - JOUR

T1 - Positive L p -Bounded Dunkl-Type Generalized Translation Operator and Its Applications

AU - Gorbachev, D. V.

AU - Ivanov, V. I.

AU - Tikhonov, S. Yu

PY - 2019/6/15

Y1 - 2019/6/15

N2 - © 2018, Springer Science+Business Media, LLC, part of Springer Nature. We prove that the spherical mean value of the Dunkl-type generalized translation operator τ y is a positive L p -bounded generalized translation operator T t . As applications, we prove the Young inequality for a convolution defined by T t , the L p -boundedness of τ y on radial functions for p> 2 , the L p -boundedness of the Riesz potential for the Dunkl transform, and direct and inverse theorems of approximation theory in L p -spaces with the Dunkl weight.

AB - © 2018, Springer Science+Business Media, LLC, part of Springer Nature. We prove that the spherical mean value of the Dunkl-type generalized translation operator τ y is a positive L p -bounded generalized translation operator T t . As applications, we prove the Young inequality for a convolution defined by T t , the L p -boundedness of τ y on radial functions for p> 2 , the L p -boundedness of the Riesz potential for the Dunkl transform, and direct and inverse theorems of approximation theory in L p -spaces with the Dunkl weight.

KW - Convolution

KW - Dunkl transform

KW - Generalized translation operator

KW - Riesz potential

U2 - 10.1007/s00365-018-9435-5

DO - 10.1007/s00365-018-9435-5

M3 - Article

VL - 49

SP - 555

EP - 605

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

ER -