Geodesic
Generalized one-dimensional Dunkl transform in direct problems of approximation theory
Matematičeskie zametki, Tome 116 (2024) no. 2, pp. 245-260.

Voir la notice de l'article provenant de la source Math-Net.Ru

On the real line, we study the generalized Dunkl harmonic analysis depending on a parameter $r\in\mathbb{N}$. The case of $r=0$ corresponds to the usual Dunkl harmonic analysis. All constructions depend on the parameter $r\geqslant 1$. The differences and the moduli of smoothness are defined using a generalized translation operator. The Sobolev space and the $K$-functional are defined using a differential-difference operator. An approximate Jackson-type inequality is proved. The equivalence of the $K$-functional and the modulus of smoothness is established.
Keywords: generalized Dunkl transform, generalized translation operator, convolution, $K$-functional, modulus of smoothness, Jackson inequality. 
@article{MZM_2024_116_2_a6,
     author = {V. I. Ivanov},
     title = {Generalized one-dimensional {Dunkl} transform in direct problems of approximation theory},
     journal = {Matemati\v{c}eskie zametki},
     pages = {245--260},
     publisher = {mathdoc},
     volume = {116},
     number = {2},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2024_116_2_a6/}
}
TY  - JOUR
AU  - V. I. Ivanov
TI  - Generalized one-dimensional Dunkl transform in direct problems of approximation theory
JO  - Matematičeskie zametki
PY  - 2024
SP  - 245
EP  - 260
VL  - 116
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2024_116_2_a6/
LA  - ru
ID  - MZM_2024_116_2_a6
ER  - 
%0 Journal Article
%A V. I. Ivanov
%T Generalized one-dimensional Dunkl transform in direct problems of approximation theory
%J Matematičeskie zametki
%D 2024
%P 245-260
%V 116
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2024_116_2_a6/
%G ru
%F MZM_2024_116_2_a6
V. I. Ivanov. Generalized one-dimensional Dunkl transform in direct problems of approximation theory. Matematičeskie zametki, Tome 116 (2024) no. 2, pp. 245-260. http://geodesic.mathdoc.fr/item/MZM_2024_116_2_a6/

[1] S. Ben Saïd, T. Kobayashi, B. Ørsted, “Laguerre semigroup and Dunkl operators”, Compos. Math., 148:4 (2012), 1265–1336 | DOI | MR

[2] C. F. Dunkl, “Integral kernels with reflection group invariance”, Canad. J. Math., 43:6 (1991), 1213–1227 | DOI | MR

[3] M. Rösler, “Dunkl operators: theory and applications”, Orthogonal Polynomials and Special Functions (Leuven, 2002), Lecture Notes in Math., 1817, Springer-Verlag, Berlin, 2002, 93–135 | DOI | MR

[4] D. V. Gorbachev, V. I. Ivanov, S. Yu. Tikhonov, “On the kernel of the $(\kappa,a)$-generalized Fourier transform”, Forum Math. Sigma, 11 (2023), 1–25 | DOI | MR

[5] V. I. Ivanov, “Nedeformirovannoe obobschennoe preobrazovanie Danklya na pryamoi”, Matem. zametki, 114:4 (2023), 509–524 | DOI

[6] D. V. Gorbachev, V. I. Ivanov, S. Yu. Tikhonov, “Positive $L^p$-bounded Dunkl-type generalized translation operator and its applications”, Constr. Approx., 49:3 (2019), 555–605 | DOI | MR

[7] D. V. Gorbachev, V. I. Ivanov, “Fractional smoothness in $L^p$ with Dunkl weight and its applications”, Math. Notes, 106:4 (2019), 537–561 | DOI | MR

[8] S. S. Platonov, “Garmonicheskii analiz Besselya i priblizhenie funktsii na polupryamoi”, Izv. RAN. Ser. matem., 71:5 (2007), 149–196 | DOI | MR | Zbl

[9] S. S. Platonov, “Obobschennye sdvigi Besselya i nekotorye zadachi teorii priblizhenii funktsii na polupryamoi”, Sib. matem. zhurn., 50:1 (2009), 154–174 | MR

[10] V. I. Ivanov, “Operator spleteniya dlya obobschennogo preobrazovaniya Danklya na pryamoi”, Chebyshevskii sb., 24:4 (2023), 48–62 | DOI

[11] H. Bateman, A. Erdélyi, et al., Higher Transcendental Functions. II, McGraw Hill Book Company, New York, 1953 | MR