### Besov-Dunkl spaces connected with generalized Taylor formula on the real line

#### Abstract

In the present paper, we define for the Dunkl tranlation operators on the real line, the Besov-Dunkl space of functions for which the remainder in the generalized Taylor's formula has a given order. We provide characterization of these spaces by the Dunkl convolution.

#### Article information

Source
Adv. Oper. Theory, Volume 2, Number 4 (2017), 516-530.

Dates
Accepted: 11 August 2017
First available in Project Euclid: 4 December 2017

https://projecteuclid.org/euclid.aot/1512431726

Digital Object Identifier
doi:10.22034/aot.1704-1154

Mathematical Reviews number (MathSciNet)
MR3730045

Zentralblatt MATH identifier
1374.44003

#### Citation

Abdelkefi, Chokri; Rached, Faten. Besov-Dunkl spaces connected with generalized Taylor formula on the real line. Adv. Oper. Theory 2 (2017), no. 4, 516--530. doi:10.22034/aot.1704-1154. https://projecteuclid.org/euclid.aot/1512431726

#### References

• C. Abdelkefi and M. Sifi, Characterisation of Besov spaces for the Dunkl operator on the real line, J. Inequal. Pure Appl. Math. 8 (2007), Issue 3, Article 73, 11 pp.
• C. Abdelkefi, J. Ph. Anker, F. Sassi, and M. Sifi, Besov-type spaces on $\mathbb{R}^d$ and integrability for the Dunkl transform, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 019, 15 pp.
• C. Abdelkefi, Weighted function spaces and Dunkl transform, Mediterr. J. Math. 9 (2012), no. 3, 499–513 Springer.
• B. Amri, J. Ph. Anker, and M. Sifi, Three results in Dunkl analysis, Colloq. Math. 118 (2010), no. 1, 299–312.
• J. L. Ansorena and O. Blasco, Characterization of weighted Besov spaces, Math. Nachr. 171 (1995), 5–17.
• O. V. Besov, On a family of function spaces in connection with embeddings and extention theoremss, (Russian) Trudy. Mat. Inst. Steklov. 60 (1961), 42–81.
• C. F. Dunkl, Differential-difference operators associated to reflexion groups, Trans. Amer. Math. Soc. 311 (1989), no. 1, 167–183.
• D. V. Giang and F. Moricz, A new characterization of Besov spaces on the real line, J. Math. Anal. Appl. 189 (1995), no. 2, 533–551.
• L. Kamoun, Besov-type spaces for the Dunkl operators on the real line, J. Comput. Appl. Math. 199, no. 1, (2007), 299–312.
• J. Löfström and J. Peetre, Approximation theorems connected with generalized translations, Math. Ann. 181 (1969), 255–268.
• M. A. Mourou and K. Trimèche, Calderon's reproducing formula related to the Dunkl operator on the real line, Monatsh. Math. 136 (2002), no. 1, 47–65.
• M. A. Mourou, Taylor series associated with a differential-difference operator on the real line, Proceedings of the Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Rome, 2001). J. Comp. and Appl. Math., 153 (2003), 343–354.
• J. Peetre, New thoughts on Besov spaces, Duke Univ. Math. Series, Durham, NC, 1976.
• M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, Nonselfadjoint operators and related topics (Beer Sheva, 1992), 369–396, Oper. Theory Adv. Appl., 73, Birkhäuser, Basel 1994.
• M. Rösler, Bessel-Type signed hypergroup on $\mathbb{R}$, Probability measures on groups and related structures, XI (Oberwolfach, 1994), 292–304, World Sci. Publ., River Edge, NJ, 1995.
• M. Rösler, Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), no. 3, 519–541.
• M. Rösler, Dunkl operators: theory and applications Orthogonal polynomials and special functions (Leuven, 2002), 93–135, Lecture Notes in Math., 1817, Springer, Berlin, 2003.