Geodesic
Some estimates for the generalized Fourier transform associated with the Cherednik–Opdam operator on $\mathbb{R}$
Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 3, pp. 78-86 
S. El Ouadih; R. Daher; H. S. Lafdal. Some estimates for the generalized Fourier transform associated with the Cherednik–Opdam operator on $\mathbb{R}$. Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 3, pp. 78-86. http://geodesic.mathdoc.fr/item/VMJ_2018_20_3_a6/
@article{VMJ_2018_20_3_a6,
     author = {S. El Ouadih and R. Daher and H. S. Lafdal},
     title = {Some estimates for the generalized {Fourier} transform associated with the {Cherednik{\textendash}Opdam} operator on $\mathbb{R}$},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {78--86},
     year = {2018},
     volume = {20},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2018_20_3_a6/}
}
TY  - JOUR
AU  - S. El Ouadih
AU  - R. Daher
AU  - H. S. Lafdal
TI  - Some estimates for the generalized Fourier transform associated with the Cherednik–Opdam operator on $\mathbb{R}$
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2018
SP  - 78
EP  - 86
VL  - 20
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VMJ_2018_20_3_a6/
LA  - en
ID  - VMJ_2018_20_3_a6
ER  - 
%0 Journal Article
%A S. El Ouadih
%A R. Daher
%A H. S. Lafdal
%T Some estimates for the generalized Fourier transform associated with the Cherednik–Opdam operator on $\mathbb{R}$
%J Vladikavkazskij matematičeskij žurnal
%D 2018
%P 78-86
%V 20
%N 3
%U http://geodesic.mathdoc.fr/item/VMJ_2018_20_3_a6/
%G en
%F VMJ_2018_20_3_a6

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

In the classical theory of approximation of functions on $\mathbb{R}^+$, the modulus of smoothness are basically built by means of the translation operators $f \to f(x+y)$. As the notion of translation operators was extended to various contexts (see [2] and [3]), many generalized modulus of smoothness have been discovered. Such generalized modulus of smoothness are often more convenient than the usual ones for the study of the connection between the smoothness properties of a function and the best approximations of this function in weight functional spaces (see [4] and [5]). In [1], Abilov et al. proved two useful estimates for the Fourier transform in the space of square integrable functions on certain classes of functions characterized by the generalized continuity modulus, using a translation operator. In this paper, we also discuss this subject. More specifically, we prove some estimates (similar to those proved in [1]) in certain classes of functions characterized by a generalized continuity modulus and connected with the generalized Fourier transform associated with the differential-difference operator $T^{(\alpha,\beta)}$ in $L^{2}_{\alpha,\beta}(\mathbb{R})$. For this purpose, we use a generalized translation operator.

[1] Abilov V. A., Abilova F. V., Kerimov M. K., “Some Remarks Concerning the Fourier Transform in the Space $L_{2}(\mathbb{R})$”, Comput. Math. Math. Phys., 48 (2008), 885–891 | DOI | MR | Zbl

[2] Hamma M. E., Daher R., Khadari A., “On Estimates for the Dunkl Transform in the Space $L^{2}(\mathbb{R}^{d}, w_{k}(x)\,dx)$”, Ser. Math. Inform., 28:3 (2013), 285–296 | MR | Zbl

[3] Daher R., Ouadih S. E., “Certain Problems on the Approximation of Functions by Fourier–Jacobi Sums in the Space $L_{2}^{(\alpha, \beta)}$”, Alabama J. Math., 40 (2016), 1–4 | MR

[4] Ouadih S. E., Daher R., “New Estimates for the Generalized Fourier–Bessel Transform in the Space $L^{2}_{\alpha,n}$”, South. Asian Bull. Math., 41 (2017), 209–218 | MR | Zbl

[5] Opdam E. M., “Harmonic Analysis for Certain Representations of Graded Hecke Algebras”, Acta Math., 175:1 (1995), 75–121 | DOI | MR | Zbl

[6] Anker J. P., Ayadi F., Sifi M., “Opdams Hypergeometric Functions: Product Formula and Convolution Structure in Dimension 1”, Adv. Pure Appl. Math., 3:1 (2012), 11–44 | DOI | MR | Zbl

[7] Platonov S. S., “Approximation of Functions in $L_{2}$-Metric on Noncompact Rank $1$ Symmetric Space”, St. Petersburg Mathematical J., 11:1 (2000), 183–201 | MR

[8] Abilov V. A., Abilova F. V., “Approximation of Functions by Fourier–Bessel Sums”, Russian Mathematics (Izvestiya VUZ. Matematika), 45:8 (2001), 1–7 | MR | Zbl

[9] Bray W. O., Pinsky M. A., “Growth Properties of Fourier Transforms via Module of Continuity”, J. Funct. Anal., 255 (2008), 2256–2285 | DOI | MR