Some Estimates for the Generalized Fourier Transform Associated with the Cherednik-Opdam Operator on \(\mathbb{R}\)

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2018.3.18031 Some Estimates for the Generalized Fourier Transform Associated with the Cherednik-Opdam Operator on \(\mathbb{R}\) El Ouadih S. , Daher R. , Lafdal H. S. Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 3. Abstract: 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. Keywords: Cherednik-Opdam operator, generalized Fourier transform, generalized translation. Language: English Download the full text For citation: El Ouadih S., Daher R., Lafdal H. S. Some Estimates for the Generalized Fourier Transform Associated with the Cherednik-Opdam Operator on \(\mathbb{R}\), Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no. 2, pp. 78-86. DOI 10.23671/VNC.2018.3.18031 + References 1. Abilov V. A., Abilova F. V. and Kerimov M. K. Some Remarks Concerning the Fourier Transform in the Space $L 2 (R) " role="presentation"> L 2 (R)$ , Comput. Math. Math. Phys., 2008, vol. 48, pp. 885-891. 2. Hamma M. E., Daher R. and Khadari A. On Estimates for the Dunkl Transform in the Space $L 2 (R d, w k (x) d x) " role="presentation"> L 2 (R d, w k (x) d x)$ , Ser. Math. Inform., 2013, vol. 28, no. 3, pp. 285-296. 3. Daher R., Ouadih S. E. Certain Problems on the Approximation of Functions by Fourier-Jacobi Sums in the Space $L 2 (α, β) " role="presentation"> L (α, β) 2$ , Alabama J. Math., 2016, vol. 40, pp. 1-4. 4. Ouadih S. E., Daher R. New Estimates for the Generalized Fourier-Bessel Transform in the Space $L α, n 2 " role="presentation"> L 2 α, n$ , South. Asian Bull. Math., 2017, vol. 41, pp. 209-218. 5. Opdam E. M. Harmonic Analysis for Certain Representations of Graded Hecke Algebras, Acta Math., 1995, vol. 175, no. 1, pp. 75-121. 6. Anker J. P., Ayadi F., and Sifi M. Opdams Hypergeometric Functions: Product Formula and Convolution Structure in Dimension 1, Adv. Pure Appl. Math., 2012, vol. 3, no. 1, pp. 11-44. 7. Platonov S. S. Approximation of Functions in $L 2 " role="presentation"> L 2$ -Metric on Noncompact Rank 1 Symmetric Space, St. Petersburg Mathematical J., 2000, vol. 11, no. 1, pp. 183–201. 8. Abilov V. A., Abilova F. V. Approximation of Functions by Fourier-Bessel Sums, Russian Mathematics (Izvestiya VUZ. Matematika), 2001, vol. 45, no. 8, pp. 1-7. 9. Bray W. O. and Pinsky M. A. Growth Properties of Fourier Transforms via Module of Continuity, J. Funct. Anal., 2008, vol. 255, pp. 2256-2285. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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