scholarly journals Bessel Transform of -Bessel Lipschitz Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Radouan Daher ◽  
Mohamed El Hamma
Keyword(s):  
Lipschitz Condition ◽  
Translation Operator ◽  
Lipschitz Functions ◽  
Generalized Translation Operator ◽  
Generalized Translation ◽  
Bessel Transform

Using a generalized translation operator, we obtain an analog of Theorem 5.2 in Younis (1986) for the Bessel transform for functions satisfying the -Bessel Lipschitz condition in .

scholarly journals Characterization of Dini Lipschitz Functions in Terms of Their Helgason Transform

2016 ◽  
Vol 54 (2) ◽  
pp. 117-130
Author(s):  
Salah El Ouadih ◽  
Radouan Daher
Keyword(s):  
Fourier Transform ◽  
Lipschitz Condition ◽  
Symmetric Spaces ◽  
Translation Operator ◽  
Lipschitz Functions ◽  
Generalized Translation Operator ◽  
Generalized Translation ◽  
Rank One ◽  
Riemannian Symmetric Spaces

Abstract In this paper, using a generalized translation operator, we obtain an analog of Younis Theorem 5.2 in [6] for the Helgason Fourier transform of a set of functions satisfying the Dini Lipschitz condition in the space L2 for functions on noncompact rank one Riemannian symmetric spaces.

Characterization of Dini–Lipschitz functions for the Helgason Fourier transform on rank one symmetric spaces

2016 ◽  
Vol 7 (4) ◽  
Author(s):  
Salah El Ouadih ◽  
Radouan Daher
Keyword(s):  
Fourier Transform ◽  
Symmetric Spaces ◽  
Translation Operator ◽  
Lipschitz Functions ◽  
Generalized Translation Operator ◽  
Generalized Translation ◽  
Rank One

AbstractIn this paper, using a generalized translation operator, we obtain an analog of Younis’ theorem, [

On the K-functional in learning theory

2019 ◽  
Vol 18 (03) ◽  
pp. 423-446
Author(s):  
Bao-Huai Sheng ◽  
Jian-Li Wang
Keyword(s):  
Learning Theory ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Approximation Error ◽  
Translation Operator ◽  
Modulus Of Smoothness ◽  
Generalized Translation Operator ◽  
Generalized Translation ◽  
Approximation Errors ◽  
Derivatives Of

[Formula: see text]-functionals are used in learning theory literature to study approximation errors in kernel-based regularization schemes. In this paper, we study the approximation error and [Formula: see text]-functionals in [Formula: see text] spaces with [Formula: see text]. To this end, we give a new viewpoint for a reproducing kernel Hilbert space (RKHS) from a fractional derivative and treat powers of the induced integral operator as fractional derivatives of various orders. Then a generalized translation operator is defined by Fourier multipliers, with which a generalized modulus of smoothness is defined. Some general strong equivalent relations between the moduli of smoothness and the [Formula: see text]-functionals are established. As applications, some strong equivalent relations between these two families of quantities on the unit sphere and the unit ball are provided explicitly.

scholarly journals Некоторые оценки для обобщенного преобразования Фурье, ассоциированного с оператором Чередника - Опдама

2018 ◽  
Author(s):  
H.S. Lafdal ◽  
R. Daher ◽  
El.O. Salah
Keyword(s):  
Fourier Transform ◽  
Translation Operator ◽  
Modulus Of Smoothness ◽  
Continuity Modulus ◽  
Generalized Translation Operator ◽  
Generalized Continuity ◽  
Generalized Translation ◽  
Translation Operators ◽  
Alpha Beta ◽  
Generalized Continuity Modulus

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.

Titchmarsh's theorem of Hankel transform

2019 ◽  
pp. 11-24
Author(s):  
Mohamed-Ahmed Boudref
Keyword(s):  
Number Theory ◽  
Data Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Lipschitz Condition ◽  
Analytic Number Theory ◽  
Hankel Transform ◽  
Partial Differential ◽  
Analytic Number ◽  
Bessel Transform

Hankel transform (or Fourier-Bessel transform) is a fundamental tool in many areas of mathematics and engineering, including analysis, partial differential equations, probability, analytic number theory, data analysis, etc. In this article, we prove an analog of Titchmarsh's theorem for the Hankel transform of functions satisfying the Hankel-Lipschitz condition.

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