# On estimates for the Fourier-Bessel integral transform in the space L2(ℝ+)

22 Jul 2009-Computational Mathematics and Mathematical Physics (SP MAIK Nauka/Interperiodica)-Vol. 49, Iss: 7, pp 1103-1110

TL;DR: In this paper, the Fourier-Bessel integral transform in L 2 (ℝ+) was shown to have a generalized modulus of continuity, and two estimates useful in applications were proved.

Abstract: Two estimates useful in applications are proved for the Fourier-Bessel integral transform in L2(ℝ+) as applied to some classes of functions characterized by a generalized modulus of continuity.

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TL;DR: Using a generalized translation operator, this article obtained an analog of Theorem 5.2 in Younis (1986) for the Bessel transform for functions satisfying the -Bessel Lipschitz condition in.

Abstract: 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 .

11 citations

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01 Nov 2012

TL;DR: In this article, Titchmarsh proved some theorems on the classical Fourier transform of functions satisfying conditions related to the Cauchy-Lipschitz conditions on the Euclidean space R.

Abstract: In [1] Titchmarsh proved some theorems on the classical Fourier transform of functions satisfying conditions related to the Cauchy-Lipschitz conditions on the Euclidean space R. In this paper we extend one those theorems for the Bessel transform for function on half-line [0,1) in a weighted Lp-metric are studied with the use of Bessel generalized translation.
Keywords: Bessel operator; Bessel transform; Bessel generalized translation.
2010 Mathematics Subject Classification: 42B10; 42A38; 42B37.

5 citations

### Cites background from "On estimates for the Fourier-Bessel..."

...0 f̂(λ)jα(λt)λ dλ The following relation connect the Bessel generalized translation, and the Bessel transform in [2], we have...

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TL;DR: In this paper, the Fourier-Bessel transform in the space of two-variable functions characterized by a generalized modulus of continuity was shown to be a generalization of the Hankel transform.

Abstract: Two estimates useful in applications are proved for the Fourier-Bessel (or Hankel) transform in the space \(\mathbb{L}_2 \left( {\mathbb{R}_ + ^2 } \right)\) for some classes of two-variable functions characterized by a generalized modulus of continuity.

4 citations

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TL;DR: In this article, a generalized spherical mean operator was used to generalize Titchmarsh's theorem for the Dunkl transform for functions satisfying the Lipschitz condition in L 2 (Rd;wk), where wk is a weight function invariant under the action of an associated reection group.

Abstract: Using a generalized spherical mean operator, we obtain the generalizationof Titchmarsh's theorem for the Dunkl transform for functions satisfyingthe Lipschitz condition in L2(Rd;wk), where wk is a weight function invariantunder the action of an associated reection groups.

3 citations

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TL;DR: In this article, a generalized version of Titchmarsh's Theorem for the Dunkl transform for functions satisfying the Lipschitz Dunkl condition in the space is presented.

Abstract: In this paper, using a generalized Dunkl translation operator, we obtain a generalization of Titchmarsh's Theorem for the Dunkl transform for functions satisfying the $(\psi,p)$ -Lipschitz Dunkl condition in the space $\mathrm{L}_{p,\alpha}=\mathrm{L}^{p}(\mathbb{R},|x|^{2\alpha+1}dx)$ , where $\alpha>-\frac{1}{2}$ .

2 citations

##### References

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01 Jan 1944

TL;DR: The tabulation of Bessel functions can be found in this paper, where the authors present a comprehensive survey of the Bessel coefficients before and after 1826, as well as their extensions.

Abstract: 1. Bessel functions before 1826 2. The Bessel coefficients 3. Bessel functions 4. Differential equations 5. Miscellaneous properties of Bessel functions 6. Integral representations of Bessel functions 7. Asymptotic expansions of Bessel functions 8. Bessel functions of large order 9. Polynomials associated with Bessel functions 10. Functions associated with Bessel functions 11. Addition theorems 12. Definite integrals 13. Infinitive integrals 14. Multiple integrals 15. The zeros of Bessel functions 16. Neumann series and Lommel's functions of two variables 17. Kapteyn series 18. Series of Fourier-Bessel and Dini 19. Schlomlich series 20. The tabulation of Bessel functions Tables of Bessel functions Bibliography Indices.

9,584 citations

01 Aug 1971

TL;DR: The theory of embeddings of classes of differentiable functions of several variables has been intensively expanded during the past two decades, and a number of its fundamental problems have been resolved as discussed by the authors.

Abstract: : The theory of embeddings of classes of differentiable functions of several variables has been intensively expanded during the past two decades, and a number of its fundamental problems have been resolved. But till now these results are to be found in journal articles. This book presents the complete theory of embeddings of the main classes (W(sub p sup r),H(sub p sup r),B(sub p theta sup r),L(sub p sup r)) of differentiable functions given for the entire n-dimensional space R sub n. The reader will find in the book the inequalities between partial derivatives in the various contexts that have found application in mathematical physics. Emphasis is placed on problems of compactness, integral representations of functions of these classes, and problems of the isomorphisms of these classes. In the book the author chiefly employs the method of approximation with exponential type integral functions and trigonometric polymonials. The theory of approximation suitably adapted for these ends is set forth at the outset of the volume. Use of the Bessel-Macdonald integral operator is also essential. The reader will even find in the book remarks given without proof on the embedding of classes of differentiable functions specified for the domains G belongs to R sup n. The reader must be familiar with the fundamentals of Lesbesgue integral theory. The book widely employs the concept of the generalized function, but it is clarified with proofs to the extent that this is necessary. (Author)

1,024 citations