Title: An Analog of Titchmarsh's Theorem for the Jacobi-Dunkl Transform in the Space L2α,β(R)
Author(s): A. Abouelaz, A. Belkhadir, R. Daher
Pages: 15-21
Cite as:
A. Abouelaz, A. Belkhadir, R. Daher, An Analog of Titchmarsh's Theorem for the Jacobi-Dunkl Transform in the Space L2α,β(R), Int. J. Anal. Appl., 8 (1) (2015), 15-21.

Abstract


In this paper, using a generalized Jacobi-Dunkl translation operator, we prove an analog of Titchmarsh's theorem  for functions satisfying the Jacobi-Dunkl Lipschitz  condition in $ L^{2}(\R,A_{\alpha ,\beta}(t)dt), \alpha \geq \beta\geq-\frac{1}{2}, \alpha \neq -\frac{1}{2}.$

Full Text: PDF

 

References


  1. Ben Mohamed. H and Mejjaoli. H, Distributional Jacobi-Dunkl transform and application, Afr. Diaspora J. Math 1 (2004), 24–46.

  2. Ben Mohamed. H, The Jacobi-Dunkl transform on R and the convolution product on new spaces of distributions , Ramanujan J. 21 (2010), 145–175.

  3. Ben Salem. N and Ould Ahmed Salem. A , Convolution structure associated with the JacobiDunkl operator on R., Ramanujan J. 12(3) (2006), 359–378.

  4. Bray. W O and Pinsky. M A, Growth properties of Fourier transforms via moduli of continuity , Journal of Functional Analysis. 255 (2008), 2256–2285.

  5. Chouchane. F, Mili. M and Trim`eche. K, Positivity of the intertwining opertor and harmonic analysis associated with the Jacobi-Dunkl operator on R, J. Anal. Appl. 1(4) (2003), 387–412.

  6. Koornwinder. T H , Jacobi functions and analysis on noncompact semi-simple Lie Groups, in: Askey. R A , Koornwinder. T H and Schempp. W (eds) Special Functions: Group Theoritical Aspects and Applications. D. Reidel, Dordrecht (1984).

  7. Koornwinder. T H , A new proof of a Paley-Wiener type theorems for the Jacobi transform, Ark. Math. 13 (1975), 145–159.

  8. Platonov. S S, Approximation of functions in L2-metric on noncompact rank 1 symetric spaces, Algebra Analiz. 11 (1) (1999), 244–270.

  9. Platonov. S S, The Fourier transform of functions satisfying the Lipschitz condition on rank 1 symetric spaces, Siberian Math. J. 46(6) (2005), 1108–1118.

  10. Titchmarsh. E C, Introduction to the Theory of Fourier Integrals , Claredon, Oxford. 1948, Komkniga, Moscow. 2005.