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Paley–Wiener Theorem for a Generalized Fourier Transform Associated to a Dunkl-Type Operator

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Abstract

In this paper, we obtain several versions of the real Paley-Wiener theorems for a generalized Fourier transform associated to a Dunkl-type differential-difference operator on the real line, which is related to Lions’ second-order singular differential operator, and includes Dunkl operator associated with the reflection group \({\mathbb {Z}}_2\) on \({\mathbb {R}}\) as a special case.

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References

  1. Andersen, N.B.: Real Paley-Wiener theorems. Bull. Lond. Math. Soc. 36, 504–508 (2004)

    Article  MathSciNet  Google Scholar 

  2. Andersen, N.B.: Real Paley-Wiener theorems for the Dunkl transform on \({{\mathbb{R}}}\). Integral Transforms Spec. Funct. 17(8), 543–547 (2006)

    Article  MathSciNet  Google Scholar 

  3. Andersen, N.B.: Real Paley-Wiener theorems for the Hankel transform. J. Fourier Anal. Appl. 12(1), 17–25 (2006)

    Article  MathSciNet  Google Scholar 

  4. Andersen, N.B., de Jeu, M.: Elementary proofs of Paley-Wiener theorems for the Dunkl transform on the real line. Int. Math. Res. Not. 30, 1817–1831 (2005)

    Article  MathSciNet  Google Scholar 

  5. Andersen, N.B., de Jeu, M.: Real Paley-Wiener theorems and local spectral radius formulas. Trans. Am. Math. Soc. 362(7), 3613–3640 (2010)

    Article  MathSciNet  Google Scholar 

  6. Bang, H.H.: A property of infinitely differentiable functions. Proc. Am. Math. Soc. 108, 73–76 (1990)

    Article  MathSciNet  Google Scholar 

  7. Bang, H.H.: Functions with bounded spectrum. Trans. Am. Math. Soc. 347, 1067–1080 (1995)

    Article  MathSciNet  Google Scholar 

  8. Ben Salem, N., Ould Ahmed Salem, A.: Convolution structure associated with the Jacobi-Dunkl operator on \({mathbb{R}}\). Ramanujan J. 12(3), 359–378 (2006)

    Article  MathSciNet  Google Scholar 

  9. Chen, Q.H., Li, L.Q., Ren, G.B.: Generalized Paley-Wiener theorems. Int. J. Wavelets Multiresolut. Inf. Process. 10(2), 1250020 (2010)

    Article  MathSciNet  Google Scholar 

  10. Chettaoui, C., Trimèche, K.: New type Paley-Wiener theorems for the Dunkl transform on \({\mathbb{R}}\). Integral Transforms Spec. Funct. 14, 97–115 (2003)

    Article  MathSciNet  Google Scholar 

  11. Chouchene, F., Mili, M., Trimèche, K.: Positivity of the intertwining operator and harmonic analysis associated with the Jacobi-Dunkl operator on \({\mathbb{R}}\). Anal. Appl. 1(4), 387–412 (2003)

    Article  MathSciNet  Google Scholar 

  12. Dunkl, C.F.: Differential-difference operators associated to reflection group. Trans. Am. Math. Soc. 311(1), 167–183 (1989)

    Article  MathSciNet  Google Scholar 

  13. Fei, M., Xu, Y., Yan, J.: Real Paley-Wiener theorem for the quaternion Fourier transform. Complex Var. Elliptic Equ. 62(8), 1072–1080 (2017)

    Article  MathSciNet  Google Scholar 

  14. Fu, Y.X., Li, L.Q.: Paley-Wiener and Boas theorems for the quaternion Fourier transform. Adv. Appl. Clifford Algebr. 23, 837–848 (2013)

    Article  MathSciNet  Google Scholar 

  15. Fu, Y.X., Li, L.Q.: Real Paley-Wiener theorems for the Clifford Fourier transform. Sci. China Math. 57(11), 2381–2392 (2014)

    Article  MathSciNet  Google Scholar 

  16. Li, S., Leng, J., Fei, M.: Paley-Wiener-type theorems for the Clifford-Fourier transform. Math. Methods Appl. Sci. 42(18), 6101–6113 (2019)

    Article  MathSciNet  Google Scholar 

  17. Li, S., Leng, J., Fei, M.: Spectrums of functions associated to the fractional Clifford-Fourier transform. Adv. Appl. Clifford Algebr. 30(1), 6 (2020)

    Article  MathSciNet  Google Scholar 

  18. Mejjaoli, H.: Sobolev spaces associated with a Dunkl type differential-difference operator on the real line and applications. J. Pseudo-Differ. Oper. Appl. 6, 33–68 (2015)

    Article  MathSciNet  Google Scholar 

  19. Mejjaoli, H.: Uncertainty principles for the generalized Fourier transform associated to a Dunkl-type operator. Appl. Anal. 95(9), 1930–1956 (2016)

    Article  MathSciNet  Google Scholar 

  20. Mejjaoli, H., Trimèche, K.: Spectrum of functions for the Dunkl transform on \({\mathbb{R}}^d\). Fract. Calc. Appl. Anal. 10, 19–38 (2007)

    MathSciNet  MATH  Google Scholar 

  21. Mourou, M.A., Trimèche, K.: Transmutation operators and Paley-Wiener theorem associated with a singular differential-difference operator on the real line. Anal. Appl. 1(1), 43–70 (2003)

    Article  MathSciNet  Google Scholar 

  22. Trimèche, K.: Inversion of the Lions transmutation operators using generalized wavelets. Appl. Comput. Harmon. Anal. 4, 97–112 (1997)

    Article  MathSciNet  Google Scholar 

  23. Trimèche, K.: The transmutation operators relating to a Dunkl type operator on \({\mathbb{R}}\) and their positivity. Mediterr. J. Math. 12, 349–369 (2015)

    Article  MathSciNet  Google Scholar 

  24. Paley, R., Wiener, N.: In: Amer. Math. Soc., Colloq. Publ. Ser., (ed.) The Fourier transforms in the complex domain, vol. 19. , Providence, RI (1934)

  25. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ. Press, Princeton (1971)

    MATH  Google Scholar 

  26. Tuan, V.K.: On the Paley-Wiener theorem, in Theory of Functions and Applications, pp. 193–196. In: Louys Publishing House, Yerevan, Collection of Works dedicated to the Memory of Mkhitar M. Djrbashian (1995)

  27. Tuan, V.K.: Paley-Wiener-Type theorems. Frac. Cal. Appl. Anal. 2(2), 135–143 (1999)

    MathSciNet  MATH  Google Scholar 

  28. Tuan, V.K.: On the supports of functions. Numer. Funct. Anal. Optim. 20, 387–394 (1999)

    Article  MathSciNet  Google Scholar 

  29. Tuan, V.K.: A real-variable Paley-Wiener theorem for the Dunkl transform, Abstract and Applied Analysis, 365–371. World Sci. Publ, River Edge (2004)

    Google Scholar 

  30. Tuan, V.K., Ismail, A., Saigo, M.: Plancherel and Paley-Wiener theorems for an index integral transform. J. Korean Math. Soc. 37, 545–563 (2000)

    MathSciNet  MATH  Google Scholar 

  31. Tuan, V.K., Zayed, A.I.: Paley-Wiener-type theorems for a class of integral transforms. J. Math. Anal. Appl. 266(1), 200–226 (2002)

    Article  MathSciNet  Google Scholar 

  32. Zayed, A.I., Tuan, V.K.: Paley-Wiener-type theorem for a class of integral transforms arising from a singular Dirac system. Z. Anal. Anwendungen 19, 695–712 (2000)

    Article  MathSciNet  Google Scholar 

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  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, People’s Republic of China

    Minggang Fei & Lan Yang

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  1. Minggang Fei
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  2. Lan Yang
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The authors were partially supported by “the Fundamental Research Funds for the Central Universities” with No. ZYGX2019J091.

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Fei, M., Yang, L. Paley–Wiener Theorem for a Generalized Fourier Transform Associated to a Dunkl-Type Operator. Mediterr. J. Math. 18, 190 (2021). https://doi.org/10.1007/s00009-021-01846-x

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  • DOI: https://doi.org/10.1007/s00009-021-01846-x

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