New Paley–Wiener Theorems

Abstract

In this paper, for an arbitrary fixed compact set K, we find necessary and sufficient conditions on the Taylor expansion coefficients of entire functions of exponential type so that these functions are the Fourier image of distributions supported in K. In other words, we state the Paley–Wiener theorem in the language of Taylor expansion coefficients.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Abreu, L.D.: Real Paley–Wiener theorems for the Koornwinder–Swarttouw–Hankel transform. J. Math. Anal. Appl 334, 223–231 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Abreu, L.D., Bouzeffour, F.: A Paley–Wiener theorem for the Askey–Wilson function transform. Proc. Amer. Math. Soc 138, 2853–2862 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Amri, B., Anker, J.P., Sifi, M.: Three results in Dunkl analysis. Colloq. Math. 118, 299–312 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Andersen, N.B.: Real Paley–Wiener theorems for the inverse Fourier transform on a Riemannian symmetric space. Pacific J. Math. 213, 1–13 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Andersen, N.B.: A simple proof of a Paley–Wiener type theorem for the Chébli–Trimèche transform. Publ. Math. Debrecen 64, 473–479 (2004)

    MathSciNet  MATH  Google Scholar 

  6. 6.

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

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Andersen, N.B.: On the range of the Chébli–Trimèche transform. Monatsh. Math. 144, 193–201 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

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

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Andersen, N.B., de Jeu, M.: Elementary proofs of Paley–Wiener theorems for the Dunkl transform on the real line. Int. Math. Res. Notices 30, 1817–1831 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

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

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Arthur, J.: A Paley–Wiener theorem for real reductive groups. Acta Math. 150, 1–89 (1983)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Arthur, J.: On a family of distributions obtained from Eisenstein series. I. Application of the Paley–Wiener theorem. Am. J. Math. 104, 1243–1288 (1982)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    van den Ban, E.P., Schlichtkrull, H.: A Paley–Wiener theorem for reductive symmetric spaces. Ann. Math. 164, 879–909 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Bang, H.H.: Theorems of the Paley–Wiener–Schwartz type. Trudy Mat. Inst. Steklov 214, 298–319 (1996)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Bang, H.H.: Nonconvex cases of the Paley–Wiener–Schwartz theorems. Doklady Akad. Nauk 354, 165–168 (1997)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Bang, H.H.: A property of entire functions of exponential type. Analysis 15, 17–23 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

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

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Bang, H.H.: The existence of a point spectral radius of pseudodifferential operators. Doklady Akad. Nauk. 348, 740–742 (1996)

    MathSciNet  Google Scholar 

  19. 19.

    Bang, H.H.: The study of the properties of functions belonging to an Orlicz space depending on the geometry of their spectra. Izv. Akad. Nauk Ser. Mat. 61, 163–198 (1997)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Bang, H.H.: A study of the properties of functions depending on the geometry of their spectrum. Doklady Akad. Nauk. 355, 740–743 (1997)

    MathSciNet  Google Scholar 

  21. 21.

    Bang, H.H.: Investigation of the properties of functions in the space \(N_\Phi \) depending on the geometry of their spectrum. Doklady Akad. Nauk. 374, 590–593 (2000)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Bang, H.H., Huy, V.N.: The Paley–Wiener theorem in the language of Taylor expansion coefficients. Doklady Akad. Nauk. 446, 497–500 (2012)

    MATH  Google Scholar 

  23. 23.

    Betancor, J.J., Betancor, J.D., Mendez, J.M.R.: Paley–Wiener type theorems for Chébli–Trimèche transforms. Publ. Math. 60, 347–358 (2002)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Björck, G.: Linear partial differential operators and generalized distributions. Ark. Mat. 6, 351–407 (1966)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Braun, R., Meise, R., Taylor, B.: Ultradifferentiable functions and Fourier analysis. Results Math. 17, 206–237 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Chabeh, W., Mourou, M.A.: Inversion of the generalized Dunkl Intertwining operator on \({\mathbb{R}}\) and its dual using generalized wavelets. Eur. J. Pure Appl. Math. 3, 958–979 (2010)

    MathSciNet  MATH  Google Scholar 

  27. 27.

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

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Cho, J.G., Kim, K.W.: Real version of Paley–Wiener–Schwartz theorem for ultradistributions with ultradifferentiable singular support. Bull. Korean Math. Soc. 36, 483–493 (1999)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Christensen, O.: A Paley–Wiener theorem for frames. Proc. Am. Math. Soc. 123, 2199–2202 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Chung, S.Y., Kim, D.: Paley-Wiener type theorems for the temperature transform. Integral Transforms Spec. Funct. 11, 151–162 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Chung, J., Chung, S.Y., Kim, D.: Real version of Paley–Wiener theorem for hyperfunctions and ultradistributions. Integral Transforms Spec. Funct. 10, 201–210 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Delorme, P., Mezo, P.: A twisted invariant Paley–Wiener theorem for real reductive groups. Duke Math. J. 144, 341–380 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Eguchi, M., Hashizume, M., Okamoto, K.: The Paley–Wiener theorem for distributions on symmetric spaces. Hiroshima Math. J. 3, 109–120 (1973)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Gangolli, R.: On the plancherel formula and the Paley–Wiener theorem for spherical functions on semisimple Lie groups. Ann. Math. Second Ser. 93, 150–165 (1971)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Helgason, S.: An analogue of the Paley–Wiener theorem for the Fourier transform on certain symmetric spaces. Math. Ann. 165, 297–308 (1966)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin (1983)

    Google Scholar 

  37. 37.

    Hörmander, L.: The Analysis of Linear Partial Differential Operators III. Springer, Berlin (1985)

    Google Scholar 

  38. 38.

    de Jeu, M.: Some remarks on a proof of geometrical Paley–Wiener theorems for the Dunkl transform. Integral Transforms Spec. Funct. 18, 383–385 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    de Jeu, M.: Paley–Wiener theorems for the Dunkl transform. Trans. Am. Math. Soc. 358, 4225–4250 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Khrennikov, AYu., Petersson, H.: A Paley–Wiener theorem for generalized entire functions on infinite-dimensional spaces. Izv. RAN. Ser. Mat. 65, 201–224 (2001)

    MATH  Article  Google Scholar 

  41. 41.

    Komatsu, H.: Ultradistributions: I. Structure theorems and characterization. J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 20, 25–105 (1973)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Lee, S., Chung, S.Y.: The Paley–Wiener theorem by the heat kernel method. Bull. Korean Math. Soc. 35, 441–453 (1998)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Lifly, E., Tikhonov, S.: Weighted Paley–Wiener theorem on the Hilbert transform. C. R. Math. 348, 1253–1258 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Lutsenko, V.I., Yulmukhametov, R.S.: Generalization of the Paley–Wiener theorem in weighted spaces. Math. Notes 48, 1131–1136 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Maergoiz, L.S.: An analog of the Paley–Wiener theorem for entire functions of the space \( W^p_\sigma, 1\le p \le 2\), and some applications. Comput. Methods Funct. Theory 6, 459–469 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Michalik, S.: Laplace ultradistributions supported by a cone. Banach Center Publ. 88, 229–241 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Musin, I.K.: On the Fourier–Laplace representation of analytic functions in tube domains. Collect. Math. 45, 301–308 (1994)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Nessel, R., Wilmes, G.: Nikolskii-type, on, inequalities for orthogonal expansions. In: Approximation Theory II, 479–484, Academic Press, New York (1976)

  49. 49.

    Olafsson, G., Schlichtkrull, H.: A local Paley–Wiener theorem for compact symmetric spaces. Adv. Math. 218, 202–215 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Othmani, Y., Trimèche, K.: Real Paley–Wiener theorems associated with the Weinstein operator. Mediterr. J. Math. 3, 105–118 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Paley, R., Wiener, N.: Fourier Transform in the Complex Domain. American Mathematical Society. Colloquium Publications XIX, New York (1934)

    Google Scholar 

  52. 52.

    Pasquale, A., Branson, T., Olafsson, G.: The Paley–Wiener theorem for the Jacobi transform and the local Huygens’ principle for root systems with even multiplicities. Indag. Math. (N.S) 16, 429–442 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Pasquale, A., Branson, T., Olafsson, G.: The Paley–Wiener theorem and the local Huygens’ principle for compact symmetric spaces: the even multiplicity case. Indag. Math. (N.S) 16, 393–428 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Plansherel, M., Polya, G.: Fonctions entières et entégrales de Fourier multiples. Comment. Math. Helvetici. I. 9, 224–248 (1937)

    Article  Google Scholar 

  55. 55.

    Plansherel, M., Polya, G.: Fonctions entières et entégrales de Fourier multiples. Comment. Math. Helvetici. II 10, 110–163 (1938)

    Article  Google Scholar 

  56. 56.

    Rachdi, L.T., Rouz, A.: On the range of the Fourier transform connected with Riemann–Liouville operator. Ann. Mathématiques Blaise Pascal 16, 355–397 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Self-adjointness, Academic Press, New York, II. Fourier Analysis (1975)

  58. 58.

    Roumieu, C.: Ultra-distributions dé finies sur \({\mathbb{R}}^n\) et sur certaines classes de variétés differentiables. J. Anal. Math. 10, 153–192 (1962)

    MathSciNet  MATH  Article  Google Scholar 

  59. 59.

    Saitoh, S.: Generalizations of Paley–Wiener’s theorem for entire functions of exponential type. Proc. Am. Math. Soc 99, 465–471 (1987)

    MathSciNet  MATH  Google Scholar 

  60. 60.

    Schwartz, L.: Transformation de Laplace des distributions, Comm. Sém. Math. Univ. Lund, 196–206 (1952)

  61. 61.

    Sigurdson, S.: Growth properties of analytic and plurisubharmonic function of finite order. Math. Scand. 59, 235–304 (1986)

    MathSciNet  Article  Google Scholar 

  62. 62.

    Stan, A.: Paley–Wiener theorem for white noise analysis. J. Funct. Anal. 173, 308–327 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  63. 63.

    Stein, E.M.: Functions of exponential type. Ann. Math. 65, 582–592 (1957)

    MathSciNet  MATH  Article  Google Scholar 

  64. 64.

    Suwa, M., Yoshino, K.: A proof of Paley–Wiener theorem for Fourier hyperfunctions with support in a proper convex cone by the heat kernel method. Complex Var. Elliptic Equ. 53, 833–841 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  65. 65.

    Suwa, M., Yoshino, K.: A characterization of tempered distributions with support in a cone by the Heat kernel method and its applications. J. Math. Sci. Univ. Tokyo 11, 75–90 (2004)

    MathSciNet  MATH  Google Scholar 

  66. 66.

    Thangavelu, S.: A Paley–Wiener theorem for the inverse Fourier transform on some homogeneous spaces. Hiroshima Math. J. 37, 145–159 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  67. 67.

    Tuan, V.K.: New type Paley–Wiener theorems for the modified multidimensional Mellin transform. J. Fourier Anal. Appl. 4, 317–328 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  68. 68.

    Tuan, V.K.: Paley–Wiener-type theorems. Fract. Calc. Appl. Anal. 2, 135–143 (1999)

    MathSciNet  MATH  Google Scholar 

  69. 69.

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

    MathSciNet  MATH  Article  Google Scholar 

  70. 70.

    Tuan, V.K.: A Real-Variable Paley–Wiener Theorem for the Dunkl Transform, Abstract and Applied Analysis, pp. 365–371. World Scientific Publications, River Edge (2004)

    Google Scholar 

  71. 71.

    Tuan, V.K.: Paley–Wiener and Boas theorems for singular Sturm-Liouville integral transforms. Adv. Appl. Math 29, 563–580 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  72. 72.

    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 

  73. 73.

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

    MathSciNet  MATH  Article  Google Scholar 

  74. 74.

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

    MathSciNet  MATH  Article  Google Scholar 

  75. 75.

    Tuan, V.K., Yakubovich, S.B.: Donoho–Stark and Paley–Wiener theorems for the G-transform. Adv. Appl. Math. 45, 108–124 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  76. 76.

    Yang, Y., Qian, T.: An elementary proof of the Paley–Wiener theorem in \(\mathbb{C}^m\). Complex Var. 51, 599–609 (2006)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.02-2018.300.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Vu Nhat Huy.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Tao Qian.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bang, H.H., Huy, V.N. New Paley–Wiener Theorems. Complex Anal. Oper. Theory 14, 47 (2020). https://doi.org/10.1007/s11785-020-01005-2

Download citation

Keywords

  • Paley–Wiener theorem
  • Generalized functions

Mathematics Subject Classification

  • 42B10
  • 47A11