Solving dual integral equations on Lebesgue spaces

Óscar Ciaurri; José Guadalupe; Mario Pérez; Juan Varona

Studia Mathematica (2000)

  • Volume: 142, Issue: 3, page 253-267
  • ISSN: 0039-3223

Abstract

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We study dual integral equations associated with Hankel transforms, that is, dual integral equations of Titchmarsh’s type. We reformulate these equations giving a better description in terms of continuous operators on L p spaces, and we solve them in these spaces. The solution is given both as an operator described in terms of integrals and as a series ∑ n = 0 ∞ c n J μ + 2 n + 1 which converges in the L p -norm and almost everywhere, where J ν denotes the Bessel function of order ν. Finally, we study the uniqueness of the solution.

How to cite

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Ciaurri, Óscar, et al. "Solving dual integral equations on Lebesgue spaces." Studia Mathematica 142.3 (2000): 253-267. <http://eudml.org/doc/216802>.

@article{Ciaurri2000,
abstract = {We study dual integral equations associated with Hankel transforms, that is, dual integral equations of Titchmarsh’s type. We reformulate these equations giving a better description in terms of continuous operators on $L^p$ spaces, and we solve them in these spaces. The solution is given both as an operator described in terms of integrals and as a series $∑_\{n=0\}^\{∞\} c_n J_\{μ+2n+1\}$ which converges in the $L^p$-norm and almost everywhere, where $J_ν$ denotes the Bessel function of order ν. Finally, we study the uniqueness of the solution.},
author = {Ciaurri, Óscar, Guadalupe, José, Pérez, Mario, Varona, Juan},
journal = {Studia Mathematica},
keywords = {Fourier series; Hankel transform; Bessel functions; dual integral equations; dual integral equations of Titchmarsh type; uniqueness; Jacobi polynomials; Fourier-Neumann series; convergence; almost everywhere convergence},
language = {eng},
number = {3},
pages = {253-267},
title = {Solving dual integral equations on Lebesgue spaces},
url = {http://eudml.org/doc/216802},
volume = {142},
year = {2000},
}

TY - JOUR
AU - Ciaurri, Óscar
AU - Guadalupe, José
AU - Pérez, Mario
AU - Varona, Juan
TI - Solving dual integral equations on Lebesgue spaces
JO - Studia Mathematica
PY - 2000
VL - 142
IS - 3
SP - 253
EP - 267
AB - We study dual integral equations associated with Hankel transforms, that is, dual integral equations of Titchmarsh’s type. We reformulate these equations giving a better description in terms of continuous operators on $L^p$ spaces, and we solve them in these spaces. The solution is given both as an operator described in terms of integrals and as a series $∑_{n=0}^{∞} c_n J_{μ+2n+1}$ which converges in the $L^p$-norm and almost everywhere, where $J_ν$ denotes the Bessel function of order ν. Finally, we study the uniqueness of the solution.
LA - eng
KW - Fourier series; Hankel transform; Bessel functions; dual integral equations; dual integral equations of Titchmarsh type; uniqueness; Jacobi polynomials; Fourier-Neumann series; convergence; almost everywhere convergence
UR - http://eudml.org/doc/216802
ER -

References

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  12. [12] K. Stempak and W. Trebels, Hankel multipliers and transplantation operators, Studia Math. 126 (1997), 51-66. Zbl1030.42009
  13. [13] G. Szegő, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, RI, 1975. 
  14. [14] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford Univ. Press, Oxford, New York, 1937. Zbl0017.40404
  15. [15] C. J. Tranter, Integral Transforms in Mathematical Physics, 3th ed., Methuen, London, 1966. Zbl0135.16002
  16. [16] C. J. Tranter, Bessel Functions with Some Physical Applications, English Univ. Press, London, 1968. Zbl0174.36203
  17. [17] J. L. Varona, Fourier series of functions whose Hankel transform is supported on [0,1], Constr. Approx. 10 (1994), 65-75. Zbl0803.42015
  18. [18] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, 1966. Zbl0174.36202

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