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2000 | 142 | 3 | 253-267
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Solving dual integral equations on Lebesgue spaces

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Czasopismo
Rocznik
Tom
142
Numer
3
Strony
253-267
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-09-24
poprawiono
2000-04-17
Twórcy
  • Departamento de Matemáticas y Computación, Universidad de La Rioja, Edificio J. L. Vives, Calle Luis de Ulloa s/n, 26004 Logro no, Spain
autor
  • Departamento de Matemáticas, Universidad de Zaragoza, Edificio de Matemáticas, Ciudad Universitaria s/n, 50009 Zaragoza, Spain
  • Departamento de Matemáticas y Computación, Universidad de La Rioja, Edificio J. L. Vives, Calle Luis de Ulloa s/n, 26004 Logro no, Spain
Bibliografia
  • [1] Ó. Ciaurri, J. J. Guadalupe, M. Pérez and J. L. Varona, Mean and almost everywhere convergence of Fourier-Neumann series, J. Math. Anal. Appl. 236 (1999), 125-147.
  • [2] A. J. Durán, On Hankel transform, Proc. Amer. Math. Soc. 110 (1990), 417-424.
  • [3] A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953.
  • [4] A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Tables of Integral Transforms, Vol. II, McGraw-Hill, New York, 1954.
  • [5] C. S. Herz, On the mean inversion of Fourier and Hankel transforms, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 996-999.
  • [6] P. Heywood and P. G. Rooney, A weighted norm inequality for the Hankel transformation, Proc. Roy. Soc. Edinburgh Sect. A 99 (1984), 45-50.
  • [7] H. Hochstadt, Integral Equations, Wiley, New York, 1973.
  • [8] B. Muckenhoupt, Mean convergence of Jacobi series, Proc. Amer. Math. Soc. 23 (1969), 306-310.
  • [9] P. G. Rooney, A technique for studying the boundedness and extendability of certain types of operators, Canad. J. Math. 25 (1973), 1090-1102.
  • [10] I. N. Sneddon, Fourier Transforms, McGraw-Hill, New York, 1951. Republication: Dover, New York, 1995.
  • [11] K. Stempak, Transplanting maximal inequalities between Laguerre and Hankel multipliers, Monatsh. Math. 122 (1996), 187-197.
  • [12] K. Stempak and W. Trebels, Hankel multipliers and transplantation operators, Studia Math. 126 (1997), 51-66.
  • [13] G. Szegő, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, RI, 1975.
  • [14] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford Univ. Press, Oxford, New York, 1937.
  • [15] C. J. Tranter, Integral Transforms in Mathematical Physics, 3th ed., Methuen, London, 1966.
  • [16] C. J. Tranter, Bessel Functions with Some Physical Applications, English Univ. Press, London, 1968.
  • [17] J. L. Varona, Fourier series of functions whose Hankel transform is supported on [0,1], Constr. Approx. 10 (1994), 65-75.
  • [18] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, 1966.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv142i3p253bwm
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