Integral transforms of the Kontorovich-Lebedev convolution type

• Autores: Semen B. Yakubovich
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 54, Fasc. 2, 2003, págs. 99-110
• Idioma: inglés
• Títulos paralelos:
  • Transformadas integrales del tipo de convolución de Kontorovich-Lebedev
• Enlaces
  • Texto completo
• Resumen

  • We deal with a class of integral transformations of the form \begin{flushleft}$f(x) \rightarrow\frac{1}{2x}\prod_{n=1}^\infty(1+\frac{x(x-\frac{d}{dx}-\frac{d^2} {dx^2})}{(2n-1)^2})\int_{\mathbb{R}_+^2}e^{-{\frac{1}{2} (x\frac{u^2+y^2} {uy}+ \frac{yu}{x})}_{f(u)h(y)dudy,x\in\mathbb{R}+}}$ \end{flushleft}in $L_2(\mathbb {R}_+;xdx)$, which is associated with the Kontorovich-Lebedev operator \begin{center} $K_{i\tau}[f]=\int_0^\infty K_{i\tau}(x)f(x)dx,\tau\in\mathbb{R}+$.

    \end{center} Necessary and sufficient conditions on h to establish that the transformation is unitary in $L_2(\mathbb{R}+;xdx)$ are obtained. A reciprocal inversion formula and an example of the unitary convolution transformation are given.

• Referencias bibliográficas
  • N.I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965.
  • C. Berg, Indeterminate moment problems and the theory of entire functions, J. Comput. Appl. Math. 65 (1995), 27–55.
  • C. Berg and A.J. Duran, The index of determinacy for measures and the $\ell^2$-norm of orthonormal polynomials, Trans. Amer. Math. Soc. 347...
  • C. Berg and A.J. Duran, When does a discrete differential perturbation of a sequence of orthonormal polynomials belong to $\ell^2$?, J. Funct....
  • C. Berg and A.J. Duran, Orthogonal polynomials, $\ell^2$-spaces and entire functions, Math. Scand. 79 (1996), 209–223.
  • C. Berg and A.J. Duran, Measures with finite index of determinacy or a mathematical model for Dr. Jekyll and Mr. Hyde, Proc. Amer. Math. Soc....
  • C. Berg and J.P.R. Christensen, Density questions in the classical theory of moments, Ann. Inst. Fourier (Grenoble) 31 (1981), 99–114.
  • C. Berg, J.P.R. Christensen, and C.U. Jensen, A remark on the multidimensional moment problem, Math. Ann. 243 (1979), 163–169.
  • C. Berg, J.P.R. Christensen, and P. Ressel, Harmonic analysis on semigroups. Theory of positive definite and related functions, Springer-Verlag,...
  • T.M. Bisgaard, Extensions of Hamburger's theorem, Semigroup Forum 57 (1998), 397–429.
  • T.M. Bisgaard and P. Ressel, Unique disintegration of arbitrary positive definite functions on \*-divisible semigroups, Math. Z. 200 (1989),...
  • G. Choquet, Lectures on analysis, W\.A. Benjamin, Inc., New York-Amsterdam, 1969.
  • H.L. Hamburger, Über eine Erweiterung des Stieltjesschen Momentenproblems, Math. Ann. 81 (1920), 235–319; 82 (1921), 120–164, 168–187.
  • Y. Nakamura and N. Sakakibara, Perfectness of certain subsemigroups of a perfect semigroup, Math. Ann. 287 (1990), 213–220.
  • K. Schmüdgen, An example of a positive polynomial which is not a sum of squares of polynomials. A positive, but not strongly positive functional,...
  • C. Fox, The $G$- and $H$-functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98 (1961), 395–429.
  • I.I. Hirschman and D.V. Widder, The Convolution Transform, Princeton University Press, Princeton, N.J., 1955.
  • F. Al-Musallam and V.K. Tuan, Integral transforms related to a generalized convolution, Results Math. 38 (2000), 197–208.
  • A.P. Prudnikov, Yu.A. Brychkov, and O.I. Marichev, Integrals and Series. Vol. 2, Special Functions, Gordon and Breach Science Publishers,...
  • I.N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York, 1972.
  • E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Chelsea Publishing Co., New York, 1986.
  • V.K. Tuan, Integral transforms of Fourier cosine convolution type, J. Math. Anal. Appl. 229 (1999), 519–529.
  • G.N. Watson, General transforms, Proc. London Math. Soc. (2) 35 (1933), 156–199.
  • S.B. Yakubovich, On the convolution for the Kontorovich-Lebedev transformation and its applications to integral equations, Dokl. Akad. Nauk...
  • S.B. Yakubovich, The Kontorovich-Lebedev type transforms and their convolutions, Complex analysis and generalized functions (Varna, 1991),...
  • S.B. Yakubovich and A.I. Moshinsky, Integral equations and convolutions associated with transformations of Kontorovich-Lebedev type, Differentsial'nye...
  • S.B. Yakubovich and Yu.F. Luchko, Operational properties of convolution for the Kontorovich-Lebedev transformation, Dokl. Akad. Nauk Belarusii...
  • S.B. Yakubovich and Yu.F. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions, Mathematics and its Applications 287,...
  • S.B. Yakubovich, Index Transforms, World Scientific Publishing Co., Inc., River Edge, N.J., 1996.
  • S.B. Yakubovich, Index Transforms and Convolution Method, Dr. Sc. Thesis, Minsk, 1996.