On a testing-function space for distributions associated with the Kontorovich-Lebedev transform

• Autores: Semen B. Yakubovich
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 57, Fasc. 3, 2006, págs. 279-293
• Idioma: español
• Enlaces
  • Texto completo
• Resumen

  • We construct a testing-function space, which is equipped with the topology that is generated by $L_{v,p}$-multinorm of the differential operator $A_x = x^2-{x {d\over dx} {[x {d\over dx}]}}$, and its $k$-th iterates $A^k_x$, where $k$ = 0, 1, . . . , and $A^0_x\varphi = \varphi$. Comparing with other testing-function spaces, we introduce in its dual the Kontorovich-Lebedev transformation for distributions with respect to a complex index. The existence, uniqueness, imbedding and inversion properties are investigated. As an application we find a solution of the Dirichlet problem for a wedge for the harmonic type equation in terms of the Kontorovich-Lebedev integral.

• Referencias bibliográficas
  • G. Bugle, Die Kontorovich-Lebedev Transformation und die Mehler-Fock Transformation für Klassen verallgemeinerter Funktionen, Dissertation,...
  • A. Erdélyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Transcendental Functions, II, McGraw-Hill Book Company, Inc., New York,...
  • H.J. Glaeske and A. Hess, A convolution connected with the Kontorovich-Lebedev transform, Math. Z. 193 (1986), 67–78.
  • B. Lisena, On the generalized Kontorovich-Lebedev transform, Rend. Mat. Appl. (7) 9 (1989), 87–101.
  • R.S. Pathak and J.N. Pandey, The Kontorovich-Lebedev transformation of distributions, Math. Z. 165 (1979), 29–51.
  • A.P. Prudnikov, Yu. A. Brychkov, and O.I. Marichev, Integrals and Series, I: Elementary Functions, II: Special Functions, Gordon and Breach...
  • S.B. Yakubovich, Index Transforms, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
  • S.B. Yakubovich, On the Kontorovich-Lebedev transformation, J. Integral Equations Appl. 15 (2003), 95–112.
  • S.B. Yakubovich, The Kontorovich-Lebedev transformation on Sobolev type spaces, Sarajevo Journ. Math. 1(14) (2005), 211–234.
  • S.B. Yakubovich and B. Fisher, On the theory of the Kontorovich-Lebedev transformation on distributions, Proc. Amer. Math. Soc. 122 (1994),...
  • A.I. Zayed, Handbook of Function and Generalized Function Transformations, CRC Press, Boca Raton, FL, 1996.
  • A.H. Zemanian, The Kontorovich-Lebedev transformation on distributions of compact support and its inversion, Math. Proc. Cambridge Philos....
  • A.H. Zemanian, Generalized Integral Transformations, Dover Publications, Inc., New York, 1987.