Opuscula Math. 26, no. 1 (2006), 161-172

 
Opuscula Mathematica

A distribution associated with the Kontorovich-Lebedev transform

Semyon B. Yakubovich

Abstract. We show that in a sense of distributions \[\lim_{\varepsilon\to 0+} {1\over \pi^2} \tau\sinh\pi\tau \int_{\varepsilon}^{\infty} K_{i\tau}(y)K_{ix}(y){dy\over y} =\delta(\tau-x),\] where \(\delta\) is the Dirac distribution, \(\tau\), \(x\in\mathbb{R}\) and \(K_{\nu}(x)\) is the modified Bessel function. The convergence is in \(\mathcal{E}^{\prime}(\mathbb{R})\) for any even \(\varphi(x)\in\mathcal{E}(\mathbb{R})\) being a restriction to \(\mathbb{R}\) of a function \(\varphi(z)\) analytic in a horizontal open strip \(G_a=\{z\in\mathbb{C}\colon\,|\text{Im}\,z|\lt a, \ a\gt 0\}\) and continuous in the strip closure. Moreover, it satisfies the condition \(\varphi(z)=O\bigl(|z|^{-\text{Im}\,z-\alpha}e^{-\pi|\text{Re}\,z|/2}\bigr)\), \(|\text{Re}\,z|\to\infty\), \(\alpha\gt 1\) uniformly in \(\overline{G_a}\). The result is applied to prove the representation theorem for the inverse Kontorovich-Lebedev transformation on distributions.

Keywords: Kontorovich-Lebedev transform, distributions, modified Bessel functions.

Mathematics Subject Classification: 46F12, 44A15, 33C10.

Full text (pdf)

  • Semyon B. Yakubovich
  • University of Porto, Faculty of Sciences, Department of Pure Mathematics, Campo Alegre st., 687, 4169-007 Porto, Portugal
  • Received: 2006-01-26.
Opuscula Mathematica - cover

Cite this article as:
Semyon B. Yakubovich, A distribution associated with the Kontorovich-Lebedev transform, Opuscula Math. 26, no. 1 (2006), 161-172

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.