Opuscula Mathematica

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

A distribution associated with the Kontorovich-Lebedev transform


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.
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.

RSS Feed

horizontal rule

ISSN 1232−9274, e-ISSN 2300−6919, DOI https://doi.org/10.7494/OpMath
Copyright © 2003−2017 OPUSCULA MATHEMATICA
Contact: opuscula@agh.edu.pl
Made by Tomasz Zabawa

horizontal rule

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.