Opuscula Math. 29, no. 3 (2009), 313-329

Opuscula Mathematica

A double index transform with a product of Macdonald's functions revisited

Semyon B. Yakubovich

Abstract. We prove an inversion theorem for a double index transform, which is associated with the product of Macdonald's functions \(K_{i \tau}(\sqrt{x^2+y^2}-y) K_{i \tau}(\sqrt{x^2+y^2}+y)\), where \((x, y) \in \mathbb{R}_+ \times \mathbb{R}_+\) and \(i \tau, \tau \in \mathbb{R}_+\) is a pure imaginary index. The results obtained in the sequel are applied to find particular solutions of integral equations involving the square and the cube of the Macdonald function \(K_{i \tau}(t)\) as a kernel.

Keywords: Macdonald function, index transform, Kontorovich-Lebedev transform, double Mellin transform, Plancherel theorem, Parseval equality.

Mathematics Subject Classification: 44A20, 33C10, 35C15.

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  • Semyon B. Yakubovich
  • University of Porto, Department of Pure Mathematics, Faculty of Science, Campo Alegre st., 687, 4169-007 Porto, Portugal
  • Received: 2009-06-08.
  • Accepted: 2009-06-13.
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Cite this article as:
Semyon B. Yakubovich, A double index transform with a product of Macdonald's functions revisited, Opuscula Math. 29, no. 3 (2009), 313-329, http://dx.doi.org/10.7494/OpMath.2009.29.3.313

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