A Convolution Related to the Inverse Kontorovich-Lebedev Transform
DOI:
https://doi.org/10.5644/SJM.03.2.08Keywords:
Kontorovich-Lebedev transform, modified Bessel function, convolution integral equations, Plancherel theorem, multiplication theoremAbstract
We establish the mapping properties in the space $L_2({\mathbb
R}_+;\frac{dt}{t\sinh \nu t})$, $0<\nu\le\pi$ for a convolution related to the transformation
\[
F(x)=\int\limits_0^\infty f(t) K_{it}(x)\,dt, \quad x\in{\mathbb R}_{+}
\]
involving the modified Bessel function $K_{it}(x)$ as a kernel. It is shown that the convolution operator of two $L_2$-functions exists as a Lebesgue integral and represents a continuous function on ${\mathbb R}_+$. As a consequence, we get the multiplication theorem for two modified Bessel functions of different subscripts. Further applications to the corresponding class of convolution integral equations are obtained.
2000 Mathematics Subject Classification. 44A15, 44A05, 44A35, 33C10, 45A05
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References
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