A class of index transforms generated by the Mellin and Laplace operators

作者：Yakubovich Semyon^*

来源：Journal of Mathematical Analysis and Applications, 2013, 403(2): 333-343.

DOI：10.1016/j.jmaa.2012.12.009

• Mellin transform
• Laplace transform
• Kontorovich-Lebedev transform
• Modified Bessel functions
• Hypergeometric functions

Classical integral representation of the Mellin type kernel %26lt;br%26gt;x(-z) = 1/Gamma(Z) integral(infinity)(0) e(-xt) t(z-1)dt, x %26gt; 0, Re z %26gt; 0, %26lt;br%26gt;in terms of the Laplace integral gives an idea to construct a class of non-convolution (index) transforms with the kernel %26lt;br%26gt;k(z)(+/-)(x) = integral(infinity)(0) e(-xt +/- 1)/r(t) t(z-1)dt, x %26gt; 0, %26lt;br%26gt;where r(t) not equal 0, t is an element of R+ admits a power series expansion, which has an infinite radius of convergence and the integral converges absolutely in a half-plane of the complex plane z. Particular examples give the Kontorovich-Lebedev-like transformation and new transformations with hypergeometric functions as kernels. Mapping properties and inversion formulas are obtained. Finally we prove a new inversion theorem for the modified Kontorovich-Lebedev transform.

• 出版日期2013-7-15