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Details for article 45 of 91 found articles
| Title: |
GENERALIZED FRACTIONAL INTEGRAL TRANSFORMS WITH GAUSS FUNCTION KERNELS AS G-TRANSFORMS |
| Author: |
Kilbas, Anatoly A. Repin, Oleg A. Saigo, Megumi |
| Appeared in: |
Integral transforms and special functions |
| Paging: | Volume 13 (2002) nr. 3 pages 285-307 |
| Year: |
2002 |
| Contents: |
The paper is devoted to the study of the generalized fractional integral transforms (I^{\alpha,\beta,\eta}_{0+}f)(x)={x^{-\alpha -\beta}\over \Gamma (\alpha)} \vint^{x}_{0}(x-t)^{\alpha -1}\ _{2}F_{1}\left(\alpha +\beta,-\eta;\alpha; 1-{t\over x}\right)f(t)\;{\rm d}t and (I^{\alpha,\beta,\eta}_{-}f)(x)={1\over \Gamma (\alpha)} \vint^{\infty}_{x}(t-x)^{\alpha -1}\ _{2}F_{1}\left(\alpha +\beta,-\eta;\alpha; 1-{x\over t}\right)t^{-\alpha -\beta}f(t)\;{\rm d}t , with \alpha, \beta, \eta \in {\bf C} \ ({\rm Re}(\alpha )\gt 0) involving the Gauss hypergeometric function _{2}F_{1}(\alpha +\beta,-\eta;\alpha;z) in the kernels, and of two their modifications. It is proved that the considered fractional constructions can be represented as the integral transforms involving Meijer's G -function as kernels. On the basis of these representations mapping properties such as the boundedness, the representation and the range of I^{\alpha,\beta,\eta}_{0+} and I^{\alpha,\beta,\eta}_{-} are proved in the space {\cal L}_{\nu,r} of Lebesgue measurable functions f on {\bf R}_{+}=(0,\infty) such that \vint^{\infty}_{0}\vert t^{\nu}f(t) \vert^{r}{{\rm d}t\over t}\lt \infty \;\;(1\leq r \lt\infty),\quad \mathop{\rm ess\ sup}_{t\gt 0}[t^{\nu}\vert\, f(t)\vert ]\lt\infty \;\; (r=\infty) , for \nu \in {\bf R}=(-\infty,\infty) , coinciding with the space L^{r}(0,\infty) when \nu =1/r . Similar results are obtained for two modifications of the transforms I^{\alpha,\beta,\eta}_{0+}f and I^{\alpha,\beta,\eta}_{-}f and for Liouville and Kober'fractional integral transforms. |
| Publisher: |
Taylor & Francis |
| Source file: |
Elektronische Wetenschappelijke Tijdschriften |
Details for article 45 of 91 found articles
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