The Obrechkoff transform on spaces of generalized functions. (English) Zbl 0989.46025
The integral transformation introduced by N. Obreshkov [Izv. Mat. Institut Bulg. Akad. Sci., Sofia, 3-28 (1958, in Bulgarian); English transl. in East J. Approx. 3, No. 1, 89-110 (1997; Zbl 0896.45003)] stands for a generalization of the classical Laplace and Meijer transforms, and is closely related to a general differential operator of Bessel-type (also called “hyper-Bessel differential operator”):
\[
B= x^{\alpha_0}{d\over dx} x^{\alpha_1}{d\over dx}\cdots x^{\alpha_{m-1}}{d\over dx} x^{\alpha_m}= x^{-\beta} \prod^m_{j=1} \Biggl(x{d\over dx}+ \beta\gamma_k\Biggr)
\]
with \(m\in\mathbb{N}\), \(\beta= m-(\alpha_0+ \alpha_1+\cdots+ \alpha_m)> 0\), \(\gamma_k= (\alpha_k+ \alpha_{k+1}+\cdots+ \alpha_m- m+ k)/\beta\), \(k= 1,\dots, m\).
In the present paper the authors study the Obreshkov transform in the following form: \[ O\{f(t); z\}=\beta z^{-\beta(\gamma_m+ 1)+ 1} \int^\infty_0 G^{m,0}_{0,m}((zt)^\beta\mid (\gamma_k+ 1-1/\beta)^m_1) f(t) dt, \] where \(G^{m,0}_{0,m}\) is an interesting particular case of the Meijer-\(G\) function [see V. Kiryakova, “Generalized fractional calculus and applications”, Pitman Res. Notes Math. Ser. 301 (1994; Zbl 0882.26003)] and analyze \(O\{f(t); z\}\) on the McBride functional space \(F_{p,\mu}\) (\(\mu\in\mathbb{C}\), \(1\leq p<\infty\)) [see A. C. McBridge, “Pitman Res. Notes Math. 31 (1979; Zbl 0423.46029)] by means of the kernel method. Properties of the generalized transform, such as analyticity, boundedness and an inversion formula are established, thus extending results given by A. Baier and H.-J. Glaeske [Math. Nachr. 159, 311-322 (1992; Zbl 0774.46025)] and J. J. Betancor and L. Rodriguez-Mesa [Math. Nachr. 185, 21-31 (1997; Zbl 0902.46019)].
In the present paper the authors study the Obreshkov transform in the following form: \[ O\{f(t); z\}=\beta z^{-\beta(\gamma_m+ 1)+ 1} \int^\infty_0 G^{m,0}_{0,m}((zt)^\beta\mid (\gamma_k+ 1-1/\beta)^m_1) f(t) dt, \] where \(G^{m,0}_{0,m}\) is an interesting particular case of the Meijer-\(G\) function [see V. Kiryakova, “Generalized fractional calculus and applications”, Pitman Res. Notes Math. Ser. 301 (1994; Zbl 0882.26003)] and analyze \(O\{f(t); z\}\) on the McBride functional space \(F_{p,\mu}\) (\(\mu\in\mathbb{C}\), \(1\leq p<\infty\)) [see A. C. McBridge, “Pitman Res. Notes Math. 31 (1979; Zbl 0423.46029)] by means of the kernel method. Properties of the generalized transform, such as analyticity, boundedness and an inversion formula are established, thus extending results given by A. Baier and H.-J. Glaeske [Math. Nachr. 159, 311-322 (1992; Zbl 0774.46025)] and J. J. Betancor and L. Rodriguez-Mesa [Math. Nachr. 185, 21-31 (1997; Zbl 0902.46019)].
Reviewer: Nácere Hayek (La Laguna)
MSC:
| 46F12 | Integral transforms in distribution spaces |
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
Keywords:
hyper-Bessel differential operator; Meijer transforms; general differential operator of Bessel-type; Obreshkov transform; McBride functional space; kernel method; analyticity; boundedness; inversion formulaReferences:
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