The generalized integral Fourier transform

1Virchenko, NO
2Chetvertak, MO
1NTU of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2National Technical University of Ukraine "Kyiv Polytechnic Institute"
Dopov. Nac. akad. nauk Ukr. 2015, 8:7-12
https://doi.org/10.15407/dopovidi2015.08.007
Section: Mathematics
Language: Ukrainian
Abstract: 
The generalized integral Fourier transform \begin{gather*}
\widetilde{f}(\alpha)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{\infty}
f(x)e^{i{\alpha}x}{_{1}\Phi^{\tau,\beta}_{1}}(a;c;-r({\alpha}x))\,dx,
\end{gather*}
where $\rm{Re}\,\it{c}> \rm{Re}\,\it{a}>0$, $ ({\tau,\beta})\subset R $, $ \tau-\beta<1$, $r>0$, and  ${{_{1}\Phi}^{\tau,\beta}_{1}}(\ldots )$ is the $(\tau,\beta)$-confluent hypergeometric function, is introduced. The inversion formula of this integral transform is proved. The basic properties of a new integral Fourier transform and some examples are given.
Keywords: confluent hypergeometric function, Fourier’ integral transform, generalized integral transform
References: 
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