On the inversion of some integral transforms by the use of the generalized oblate spheroidal wave functions

ABSTRACT

In this paper, we introduce a new set of functions, which have the property of the completeness over a finite and infinite intervals. This family of functions, denoted for simplicity GOSWFs, are a generalization of the oblate spheroidal wave functions. They generalize also the Jacobi polynomials in some sense. The GOSWFs are nothing but the eigenfunctions of the finite weighted bilateral Laplace transform ${\mathcal{F}}_c^{(\alpha ,\beta )}.$ We compute this functions by the use the Gaussian quadrature method. As an application, we use the GOSWFs to invert the finite bilateral Laplace transform as well as the finite Laplace transform and the Mellin transform. Finally, we provide the reader by some numerical examples that illustrate the theoretical results. We reconstruct signals without noise and others with noise. We have used three kinds of noisy input signals: Sinusoidal noise given by $\eta (x)=\delta sin(\mu x),$ Here $\delta$ and $\mu$ are two real parameters. The second type of noise is a functional noise given by: $\eta (x)=0.001(x-0.01x^2-exp(-x)).$ The third kind of the signal noise is the random noise which is normally distributed with mean 0 and standard deviation $\delta .$

KEYWORDS:

• Jacobi polynomials
• oblate spheroidal wave functions
• bilateral weighted Laplace transform
• generalized Oblate spheroidal wave functions

AMS SUBJECT CLASSIFICATIONS:

• Primary: 45A05
• 33E30
• Secondary: 42C10
• 34L10

Acknowledgements

The authors would like to thank Deanship of Scientific Research of Majmaah University, Saudi Arabia for the financial grant received in conducting this research.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Deanship of Scientific Research of Majmaah University, Saudi Arabia.