  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
sens. The GOSWFs are nothing but the eigenfunctions of the finite weighted
bilateral Laplace transform $\mathcal{F}_c^{(\alpha,\beta)}.$ We compute this
functions by two methods: In the first one we use a differential operator
$\mathcal{D}$ which commutes with $\mathcal{F}_c^{(\alpha,\beta)}.$ In the
second one we use the Gaussian quadrature method. As an application, we use the
GOSWFs to invert the finite bilateral Laplace transform. We also use the GOSWFs
to approximate bilateral weighted Laplace bandlimited functions and we show
that they are more advantageous then other classical basis of
$L^2((-1,1),(1-x)^\alpha(1+x)^\beta)dx$. Finally, we provide the reader by some
numerical examples that illustrate the theoretical results.
