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For two real numbers $c>0, \alpha> -1,$ we study some spectral properties ofthe weighted finite bilateral Laplace transform operator, defined over thespace $E=L^2(I,\omega_{\alpha}),$ $I=[-1,1],$$\omega_{\alpha}(x)=(1-x^2)^{\alpha},$ by ${\displaystyle \mathcal L_c^{\alpha}f(x)= \int_I e^{cxy} f(y) \omega_{\alpha}(y)\, dy}.$ In particular, we use atechnique based on the Min-Max theorem to prove that the sequence of theeigenvalues of this operator has a super-exponential decay rate to zero.Moreover, we give a lower bound with a magnitude of order $e^c,$ for thelargest eigenvalue of the operator $\mathcal L_c^{\alpha}.$ Also, we give somelocal estimates and bounds of the eigenfunctions $\varphi_{n,c}^{\alpha}$ of$\mathcal L_c^{\alpha}.$ Moreover, we show that these eigenfunctions are goodcandidates for the spectral approximation of a function that can be written asa weighted finite Laplace transform of an other$L^2(I,\omega_{\alpha})-$function.  Finally, we give some numerical examples that illustrate the differentresults of this work. In particular, we provide an example that illustrate theLaplace based spectral method, for the inversion of the finite Laplacetransform.

Weighted Finite Laplace Transform Operator: Spectral Analysis and Quality of Approximation by its Eigenfunctions