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

For two real numbers $c>0, \alpha> -1,$ we study some spectral properties of the weighted finite bilateral Laplace transform operator, defined over the space $E=L^2(I,\omega_{\alpha}),$ $I=[-1,1],$ $\omega_{\alpha}(x)=(1-x^2)^{\alpha},$ by ${\displaystyle \mathcal L_c^{\alpha} f(x)= \int_I e^{cxy} f(y) \omega_{\alpha}(y)\, dy}.$ In particular, we use a technique based on the Min-Max theorem to prove that the sequence of the eigenvalues 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 the largest eigenvalue of the operator $\mathcal L_c^{\alpha}.$ Also, we give some local estimates and bounds of the eigenfunctions $\varphi_{n,c}^{\alpha}$ of $\mathcal L_c^{\alpha}.$ Moreover, we show that these eigenfunctions are good candidates for the spectral approximation of a function that can be written as a weighted finite Laplace transform of an other $L^2(I,\omega_{\alpha})-$function. Finally, we give some numerical examples that illustrate the different results of this work. In particular, we provide an example that illustrate the Laplace based spectral method, for the inversion of the finite Laplace transform.