Weighted finite Laplace transform operator: spectral analysis and quality of approximation by its eigenfunctions

For two real numbers c > 0, a > -1, we study some spectral properties of the weighted finite bilateral Laplace transform operator, defined over the space E = L2( I,.a), I = [-1, 1],.a( x) = ( 1 -x2) a, by Lac f ( x) = I ecxyf ( y).a( 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 ec, for the largest eigenvalue of the operatorLac. Also, wegive some local estimates and bounds of the eigenfunctions.a n, c ofLac. 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 L2( I,.a)-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.