Quadrature formulas for integrals transforms generated by orthogonal polynomials
Provides theoretical quadrature rules for specific integral transforms, but the work is purely analytical with no demonstrated practical impact or empirical validation.
The authors derive quadrature formulas for integral transforms generated by classical orthogonal polynomials (Hermite, Laguerre, Jacobi) using recurrence relations and Christoffel-Darboux-type functions. No numerical results or comparisons are provided.
By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, the Christoffel-Darboux-type bilinear generating function and their asymptotic expressions, we obtain quadrature formulas for integral transforms generated by the classical orthogonal polynomials. These integral transforms, related to the so-called Poisson integrals, correspond to a modified Fourier Transform in the case of the Hermite polynomials, a Bessel Transform in the case of the Laguerre polynomials and to an Appell Transform in the case of the Jacobi polynomials.