Convergence Results for an Extension of the Fourier Transform

In a preceding paper (see2), for any function f such that: $${x^k}f\left( x \right) \in L\left( {a,b} \right),\forall k \in \mathbb{N} \cup \left\{ 0 \right\},$$ , we have considered the integral transform: 1.1 $$ \hat f\left( y \right): = \int\limits_a^b {F\left( {x,y} \right)f\left( x \right)} dx, $$ related to the kernel F(x,y) which is the generating function of a set of polynomials orthogonal in (a,b) with respect to the weight W(x) (shortly O.P.S.). We have proved that the integral transforms of this kind can be considered in some sense as a generalization of the Fourier transform, since they formally verify a property which is analogous to a known property of this classical operator.