Bispectral and (glN,glM) dualities, discrete versus differential

We define integral transforms which establish a correspondence between spaces of quasi-exponentials and spaces of quasi-polynomials under certain conditions. As a corollary of the properties of our transforms we obtain a correspondence between solutions to the Bethe ansatz equations of two (glN,glM) dual quantum integrable models: one is the special trigonometric Gaudin model and the other is the special XXX model.