  Generalized eigenfunctions may be regarded as vectors of a basis in a
particular direct integral of Hilbert spaces or as elements of the antidual
space $\Phi^\times$ in a convenient Gelfand triplet
$\Phi\subseteq{\mathcal\H}\subseteq\Phi^\times$. This work presents a fit
treatment for computational purposes of transformations formulas relating
different generalized bases of eigenfunctions in both frameworks direct
integrals and Gelfand triplets. Transformation formulas look like usual in
Physics literature, as limits of integral functionals but with well defined
kernels. Several approaches are feasible. Here Vitali and martingale approaches
are showed.
