  Quantum Mechanics and Signal Processing in the line R, are strictly related
to Fourier Transform and Weyl-Heisenberg algebra. We discuss here the addition
of a new discrete variable that measures the degree of the Hermite functions
and allows to obtain the projective algebra io(2). A Rigged Hilbert space is
found and a new discrete basis in R obtained. The operators {O[R]} defined on R
are shown to belong to the Universal Enveloping Algebra UEA[io(2)] allowing, in
this way, their algebraic discussion. Introducing in the half-line a
Fourier-like Transform, the procedure is extended to R^+ and can be easily
generalized to R^n and to spherical reference systems.
