2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/149106We solve the problem of Fourier transformation for the one-dimensional $q$-deformed Heisenberg algebra. Starting from a matrix representation of this algebra we observe that momentum and position are unbounded operators in the Hilbert space. Therefore, in order to diagonalise the position operator in a momentum eigenbasis we have to study self-adjoint extensions of these operators. It turns out that there exist a whole family of such extensions for the position operator. This leads, correspondingly, to a one-parametric family of Fourier transformations. These transformations, which are related to continued fractions, are constructed in terms of $q$-deformed trigonometric functions. The entire family of the Fourier transformations turns out to be characterised by an elliptic function.19, MPI-PhT/94-36 (14.July 1994)High Energy Physics - Theorytext