From Online Dictionary of Crystallography
A function is called quasiperiodic if its Fourier transform consists of δ-peaks on positions
for basis vectors ai* in a space of dimension m. If the basis vectors form a basis for the space (n equal to the space dimension, and linearly independent basis vectors over the real numbers) then the function is lattice periodic. If n is larger than the space dimension, then the function is aperiodic.
Sometimes the definition includes that the function is not lattice periodic.
A quasiperiodic function may be expressed in a convergent trigonometric series.
It is a special case of an almost periodic function. An almost periodic function is a function f(r) such that for every small number ε there is a translation a such that the difference between the function and the function shifted over a is smaller than the chosen quantity:
A quasiperiodic function is always an almost periodic function, but the converse is not true.
The theory of almost-periodic functions goes back to the work by H. Bohr.