The description of the electron wavefunctions in atoms is generalized to the fractional Fourier series. This method introduces a continuous and infinite number of chirp basis sets with linear variation of the frequency to expand the wavefunctions, in which plane-waves are a special case. The chirp characteristics of each basis set can be adjusted through a single parameter. Thus, the basis set cutoff can be optimized variationally. The approach is tested with the expansion of the electron wavefunctions in atoms, and it is shown that chirp basis sets substantially improve the convergence in the description of the electron density. We have found that the natural oscillations of the electron core states are efficiently described in chirp-waves. © 2013 Elsevier B.V. All rights reserved.