  This article introduces an effective generalization of the polar flavor of
the Fourier Theorem based on a new method of analysis. Under the premises of
the new theory an ample class of functions become viable as bases, with the
further advantage of using the same basis for analysis and reconstruction. In
fact other tools, like the wavelets, admit specially built nonorthogonal bases
but require different bases for analysis and reconstruction (biorthogonal and
dual bases) and vectorial coordinates; this renders those systems unintuitive
and computing intensive. As an example of the advantages of the new
generalization of the Fourier Theorem, this paper introduces a novel method for
the synthesis that is based on frequency-phase series of square waves (the
equivalent of the polar Fourier Theorem but for nonorthogonal bases). The
resulting synthesizer is very efficient needing only few components, frugal in
terms of computing needs, and viable for many applications.
