  The famous Fourier theorem states that, under some restrictions, any periodic
function (or real world signal) can be obtained as a sum of sinusoids, and
hence, a technique exists for decomposing a signal into its sinusoidal
components. From this theory an entire branch of research has flourished: from
the Short-Time or Windowed Fourier Transform to the Wavelets, the Frames, and
lately the Generic Frequency Analysis. The aim of this paper is to take the
Frequency Analysis a step further. It will be shown that keeping the same
reconstruction algorithm as the Fourier Theorem but changing to a new computing
method for the analysis phase allows the generalization of the Fourier Theorem
to a large class of nonorthogonal bases. New methods and algorithms can be
employed in function decomposition on such generic bases. It will be shown that
these algorithms are a generalization of the Fourier analysis, i.e. they are
reduced to the familiar Fourier tools when using orthogonal bases. The
differences between this tool and the wavelets and frames theories will be
discussed. Examples of analysis and reconstruction of functions using the given
algorithms and nonorthogonal bases will be given. In this first part the focus
will be on vectorial decomposition, while the second part will be on phased
decomposition. The phased decomposition thanks to a single function basis has
many interesting consequences and applications.
