  According to harmonic analysis (Fourier analysis), any function $f(x)$,
periodic over the interval $[-L, L]$, which satisfies the Dirichlet conditions,
can be developed into an infinite sum (known in the literature as the
trigonometric series, and for which, for reasons which will become evident in
the course of this work, we will use the name of sinusoidal series), consisting
of the weighted components of a complete biortogonal base, formed of the
unitary function 1, of the fundamental harmonics $\sin(\pi x/L)$, even and
$\cos(\pi x/L)$, odd ($2L$-periodic functions) and of the secondary harmonics
$\sin(n\pi x/L)$ and $\cos(n\pi x/L)$ (periodic functions, with period $2L/n$,
where $n\in \mathbb{Z}^+$, positive integers). The coefficients of these
expansions (Fourier coefficients) can be calculated using Euler formulas. We
will generalize this statement and show that the function $f(x)$ can also be
developed into non-sinusoidal periodic series, formed from the sum of the
weighted components of a complete, non-orthogonal base: the unit function 1,
the fundamental quasi-harmonics $g(x)$, even and $h(x)$, odd ($2L$-periodic
functions, with zero mean value over the definition interval) and the secondary
quasi-harmonics $g_n(x)$ and $h_n(x)$ ($2L/n$-periodic functions), where $n\in
\mathbb{Z}^+$. The fundamental quasi-harmonics $g(x)$ and $h(x)$ are any
functions which admit expansions in sinusoidal series (satisfy Dirichlet
conditions, or belong to $L^2$ space). The coefficients of these expansions are
obtained with the help of certain algebraic relationships between the Fourier
coefficients of the expansions of the functions $f(x)$, $g(x)$ and $h(x)$. In
addition to their obvious theoretical importance, these types of expansions can
have practical importance in the approximation of functions and in the
numerical and analytical resolution of certain classes of differential
equations.
