A generalized Fourier series is the expansion of a square integrable function into a sum of square integrable orthogonal basis functions. The standard Fourier series uses an orthonormal basis of trigonometric functions, and the series expansion is applied to periodic functions. In contrast, a generalized Fourier series uses any set of orthogonal basis functions and can apply to any square integrable function.

Consider a set ${\displaystyle \Phi =\{\phi _{n}:[a,b]\to \mathbb {C} \}_{n=0}^{\infty }}$ of square-integrable complex valued functions defined on the closed interval ${\displaystyle [a,b]}$ that are pairwise orthogonal under the weighted inner product:

${\displaystyle \langle f,g\rangle _{w}=\int _{a}^{b}f(x){\overline {g(x)}}w(x)dx,}$

where ${\displaystyle w(x)}$ is a weight function and ${\displaystyle {\overline {g}}}$ is the complex conjugate of ${\displaystyle g}$. Then, the generalized Fourier series of a function ${\displaystyle f}$ is: $${\displaystyle f(x)=\sum _{n=0}^{\infty }c_{n}\phi _{n}(x),}$$where the coefficients are given by: $${\displaystyle c_{n}={\langle f,\phi _{n}\rangle _{w} \over \|\phi _{n}\|_{w}^{2}}.}$$

Given the space ${\displaystyle L^{2}(a,b)}$ of square integrable functions defined on a given interval, one can find orthogonal bases by considering a class of boundary value problems on the interval ${\displaystyle [a,b]}$ called regular Sturm-Liouville problems. These are defined as follows, $${\displaystyle (rf')'+pf+\lambda wf=0}$$ $${\displaystyle B_{1}(f)=B_{2}(f)=0}$$ where ${\displaystyle r,r'}$ and ${\displaystyle p}$ are real and continuous on ${\displaystyle [a,b]}$ and ${\displaystyle r>0}$ on ${\displaystyle [a,b]}$, ${\displaystyle B_{1}}$ and ${\displaystyle B_{2}}$ are self-adjoint boundary conditions, and ${\displaystyle w}$ is a positive continuous functions on ${\displaystyle [a,b]}$.

Given a regular Sturm-Liouville problem as defined above, the set ${\displaystyle \{\phi _{n}\}_{1}^{\infty }}$ of eigenfunctions corresponding to the distinct eigenvalue solutions to the problem form an orthogonal basis for ${\displaystyle L^{2}(a,b)}$ with respect to the weighted inner product ${\displaystyle \langle \cdot ,\cdot \rangle _{w}}$. We also have that for a function ${\displaystyle f\in L^{2}(a,b)}$ that satisfies the boundary conditions of this Sturm-Liouville problem, the series ${\displaystyle \sum _{n=1}^{\infty }\langle f,\phi _{n}\rangle \phi _{n}}$ converges uniformly to ${\displaystyle f}$.

A function ${\displaystyle f(x)}$ defined on the entire number line is called periodic with period ${\displaystyle T}$ if a number ${\displaystyle T>0}$ exists such that, for any real number ${\displaystyle x}$, the equality ${\displaystyle f(x+T)=f(x)}$ holds.

If a function is periodic with period ${\displaystyle T}$, then it is also periodic with periods ${\displaystyle 2T}$, ${\displaystyle 3T}$, and so on. Usually, the period of a function is understood as the smallest such number ${\displaystyle T}$. However, for some functions, arbitrarily small values of ${\displaystyle T}$ exist.