Generalized Fourier series

In mathematics, a generalized Fourier series expands a square-integrable function defined on an interval over the real line. The constituent functions in the series expansion form an orthonormal basis of an inner product space. While a Fourier series expansion consists of only trigonometric functions, a generalized Fourier series is a decomposition involving any set of functions that satisfy the Sturm-Liouville eigenvalue problem. These expansions find common use in interpolation theory. It is expressed by a series of sinusoids that can be stated in various forms. In essence, a pair of functions is considered, where t is a variable (usually time), and m and n are real multipliers of t, reflecting the length of the interval.

Definition
Consider a set of square-integrable functions with values in $$ \mathbb{F} = \Complex$$ or $$\mathbb{F} = \R$$, $$\Phi = \{\varphi_n:[a,b] \to \mathbb{F}\}_{n=0}^\infty,$$ which are pairwise orthogonal under the inner product $$\langle f, g\rangle_w = \int_a^b f(x)\,\overline{g}(x)\,w(x)\,dx,$$ where $$w(x)$$ is a weight function, and $$\overline g$$ represents complex conjugation, i.e., $$\overline{g}(x) = g(x)$$ for $$ \mathbb{F} = \R$$.

The generalized Fourier series of a square-integrable function $$f : [a, b] \to \mathbb{F}$$, with respect to Φ, is then $$f(x) \sim \sum_{n=0}^\infty c_n\varphi_n(x),$$ where the coefficients are given by $$c_n = {\langle f, \varphi_n \rangle_w\over \|\varphi_n\|_w^2}.$$

If Φ is a complete set, i.e., an orthogonal basis of the space of all square-integrable functions on [a, b], as opposed to a smaller orthogonal set, the relation $$\sim $$ becomes equality in the L2 sense, more precisely modulo $$|\cdot|_w $$ (not necessarily pointwise, nor almost everywhere).

Example (Fourier–Legendre series)
1. Trigonometric system.

Definition: a function $$f(x)$$ defined on the entire number line is called periodic if there is a number $$T>0$$ such that $$\forall x:f(x+T)=f(x)$$. The number $$T$$ is called the period of the function.

Note that if the number $$T$$ is the period of the function, then the numbers $$2T, 3T,...$$ are also the periods of this function. Usually, the period of a function is understood as its smallest period (if it exists, see point 1 in test questions and assignments).

Sequence of functions:

$$1, cos(x), sin(x), cos(2x) ,sin(2x) ,..., cos(nx) , sin(nx),...$$

This is the sequence of functions called the trigonometric system. Any linear combination of functions of a trigonometric system, including an infinite combination (that is, a series if it converges), is a periodic function with a period of 2π.

In the following we will consider the trigonometric system, as a rule, on the segment [−π, π], sometimes on the segment [0, 2π]. On any segment of length 2π (including on the segments [−π,π] and [0,2π]) the trigonometric system is an orthogonal system. This means that for any two functions of the trigonometric system, the integral of their product over a segment of length 2π is equal to zero. This integral can be treated as a scalar product in the space of functions that are integrable on a given segment of length 2π.

2. Fourier coefficients.

Let the function $$f(x)$$ be defined on the segment [−π, π]. Below in Theorem 1, we will indicate sufficient conditions under which the function $$f(x)$$ can be represented on this segment as a linear combination of functions of the trigonometric system, also referred to as expansion of the function $$f(x)$$ into a trigonometric Fourier series (converging to $$f(x)$$ at all points of the segment [−π,π] except, perhaps, for a finite number of points).

The Legendre polynomials are solutions to the Sturm–Liouville problem


 * $$ \left((1-x^2)P_n'(x)\right)'+n(n+1)P_n(x)=0.$$

As a consequence of Sturm-Liouville theory, these polynomials are orthogonal eigenfunctions with respect to the inner product above with unit weight. We can form a generalized Fourier series (known as a Fourier–Legendre series) involving the Legendre polynomials, and


 * $$f(x) \sim \sum_{n=0}^\infty c_n P_n(x),$$
 * $$c_n = {\langle f, P_n \rangle_w\over \|P_n\|_w^2}$$

As an example, we may calculate the Fourier–Legendre series for $$f(x)=\cos x$$ over $$[-1, 1]$$. We have that,



\begin{align} c_0 & = {\int_{-1}^1 \cos{x}\,dx \over \int_{-1}^1 (1)^2 \,dx} = \sin{1} \\ c_1 & = {\int_{-1}^1 x \cos{x}\,dx \over \int_{-1}^1 x^2 \, dx} = {0 \over 2/3 } =0 \\ c_2 & = {\int_{-1}^1 {3x^2 - 1 \over 2} \cos{x} \, dx \over \int_{-1}^1 {9x^4-6x^2+1 \over 4} \, dx} = {6 \cos{1} - 4\sin{1} \over 2/5 } \end{align} $$

and a series involving these terms would be


 * $$\begin{align}c_2P_2(x)+c_1P_1(x)+c_0P_0(x)&= {5 \over 2} (6 \cos{1} - 4\sin{1})\left({3x^2 - 1 \over 2}\right) + \sin1\\

&= \left({45 \over 2} \cos{1} - 15 \sin{1}\right)x^2+6 \sin{1} - {15 \over 2}\cos{1}\end{align}$$

which differ from $$\cos x$$ by approximately 0.003. It may be advantageous to use such Fourier–Legendre series since the eigenfunctions are all polynomials and hence the integrals and thus the coefficients are easier to calculate.

Coefficient theorems
Some theorems on the coefficients $$c_n$$ include:

Bessel's inequality

 * $$\sum_{n=0}^\infty |c_n|^2\leq\int_a^b|f(x)|^2w(x)\,dx.$$

Parseval's theorem
If Φ is a complete set, then
 * $$ \sum_{n=0}^\infty |c_n|^2 = \int_a^b |f(x)|^2w(x)\, dx.$$