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.^[1] 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 $\mathbb {F} =\mathbb {C}$ or $\mathbb {F} =\mathbb {R}$ , $\Phi =\{\varphi _{n}:[a,b]\to \mathbb {F} \}_{n=0}^{\infty },$ 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 $\mathbb {F} =\mathbb {R}$ .

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).

Examples

Fourier–Legendre series

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

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

The sequence of functions $1,cos(x),sin(x),cos(2x),sin(2x),...,cos(nx),sin(nx),...$ is known as the trigonometric system. Any linear combination of functions of a trigonometric system, including an infinite combination (that is, a converging infinite series), is a periodic function with a period of 2π.

On any segment of length 2π (such as 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.

Let the function $f(x)$ be defined on the segment [−π, π]. Under sufficient conditions, $f(x)$ may be represented on this segment as a linear combination of functions of the trigonometric system, also referred to as the 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. This can be written as 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, the Fourier–Legendre series may be calculated for $f(x)=\cos x$ over $[-1,1]$ . Then,

{\begin{aligned}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{aligned}}

and a series involving these terms would be

{\begin{aligned}c_{2}P_{2}(x)+c_{1}P_{1}(x)+c_{0}P_{0}(x)&={5 \over 2}(6\cos {1}-4\sin {1})\left({3x^{2}-1 \over 2}\right)+\sin 1\\&=\left({45 \over 2}\cos {1}-15\sin {1}\right)x^{2}+6\sin {1}-{15 \over 2}\cos {1}\end{aligned}}

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.