Generalized Fourier series

In mathematics, in the area of functional analysis, a generalized Fourier series corresponds to the series expansion of square-integrable functions with respect to an orthogonal basis. In contrast to a Fourier series, in which the series expansion is applied specifically to periodic functions using an orthonormal basis of trigonometric functions, i.e., an orthogonal basis of solutions to a Laplacian eigenvalue problem on an interval with periodic boundary conditions, a generalized Fourier series can apply to any functions that satisfy the boundary conditions of a Sturm-Liouville eigenvalue problem on any interval, using any orthogonal basis of solutions to such an eigenvalue problem on that interval with those boundary conditions.

Definition

Consider a set $\Phi =\{\varphi _{n}:[a,b]\to \mathbb {C} \}_{n=0}^{\infty }$ of square-integrable functions values over $\mathbb {C}$ that 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. The generalized Fourier series 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 an orthogonal basis, then the relation $\sim$ becomes equality in the L² sense. If Φ is in addition an orthonormal basis, then the denominators $\|\varphi _{n}^{2}\|_{w}^{2}=1$ for all n.

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]. Given appropriate smoothness and differentiability 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.

The Legendre polynomials $P_{n}(x)$ are solutions to the Sturm–Liouville eigenvalue 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 with unit weight. This can be written as a generalized Fourier series (known in this case as a Fourier–Legendre series) involving the Legendre polynomials, so that

$f(x)\sim \sum _{n=0}^{\infty }c_{n}P_{n}(x),$

$c_{n}={\langle f,P_{n}\rangle _{w} \over \|P_{n}\|_{w}^{2}}$

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 truncated series involving only 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 differs from $\cos x$ by approximately 0.003. In computational applications it may be advantageous to use such Fourier–Legendre series rather than Fourier series since the basis functions for the series expansion are all polynomials and hence the integrals and thus the coefficients may be easier to calculate.

Coefficient theorems

Some theorems on the series coefficients $c_{n}$ include:

Bessel's inequality

$\sum _{n=0}^{\infty }|c_{n}|^{2}\leq \int _{a}^{b}|f(x)|^{2}w(x)\,dx.$

Parseval's theorem

If Φ is a complete basis, then

$\sum _{n=0}^{\infty }|c_{n}|^{2}=\int _{a}^{b}|f(x)|^{2}w(x)\,dx.$

See also

• Banach space
• Eigenfunctions
• Fractional Fourier transform
• Function space
• Hilbert space
• Least-squares spectral analysis
• Orthogonal function
• Orthogonality
• Topological vector space
• Vector space

References

• Generalized Fourier Series at MathWorld