User:Brews ohare/Matrix (Function spaces)

This article supplements Matrix (mathematics) by providing a more detailed discussion of matrices used in connection with function spaces and basis functions as used, for example, in quantum mechanics. In this article we consider functions ψ that can be expressed in terms of a basis set {φ_j} in the form:

$\psi =\sum _{j=1}^{n}\ c_{j}\ \phi _{j}\ ,$

and operators on these functions, say O, such that:

$\chi =O\ \psi \ .$

Supposing for simplicity that the basis set is orthogonal, so:

$\int \ dx\ \phi _{i}^{*}\phi _{j}=\delta _{i,j}\ ,$

with δ_ij the Kronicker delta, a matrix corresponding to the operator O is defined by:

$O_{i,j}=\int \ dx\ \phi _{i}^{*}O\phi _{j}=(\phi _{i},\ O\phi _{j})\ .$

Here the integral over x could be a many-dimensional integration over many variables, and the * indicates complex conjugation.

Functions and function spaces

Matrices have application in connection with generalized Fourier series, and their generalizations. The basic idea is that a basis set of functions {φ_n} can be used to form functions ψ = Σ c_n φ_n that are superpositions of the basis set. This relation sets up a correspondence ψ → {c_n} that can be viewed as a column vector c. Then an operation on the function ψ results in a new function, say χ, and therefore results in a new superposition χ = Σ d_n φ_n and a new correspondence χ → {d_n} to the new column vector d. The operation is then represented by a matrix, say M:

$d_{m}=\Sigma _{n}M_{m,n}c_{n}\ ,\ \mathrm {or} \ \mathbf {d} ={\boldsymbol {M}}\cdot \mathbf {c} \ .$

Needless to say, there are a number of mathematical requirements to make sure this idea works. For example, these questions are addressed:

1. What functions ψ can be expressed by the basis set {φ_n}? This topic is that of completeness of the basis functions.
2. What limitations are placed on operators upon these functions to insure they result in new functions that also are expressible using the basis set {φ_n}?
3. What meaning attaches to infinitely many basis functions (n → ∞)?
4. What are the relative merits of one basis compared to another?
5. How are the elements M_m,n corresponding to an operation determined?

These and other issues are topics in the theory of function spaces like Hilbert space, Banach space and the like.

These ideas have important application to numerical analysis where the basis functions are used to approximate differential equations with matrix equations, for example, using the Galerkin method to introduce powerful techniques like the finite element method. The basis functions in such methods often are chosen to be localized to small regions, for example, the so-called hat functions, or in one dimension, the triangular functions.