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Mathematics for Physics

263 Orthonormal Set of Functions

A set of functions whose pairwise integrals equal one for a matching pair and zero otherwise that is used to expand a function as a series in that set.

definition [D] (Orthonormal Set of Functions) A specific system of functions within a function space where each member is normalized to unity and all distinct members are mutually orthogonal. This set applies to functions in a Hilbert space, often defined over a specific interval or domain. The set must satisfy the following condition:

  • The integral of the product of one function and the complex conjugate of another over the domain equals the Kronecker delta.

where

  • \(f(x)\) and \(g(x)\) are complex-valued functions of a real variable.
  • \(\bar{f}\) denotes the complex conjugate of the function \(f\).
  • \((f, g)\) represents the inner product defined as the integral \(\int f^*(x)g(x) dx\) over a fundamental domain.
  • \(w(x)\) is an optional positive weight function included in the integral.
  • \(Nf\) is the norm of the function, defined as the inner product of the function with itself.

Note:

  • \(\bar{f}\) is also written \(f^*\).
  • \((f, g)\) is also written \(\langle f | g \rangle\).
  • the orthogonality integral may use another function and the complex conjugate of itself.

263.1 Elementary Example

263.1.1 Simple

An orthonormal set of functions has pairwise integrals equal to \(\delta_{mn}\).

\[ \{ \varphi_{1},\ \varphi_{2} \} \]

\[ \int_{a}^{b} \varphi_{m}(x)\, \overline{\varphi_{n}(x)}\, dx = \delta_{mn} \]

\[ \delta_{11} = 1,\quad \delta_{12} = 0 \]

where

  • \(\varphi_{m}, \varphi_{n}\) are functions on \([a,b]\).
  • \(\delta_{mn}\) is the Kronecker delta.

263.1.2 General

On \([0,2\pi]\), the complex exponentials \(e^{inx}/\sqrt{2\pi}\) for several integers \(n\) form an orthonormal set.

\[ \varphi_{n}(x) = \dfrac{e^{i n x}}{\sqrt{2\pi}},\quad n \in \{ -1,\ 0,\ 1 \} \]

\[ \int_{0}^{2\pi} \varphi_{m}(x)\, \overline{\varphi_{n}(x)}\, dx = \delta_{mn} \]

where

  • \(n\) labels the functions in the set.