Orthonormal Expansions

Complete Orthonormal Expansion

A set of functions \(\{\Theta_i(t)\}_{i=1}^{\infty}\) is a complete set of orthonormal functions if the following conditions are satisfied:

• Orthonormality, which combines orthogonality (zero inner product between distinct functions) and normalization (unit length for each function).

• Completeness, which ensures that the orthonormal set forms a complete basis, allowing any function in the space to be expressed as a linear combination of the basis functions.

Orthonormality

Orthonormality is a fundamental property of a set of functions (or vectors) in a function space (or vector space).

A set of functions \(\{\Theta_i(t)\}\) is said to be orthonormal if each pair of distinct functions is orthogonal, and each function is normalized. Formally, this means:

• Orthogonality:
  Two functions \(\Theta_i(t)\) and \(\Theta_j(t)\) are orthogonal if their inner product is zero:

  \[ \langle \Theta_i, \Theta_j \rangle = \int \Theta_i(t) \Theta_j^*(t) \, dt = 0 \quad \text{for } i \neq j. \]

• Normalization:
  Each function is normalized to have unit length:

  \[ \langle \Theta_i, \Theta_i \rangle = \int |\Theta_i(t)|^2 \, dt = 1. \]

Combining these, the set \(\{\Theta_i(t)\}\) satisfies:

\[ \langle \Theta_i, \Theta_j \rangle = \delta_{ij}, \]

where \(\delta_{ij}\) is the Kronecker delta.

Characterization via the Kronecker Delta Function

The Kronecker delta function, denoted by \(\delta_{ij}\), serves as a concise way to express the orthonormality condition. It is defined as:

\[\begin{split} \delta_{ij} = \begin{cases} 1 & \text{if } i = j, \\ 0 & \text{if } i \neq j. \end{cases} \end{split}\]

In the context of orthonormal functions, the inner product between any two functions in the set is given by the Kronecker delta:

\[ \int \Theta_i(t) \Theta_j^*(t) \, dt = \delta_{ij}. \]

This equation succinctly captures both orthogonality (when \(i \neq j\)) and normalization (when \(i = j\)) in a single expression.

Completeness

Completeness ensures that the orthonormal set of functions spans the entire function space under consideration.

This means that any arbitrary function \(s(t)\) within the space can be expressed as a (possibly infinite) linear combination of the orthonormal basis functions:

\[ s(t) = \sum_{i=1}^{\infty} s_i \Theta_i(t), \]

where the coefficients \(s_i\) are determined by the inner product:

\[ s_i = \int s(t) \Theta_i^*(t) \, dt. \]

In other words, completeness guarantees that the orthonormal set is sufficient to represent any function in the space without omission.

It ensures that there are no “gaps” in the representation, making the set a complete basis for the function space.