**Orthonormal functions **are orthogonal and normalized. They are often used to find approximations for other difficult to compute functions. You can construct *any* well behaved function *f* using a complete set of orthonormal functions (Errede, 2008):

f(x) = a_{0}u_{0} + a_{1}u_{1} + a_{2}u_{2} + a_{3}u_{3} + …

A “well behaved” function on an interval a ≥ x ≥ b is univariate, finite-valued and continuous (at least piecewise continuous).

The idea of creating new functions from a linear combination of orthogonal functions is similar to constructing a three-dimensional vector by combining the vectors (Levitus, 2020):

**v**_{1}= {(1,0,0)}**v**_{1}= {(0,1,0)}**v**_{1}= {(0,0,1)}

## Properties of Orthogonal Functions

Orthonormal functions are *normalized* and *mutually orthogonal*; They are orthogonal functions with the property that the inner product of_{n} with itself is 1. Orthonormal functions are always linearly independent, which means that the maximum number of them in general n-dimensional space is equal to n. This set of linearly-independent, mutually orthogonal, normalized functions in n-dimensional space is called an **orthonormal basis function** (Bishop, 1993).

## Example: Orthonormal Functions and Representation of Signals

A set of signals can be represented by a set of orthonormal basis functions; All possible

linear combinations are called a *signal space* (which is a function-space coordinate system). The coordinate axes in this space are the orthonormal functions u<_{1}sub>1(t), u(t), …, u_{n}(t). The major benefit of performing this series expansion is that once this space has been created, any signal can be represented as a point in space, which means that you can use “usual” vector theory.

## References

Bishop, D. (1993). Group Theory and Chemistry. Dover Publications.

Errede, S. (2008). Supplemental Handout #1: Orthogonal Functions & Expansions. Retrieved November 20, 2020 from: https://courses.physics.illinois.edu/phys435/sp2010/Lecture_Notes/P435_Supp_HO_01.pdf

Gram-Schmidt. 21. Orthonormal Representation of Signals. Retrieved November 20, 2020 from: http://charleslee.yolasite.com/resources/elec321/lect_gram_schmidt.pdf

Levitus, M. (2020). Orthogonal Expansions. licensed by CC BY-NC-SA 3.0. Retrieved November 20, 2020 from: https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Book%3A_Mathematical_Methods_in_Chemistry_(Levitus)/07%3A_Fourier_Series/7.03%3A_Orthogonal_Expansions

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