Calculus How To

Orthonormal Functions

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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) = a0u0 + a1u1 + a2u2 + a3u3 + …

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):

  • v1 = {(1,0,0)}
  • v1 = {(0,1,0)}
  • v1 = {(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 ofn 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<1sub>1(t), u(t), …, un(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.


Bishop, D. (1993). Group Theory and Chemistry. Dover Publications.
Errede, S. (2008). Supplemental Handout #1: Orthogonal Functions & Expansions. Retrieved November 20, 2020 from:
Gram-Schmidt. 21. Orthonormal Representation of Signals. Retrieved November 20, 2020 from:
Levitus, M. (2020). Orthogonal Expansions. licensed by CC BY-NC-SA 3.0. Retrieved November 20, 2020 from:

Stephanie Glen. "Orthonormal Functions" From Calculus for the rest of us!

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