Basis Functions

• is an element of a particular basis for a function space

• every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors

  • i.e. function space is to the basis function as vector space is to the basis vector

Basis Functions - Examples

Given a finite-dimensional vector space (𝑉,𝐹) where 𝑉 is a set of all polynomial functions of degree 𝑘 or less

• example finite basis #1 {1, 𝑥, 𝑥², ..., 𝑥^𝑘} # "standard basis"
• example finite basis #2 {4, 6𝑥, 1𝑥², ..., 2𝑥^𝑘}

Given an infinite-dimensional vector space (𝑉,𝐹) where 𝑉 is a set of all polynomial functions of any degree

• example infinite basis #1 {1, 𝑥, 𝑥², ... } # "standard basis"
• example infinite basis #2 {4, 6𝑥, 1𝑥², ... }

Given an infinite-dimensional vector space (𝑉,𝐹) where 𝑉 is a set of functions, where each function (𝑓) outputs a field (𝐹) (i.e. 𝑓: 𝑋 → 𝐹)

• example infinite basis #1 {1, 𝑥, 𝑥², ..., 𝑐𝑜𝑠(𝑥), ..., 5^𝑥, ... }