# Basis functions in Fourier Series

I've doing a bit of self study on the Fourier Transform and I keep coming across "basis function", I have attempted to understand the theory behind it, but I'm finding a lot of explanations go into more maths concepts that I'm not fully understanding.

I'm wondering if someone can in simple english give me an explanation of "Basis Function" that I can understand?

Any help is appreciated

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Do you know linear algebra? A basis function is a function in the basis. The basis element is a function because we are in a function space, $L^2$ or whatever. – Jonas Teuwen Mar 14 '11 at 12:09

## 1 Answer

As Jonas points out, a basis function is simply a function in the basis of a function space. Most likely, you are (whether it explicitly says so or not) working in the function space $L^2[0,1]$ (or something like it), which is an infinite-dimensional Hilbert space.

The functions $\{\exp(2 \pi inx) : n \in \mathbb{Z}\}$ form a infinite (but countable) basis (sometimes called a Schauder basis) for all functions $f \in L^2[0,1]$. That is, every function (in $L^2$) can be expressed as a linear combination of complex exponentials. See, for example, here. A basis function in this sense is simply one of those complex exponentials $f(x) = e^{2 \pi i n x}$.

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