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Jean Baptiste Joseph Fourier (1798–1830) was a French mathematician who is most widely recognized for his development of what we now call Fourier Series. In ABE425, we do not solve differential equations, but we use Fourier Series to analyze signals by creating frequency spectra. Let's assume we have an interval $[0,2\pi]$ on which our signal (input function) is defined. According to Fourier we can now write our input function $f(t)$ as the sum of a constant and a sum of cosine and sine functions of varying frequency $n$ (also known as basis functions) as follows $$ f(t) = c_0 + \sum_{n=1}^{\infty} a_n \cos(nt) + b_n\sin(nt) \tag{11.1} $$

According to a textbook that my professor wrote, it is defined as shown above. But I wanted a more specific/deeper explanation of what exactly this "basis function" is. Can someone explain what basis function is with examples if possible and why it is applicable in Fourier series?

Xander Henderson
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CuriousJ
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The basis functions are the constant function $1$, $\cos (nt)$, and $\sin (nt)$. This is exactly the sense of a basis in linear algebra. It is a linearly independent set that spans the space, so every function in the space can be uniquely represented by a linear combination of them. I could make a new basis of $1, (\cos (nt)+\sin(nt)), (\cos(nt)-\sin(nt))$. The Fourier basis is more traditional and somewhat more intuitive, but mine is also a basis and might be more appropriate for some problems. It is easy to find the trasformation between my basis and the Fourier one.

To avoid issues of infinite sums, we can consider the case of the finite Fourier transform. You are given a list of values of function as $f(k)$, where $k$ is the sample number. You can think of the basis of this representation as being $f_n(k)$ where $f_n(k)$ is $1$ when $k=n$ and $0$ otherwise. When you do an FFT you transform from that basis to your frequency basis. The FFT is basically a matrix multiply, with the matrix being the basis change matrix.

Ross Millikan
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    It's not exactly like linear algebra, since linear combinations in linear algebra are defined to be finite. When you go to infinite linear combinations you have a Schauder basis and not a Hamel basis. https://en.wikipedia.org/wiki/Schauder_basis – user4894 Apr 24 '19 at 03:44
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    @user4894: that is true, but I thought it a distracting detail in this context. Others may disagree. – Ross Millikan Apr 24 '19 at 03:45
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    As @user4894 points out, there is a difference between an analytic basis and an algebraic basis. When you say that these functions are a basis in "exactly the sense of a basis in linear algebra," this is incorrect. Indeed, these functions don't span the space–the space is the closure of the span. I have no objection to making simplifications, but you should at least make those simplifications clear, and perhaps avoid using adverbs like "exactly" when you are making such simplifications. – Xander Henderson Apr 24 '19 at 04:30
  • @XanderHenderson I accepted Ross's comment that by exactly he meant "exactly, modulo complications that wouldn't be helpful to the OP." – user4894 Apr 24 '19 at 04:33