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?
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.