Fourier Cosine Series

A Fourier cosine series is simply the same idea as the sine series, but built from cosine waves plus one flat level to start from. (Same Fourier, just the other family of waves.) It's the natural choice when the ends are insulated instead of held at zero, and it's the math behind JPEG image compression. The formula $f(x) = A_0 + \sum A_n\cos\frac{n\pi x}{L}$ looks busy, but it's just two simple parts: $A_0$ (the average height of $f$, a flat line) and the sum of $A_n\cos\frac{n\pi x}{L}$ (waves that bend that flat line into the right shape, with $A_n$ saying how much of each wave to use). The coefficient formulas are the same "multiply and integrate" trick from the sine series. It's the last lesson again with one small twist, so bear with it.

Sine or cosine? On $[0, L]$ you can choose. Physical boundary conditions decide: ends held at zero call for sines; insulated ends (zero derivative) call for cosines. Those are exactly the two sets of eigenfunctions from the eigenvalue lesson, and the cosine set included the constant, which is where $A_0$ comes from.

Every cosine is even, so a cosine series builds the even extension of $f$: on $[-L, 0]$ it gives $f(-x)$, a mirror image instead of a flipped one.

A common mistake. It's tempting to compute $A_0$ with $\frac2L$ in front, like every other coefficient, because the formulas look so alike. It isn't the same: the $A_n$ get $\frac2L$ because $\int_0^L\cos^2\frac{n\pi x}{L}\,dx = \frac L2$, but the constant term has $\int_0^L 1^2\,dx = L$. So $A_0 = \frac1L\int_0^L f\,dx$, which is just the average of $f$.

The cosine series of $f(x) = x$ on $[0, \pi]$: $A_0 = \frac\pi2$, $A_n = \frac{2}{\pi n^2}\left((-1)^n - 1\right)$, so the odd terms are $-\frac{4}{\pi n^2}$. These shrink like $\frac{1}{n^2}$, faster than the sine series of the same function, because the even extension of $x$ has no jumps. Think about why: the odd extension of $x$ jumps from $L$ to $-L$, but the even extension is $|x|$, which only has corners. Corners are much easier to build out of smooth waves than jumps are. The graph further down shows how quickly it closes in.

JPEG compression uses a discrete cosine transform: images are stored as cosine coefficients, and the small high-frequency ones are thrown away.