Fourier Convergence, Gibbs Phenomenon, and Half-Range Series

Definition A function $f$ is piecewise continuous on $[a, b]$ if there are finitely many points $a = x_0 < x_1 < \cdots < x_k = b$ such that $f$ is continuous on each open interval $(x_{j-1}, x_j)$ and has finite one-sided limits at both ends of each of these intervals. It is piecewise smooth on $[a, b]$ if, in addition, $f$ is continuously differentiable on each open interval $(x_{j-1}, x_j)$ and $f'$ has finite one-sided limits at both ends of it. A periodic function is piecewise smooth if it is piecewise smooth on one period. The values of $f$ at the break points $x_j$ play no role.

Definition A series of functions converges pointwise on a set if, for each fixed $x$ in the set, its partial sums form a convergent sequence of numbers. The pointwise convergence theorem for Fourier series says the following. Let $f$ be piecewise smooth and periodic with period $2L$, and write $f(x^-)$ and $f(x^+)$ for its left and right limits at $x$. Then at every real $x$ the Fourier series of $f$ converges, and its sum is

At a point where $f$ is continuous this is $f(x)$. At a jump it is the average of the one-sided limits, the midpoint of the jump, whatever value $f$ is given at that point. At the ends $x = \pm L$ of the basic interval the limits come from the periodic extension, so the sum at $x = L$ is $\tfrac{1}{2}\bigl(f(L^-) + f(-L^+)\bigr)$, which is not $f(L^-)$ unless $f(L^-) = f(-L^+)$. If $f$ is also continuous on the whole line, the convergence is uniform.

Definition Let $f$ be periodic and piecewise smooth with a jump at $x_0$ of size $J = f(x_0^+) - f(x_0^-) \ne 0$. The Gibbs phenomenon is the overshoot of the partial sums $S_N f$ next to $x_0$. On the side of the larger one-sided limit the partial sums rise above that limit, and on the other side they fall below the smaller one. As $N \to \infty$, the peak of each overshoot nearest the jump differs from its one-sided limit by an amount that tends to $G\lvert J \rvert$, where

with the integrand taken to be $1$ at $u = 0$. So about $9$ percent of the jump remains on each side, while the peak moves toward $x_0$ and sits at a distance of about $L/N$ from it. The partial sums therefore do not converge uniformly on any interval that has $x_0$ as an endpoint, even though they converge at every point. On every closed interval that contains no jump of the periodic extension the convergence is uniform. Mean-square convergence still holds, and the two facts fit together because the overshoot occupies a strip whose width tends to zero.

Definition Let $f$ be piecewise continuous on $[0, L]$. Its even extension is the function $f_e$ on $(-L, L)$ with $f_e(x) = f(x)$ and $f_e(-x) = f(x)$ for $0 < x < L$. Its odd extension $f_o$ has $f_o(x) = f(x)$ and $f_o(-x) = -f(x)$ for $0 < x < L$, with $f_o(0) = 0$. Each is then repeated with period $2L$. The values at $x = 0$ and $x = \pm L$ affect no Fourier coefficient. The periodic even extension never has a jump at $0$ or at $L$, since its one-sided limits are $f(0^+)$ on both sides of $0$ and $f(L^-)$ on both sides of $L$. The periodic odd extension has a jump at $0$ unless $f(0^+) = 0$, and a jump at $L$ unless $f(L^-) = 0$.

Definition Let $f$ be piecewise continuous on $[0, L]$. Its half-range sine series is the Fourier series of its odd extension,

and its half-range cosine series is the Fourier series of its even extension,

for $n \ge 0$. Both use only the data on $[0, L]$. If $f$ is piecewise smooth on $[0, L]$, both series converge at every interior point $x$ to $\tfrac{1}{2}\bigl(f(x^-) + f(x^+)\bigr)$, which is $f(x)$ where $f$ is continuous. They can differ at the ends. Every term of the sine series is zero at $x = 0$ and at $x = L$, so its sum is $0$ there. The cosine series converges to $f(0^+)$ at $x = 0$ and to $f(L^-)$ at $x = L$. The sine modes satisfy the Dirichlet conditions $X(0) = X(L) = 0$, while the cosine modes, the constant among them, satisfy the Neumann conditions $X'(0) = X'(L) = 0$. The boundary conditions of a problem decide which series is useful.

Definition To differentiate a Fourier series term by term is to replace each term by its derivative, so the amplitudes of the $n$th modes are multiplied by $n\pi/L$, with sines and cosines exchanged. The result need not be the Fourier series of $f'$, and it need not converge at all. A sufficient condition is the following. If $f$ is continuous on the whole line and periodic with period $2L$, and $f'$ is piecewise smooth, then the termwise derivative of the Fourier series of $f$ is the Fourier series of $f'$, and it converges to $\tfrac{1}{2}\bigl(f'(x^-) + f'(x^+)\bigr)$ at every $x$. The reason is integration by parts. For such an $f$ it gives the coefficients $(n\pi/L)\,b_n$ and $-(n\pi/L)\,a_n$ for the cosines and sines of $f'$, exactly those of the termwise derivative, while a jump of $f$ would add boundary terms to these formulas. Termwise integration needs much less. If $f$ is piecewise continuous on $[-L, L]$, then integrating its Fourier series term by term from $-L$ to $x$ gives a series that converges to $\int_{-L}^{x} f(t)\,dt$ for every $x$ in $[-L, L]$, whether or not the Fourier series of $f$ itself converges.