Fourier series

At the jump discontinuities, the Fourier series approaches the midpoint of the one-sided limits. For the specified square wave, the left limit is $-1$ and the right limit is $1$, so the series converges to $0$ at the jumps.

Conditions: The point is a jump discontinuity.; The one-sided limits exist and are finite.

Pointwise convergence requires that for every fixed $x$, $S_N(x) \to f(x)$. This holds true except at the jump, where it converges to the midpoint.

Conditions: Comparing definitions of pointwise and uniform convergence.; Observing the error bound near the discontinuity.

The nonzero sine coefficients occur at odd frequencies and are given by the formula $4/[\pi(2k+1)]$. These coefficients are strictly positive and do not alternate in sign, meaning every odd harmonic adds constructively to the approximation of the square wave's flat sections.

Conditions: The function is the specified $2\pi$-periodic square wave.; Coefficients are indexed by $k \ge 0$ corresponding to frequency $2k+1$.

Starting with the fundamental sine wave provides the basic oscillation. Adding higher-order odd harmonics (frequency 3, 5, etc.) introduces faster oscillations.

Conditions: Using the partial sum $S_N(x)=\frac4\pi\sum_{k=0}^{N-1}\frac{\sin((2k+1)x)}{2k+1}$.; $N$ increases to include more harmonics.

The nonzero sine coefficients occur only at odd frequencies ($n=1, 3, 5, \dots$). The coefficient for the harmonic $2k+1$ is given by $\frac{4}{\pi(2k+1)}$.

Conditions: The square wave is defined as 1 on $(0, \pi)$ and -1 on $(-\pi, 0)$.; Only odd harmonics are considered.

As more terms are added, the location of the overshoot narrows toward the jump discontinuity, but the relative peak height of the overshoot does not vanish. It persists as a fixed percentage above the flat level, distinguishing pointwise convergence from uniform convergence.

Conditions: The series is the Fourier expansion of a function with a jump discontinuity.; The number of terms $N$ approaches infinity.

By starting with the fundamental sine wave and sequentially adding higher odd frequencies (3, 5, etc.), the partial sums interfere constructively in the middle of the half-periods to flatten the peaks, and destructively near the boundaries to steepen the transitions toward vertical jumps.

Conditions: The series consists of the fundamental and odd harmonics.; Terms are added sequentially to form partial sums.

The series converges pointwise because at every continuous point it approaches the function value, and at jumps it approaches the midpoint. However, it does not converge uniformly because the Gibbs overshoot near the jumps maintains a nonzero relative peak height regardless of how many terms are added, preventing the maximum error from vanishing globally.

Conditions: The function has jump discontinuities.; Convergence is analyzed over the entire period including the jumps.

At a point of discontinuity, the Fourier series converges to the midpoint of the one-sided limits. For the standard square wave jumping from -1 to 1 (or vice versa) at $x=0$, the limit of the partial sums is $\frac{-1 + 1}{2} = 0$.

Conditions: The function satisfies Dirichlet conditions.; There is a finite jump discontinuity at the point of evaluation.

Odd symmetry dictates that the function satisfies $f(-x) = -f(x)$. The constant term represents the average value of the function over a period, which is zero for an odd function.

Conditions: The function is $2\pi$-periodic.; The function is odd, meaning $f(-x) = -f(x)$.

Odd symmetry implies that $f(-x) = -f(x)$. Cosine functions are even ($\cos(-x) = \cos(x)$) and the constant term is also even.

Conditions: The function is periodic with period $2\pi$.; The function satisfies $f(-x) = -f(x)$ almost everywhere.