Fourier Series Properties

Value at Discontinuities

If $f$ has a jump discontinuity at $x_0$, the Fourier series converges to:

$\text{Fourier value} = \frac{f(x_0^-) + f(x_0^+)}{2}$

The average of left and right limits.

Example: For sawtooth wave at $x = 1$:

• $f(1^-) = 1$, $f(1^+) = -1$
• Fourier series value = $\frac{1 + (-1)}{2} = 0$

This matches the series: all $\sin(n\pi) = 0$.

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Parseval’s Theorem

For periodic $f$ with period $2L$:

$\frac{1}{2L}\int_{-L}^{L} [f(x)]^2\,dx = \frac{a_0^2}{4} + \frac{1}{2}\sum_{n=1}^{\infty}(a_n^2 + b_n^2)$

Interpretation: The mean square value equals sum of squares of amplitudes.

Application: Evaluate series.

Example (sawtooth): $f(x) = x$ on $[-1, 1]$.

Left side: $\frac{1}{2}\int_{-1}^{1} x^2\,dx = \frac{1}{3}$

Right side: Using $b_n = \frac{2(-1)^{n+1}}{n\pi}$, get $\frac{2}{\pi^2}\sum \frac{1}{n^2}$

Therefore: $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$

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Gibbs Phenomenon

Near a jump discontinuity, partial Fourier series exhibit overshoot:

• The overshoot approaches ~9% of the jump height
• As more terms are added, overshoot stays constant in height
• Overshoot moves closer to the discontinuity

Example: Square wave $f(x) = \begin{cases} -1 & -1 < x < 0 \\ 1 & 0 < x < 1 \end{cases}$

Even with many terms, partial sums overshoot near $x = 0$.

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Differentiation of Fourier Series

If $f$ is continuous and $f'$ is piecewise smooth, then:

$\frac{df}{dx} \sim \sum_{n=1}^{\infty} \left[\frac{n\pi}{L}b_n \cos\left(\frac{n\pi x}{L}\right) - \frac{n\pi}{L}a_n \sin\left(\frac{n\pi x}{L}\right)\right]$

Warning: Differentiating term-by-term can fail if $f$ has discontinuities (Gibbs phenomenon).

Example: Triangle wave differentiated gives square wave.

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Integration of Fourier Series

If Fourier series for $f$ is known:

$\int f(x)\,dx = \frac{a_0 x}{2} + \sum \left[\frac{a_n L}{n\pi}\sin\left(\frac{n\pi x}{L}\right) - \frac{b_n L}{n\pi}\cos\left(\frac{n\pi x}{L}\right)\right] + C$

Condition: Requires $a_0 = 0$ for the integral to be periodic.

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Decay of Coefficients

Function smoothness	Coefficient decay
Discontinuous	$O(1/n)$
Continuous, $f'$ discontinuous	$O(1/n^2)$
$f$ and $f'$ continuous, $f''$ discontinuous	$O(1/n^3)$
Analytic (smooth)	Exponential

Rule: Smoothness determines how quickly coefficients shrink.

Intuition: High frequencies (large $n$) are needed to capture sharp features.

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Half-Range Expansions

Sine Series (odd extension)

Extend $f(x)$ for $0 < x < L$ as an odd function of period $2L$:

$f(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{L}\right)$

$b_n = \frac{2}{L}\int_0^L f(x)\sin\left(\frac{n\pi x}{L}\right)\,dx$

Cosine Series (even extension)

Extend $f(x)$ for $0 < x < L$ as an even function of period $2L$:

$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right)$

$a_n = \frac{2}{L}\int_0^L f(x)\cos\left(\frac{n\pi x}{L}\right)\,dx$

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Convergence Conditions

Dirichlet conditions: Fourier series converges to $f(x)$ if $f$ is:

1. Single-valued
2. Finite number of maxima and minima in one period
3. Finite number of discontinuities in one period
4. Absolutely integrable over one period

At discontinuities: converges to the average value.

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Why Fourier Analysis Matters

• Signal processing: Decompose signals into frequencies
• PDEs: Solve heat and wave equations (separation of variables)
• Quantum mechanics: Superposition of wave functions
• Audio/visual compression: Encode by frequency content

Fourier series connect the “time domain” (function values) to “frequency domain” (coefficients).