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9.2 General Fourier Series and Convergence

Consider a function \( f(x) \) with period \( 2\pi \) given on the interval \( -\pi < x < \pi \).

\[ f(x) \sim \frac{1}{2} a_0 + \sum_{n=1}^{\infty} [a_n \cos(nx) + b_n \sin(nx)] \]

Where the Fourier coefficients are defined as:

\[ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) dx \]
\[ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) dx \]

Generalizing the Period

First, let's generalize the period. Now consider a period of \( 2L \) (where \( L \) is the half-period), on the interval \( -L < t < L \).

Define the transformation:

\[ t = \frac{L}{\pi} x \]
A Cartesian coordinate graph showing the linear relationship between x and t. The horizontal axis is labeled x with tick marks at -\pi and \pi. The vertical axis is labeled t with tick marks at -L and L. A green straight line passes through the origin (0,0), connecting the points (-\pi, -L) and (\pi, L), representing the function t = \frac{L}{\pi}x.
Graph of the transformation \( t = \frac{L}{\pi} x \)

Mapping points:

  • \( x = -\pi \rightarrow t = -L \)
  • \( x = \pi \rightarrow t = L \)
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\( x = \frac{\pi}{L} t \)\( dx = \frac{\pi}{L} dt \)

\[ f(t) \sim \frac{1}{2} a_0 + \sum_{n=1}^{\infty} \left[ a_n \cos\left( \frac{n \pi}{L} t \right) + b_n \sin\left( \frac{n \pi}{L} t \right) \right] \]

\[ a_n = \frac{1}{\pi} \int_{-L}^{L} f(t) \cos\left( \frac{n \pi}{L} t \right) \cdot \frac{\pi}{L} dt \]
\[ a_n = \frac{1}{L} \int_{-L}^{L} f(t) \cos\left( \frac{n \pi}{L} t \right) dt \]
\[ b_n = \frac{1}{L} \int_{-L}^{L} f(t) \sin\left( \frac{n \pi}{L} t \right) dt \]

Example

\[ f(t) = \begin{cases} -1 & -4 < t < 0 \\ 1 & 0 < t < 4 \end{cases} \]

period 8 \( (L=4) \)

A graph of a square wave function f(t) with a period of 8. The horizontal t-axis ranges from approximately -8 to 8, and the vertical f(t) axis ranges from -1 to 1. The function is a step function: it is equal to 1 for t between 0 and 4, and equal to -1 for t between -4 and 0. This pattern repeats periodically. Vertical dashed lines indicate the jumps at t = -4, 0, and 4.
Graph of the square wave function \( f(t) \) with period 8.
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Fourier Series Coefficients and Expansion

Calculating Coefficients

\[ a_0 = \frac{1}{4} \int_{-4}^{4} f(t) dt = 0 \]

Note: The integral \( \int_{-4}^{4} f(t) dt \) represents the net area under \( f(t) \).

\[ a_n = \frac{1}{4} \int_{-4}^{4} f(t) \cos\left( \frac{n \pi t}{4} \right) dt = \dots = 0 \]
\[ b_n = \frac{1}{4} \int_{-4}^{4} f(t) \sin\left( \frac{n \pi t}{4} \right) dt = \dots = \frac{2}{n \pi} \left[ 1 - \cos(n \pi) \right] \]
\[ = \frac{2}{n \pi} \left[ 1 - (-1)^n \right] = \begin{cases} \frac{4}{n \pi} & n \text{ is odd} \\ 0 & n \text{ is even} \end{cases} \]

Fourier Series Expansion

So, the Fourier series for \( f(t) \) is:

\[ f(t) \sim \sum_{n=1}^{\infty} \frac{2}{n \pi} \left[ 1 - (-1)^n \right] \sin\left( \frac{n \pi t}{4} \right) \]
\[ \sim \sum_{n \text{ odd}}^{\infty} \frac{4}{n \pi} \sin\left( \frac{n \pi t}{4} \right) \]

Expanding the first few terms:

\[ \frac{4}{\pi} \sin\left( \frac{\pi}{4} t \right) + \frac{4}{3 \pi} \sin\left( \frac{3 \pi}{4} t \right) + \frac{4}{5 \pi} \sin\left( \frac{5 \pi}{4} t \right) + \dots \]
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Fourier Series Approximation of a Square Wave

Visualizing the convergence of partial sums for a periodic function with period \( T = 8 \).

Graphical Representation

A graph titled 'Fourier Series Approximation of a Square Wave (T=8)'. The horizontal axis represents time t from -8 to 8, and the vertical axis represents f(t) from -1.0 to 1.0. A dashed gray line shows the original piecewise square wave, which alternates between 1 and -1 every 4 units. Overlaid are four Fourier series approximations: a blue curve for N=1 term (a simple sine wave), an orange curve for N=3 terms, a green curve for N=11 terms, and a red curve for N=51 terms. As N increases, the approximation becomes flatter on the plateaus and steeper at the transitions, though high-frequency oscillations (Gibbs phenomenon) remain visible near the discontinuities.
Figure 1: Comparison of partial sums \( S_N(t) \) for varying numbers of terms \( N \).

Key Observations

  • Fundamental Frequency (\( N=1 \)): The first term (blue) provides a basic sinusoidal approximation of the square wave's period.
  • Higher Order Harmonics: As more terms are added (\( N=3, 11, 51 \)), the sum more closely resembles the sharp edges of the square wave.
  • Gibbs Phenomenon: Note the persistent overshoots and oscillations near the jump discontinuities at \( t = -4, 0, 4, 8 \), which do not disappear even as \( N \) becomes large.

Mathematical Context

The square wave shown is an odd function, meaning its Fourier series consists only of sine terms: \[ f(t) = \frac{4}{\pi} \sum_{n=1,3,5,\dots}^{\infty} \frac{1}{n} \sin\left(\frac{n\pi t}{4}\right) \]

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Fourier Series Example

Function Definition

Consider the periodic function \( f(t) \) defined piecewise over one period:

\[ f(t) = \begin{cases} -1 - t & -1 < t < 0 \\ 1 - t & 0 < t < 1 \end{cases} \]

The function has a period of 2, which implies \( L = 1 \).

Visual Representation

A graph of a sawtooth-like periodic function f(t) against time t . The function consists of parallel downward-sloping line segments. For each period, the segment starts at a value of 1 when t is just above 0 and slopes down to -1 as t approaches 1. There is a jump discontinuity at every integer value of t . The graph shows several periods, with x-intercepts at t = -1, 1, and 3 , and y-intercepts at 1 and -1.
Figure 1: Plot of the periodic function \( f(t) \) showing its sawtooth behavior and discontinuities.

Fourier Coefficients Calculation

We calculate the Fourier coefficients \( a_0 \), \( a_n \), and \( b_n \) for the given function:

\[ a_0 = \frac{1}{1} \int_{-1}^{1} f(t) \, dt = 0 \]
\[ a_n = \frac{1}{1} \int_{-1}^{1} f(t) \cos(n\pi t) \, dt = \dots = 0 \]
\[ b_n = \frac{1}{1} \int_{-1}^{1} f(t) \sin(n\pi t) \, dt = \frac{2}{n\pi} \]

Fourier Series Expansion

Substituting the coefficients back into the Fourier series formula, we obtain the series representation for \( f(t) \):

\[ f(t) \sim \sum_{n=1}^{\infty} \frac{2}{n\pi} \sin(n\pi t) \sim \frac{2}{\pi} \sin(\pi t) + \frac{2}{2\pi} \sin(2\pi t) + \frac{2}{3\pi} \sin(3\pi t) + \dots \]
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Fourier Series: Sawtooth Wave Convergence (T=2)

Visualizing the approximation of a periodic sawtooth function using partial sums of its Fourier series.

A graph showing the Fourier series approximation of a sawtooth wave with period T=2 . The horizontal axis represents time t from -3 to 3, and the vertical axis represents f(t) from -1.0 to 1.0. The original sawtooth wave is shown as a dashed gray line, consisting of linear segments with a slope of 1 that jump from 1 to -1 at every even integer. Four partial sums are plotted: N=1 (blue sine wave), N=2 (orange curve with two peaks per period), N=5 (green curve showing more oscillations), and N=50 (red curve closely following the sawtooth shape but exhibiting high-frequency oscillations near the discontinuities, known as the Gibbs phenomenon).
Figure 1: Convergence of the Fourier series for a sawtooth wave as the number of terms \( N \) increases.

Key Observations

  • Fundamental Frequency (N=1): The blue curve represents the first harmonic, a simple sine wave \( \sin(\pi t) \) that captures the basic periodicity but lacks the sharp features of the sawtooth.
  • Increasing Precision: As the number of terms \( N \) increases from 2 to 50, the partial sum more closely matches the linear slope of the original function \( f(t) \).
  • Gibbs Phenomenon: Even at \( N=50 \) (red line), there are visible overshoots and high-frequency oscillations near the points of discontinuity (e.g., at \( t = -2, 0, 2 \)). This is a characteristic behavior of Fourier series when approximating functions with jump discontinuities.

Mathematical Context

The sawtooth wave with period \( T=2 \) can be represented by the Fourier series:

\[ f(t) = \frac{2}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin(n \pi t) \]

The graph illustrates the partial sums \( S_N(t) \) where the summation goes from \( n=1 \) to \( N \).

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Two main features: at discontinuity, Fourier series goes through the middle (average)

A Cartesian coordinate system showing a periodic function approximation. The graph displays a wave that oscillates significantly near vertical jumps (discontinuities), a characteristic of the Gibbs phenomenon. At the exact point of the jump on the y-axis and at subsequent periods, a distinct dot is placed at the vertical midpoint of the jump, illustrating that the Fourier series converges to the average value \frac{f(t^-) + f(t^+)}{2} at points of discontinuity.

right before discontinuity → \( f(t^-) \)

right after → \( f(t^+) \)

\[ \frac{f(t^-) + f(t^+)}{2} \]

Why? 1st example

\[ f(t) \sim \sum_{n=1}^{\infty} \frac{2}{n\pi} [1 - (-1)^n] \sin\left(\frac{n\pi t}{4}\right) \]

At \( t = 0 \), all \( \sin\left(\frac{n\pi t}{4}\right) = 0 \)

\( f(t) \) discontinuous at \( t = 0 \)

(-1 right before, 1 right after)

Gibbs phenomenon

Overshoot right before discontinuity.

Increasing \( n \) doesn't make it go away but pushes it closer to discontinuity.

Always \( \sim 9\% \) of the jump

(artifact noticeable as sound or light)