Unit 6: Introduction to Fourier series

I. Orientation — Periodic Functions and Harmonic Representation

A Fourier series represents a periodic function as a sum of constant, cosine, and sine components. Introduced through Joseph Fourier’s work on heat conduction (early nineteenth century), the method converts a function in the physical or time domain into harmonic components with discrete frequencies.

A. introduction and Euler's formulae

The central principle is that, under suitable conditions, a periodic function can be reconstructed from orthogonal trigonometric functions.

• Periodicity: A function has period (T>0) when its values repeat after every interval of length (T).

f(x + T) = f(x)

Here, (f) is the function, (x) is the independent variable, and (T) is its period. The fundamental period is the smallest positive value of (T).

• General Fourier series: For a function of period (2L), the trigonometric expansion is

f(x) ~ a₀/2 + Σₙ₌₁^∞ [aₙ cos(nπx/L) + bₙ sin(nπx/L)]

Here, (L) is half the period; (a_0) is the constant coefficient; (a_n) and (b_n) are Fourier coefficients; and (n) is a positive integer. The symbol (\sim) indicates a Fourier representation whose pointwise value depends on convergence conditions.

• Euler’s formulae for coefficients: Orthogonality isolates each harmonic over a complete interval of length (2L).

a₀ = (1/L) ∫₋Lᴸ f(x) dx

aₙ = (1/L) ∫₋Lᴸ f(x) cos(nπx/L) dx

bₙ = (1/L) ∫₋Lᴸ f(x) sin(nπx/L) dx

The integration variable is (x), and (n=1,2,3,\ldots). The term (a_0/2) equals the mean value of (f) over one period.

• Orthogonality: Distinct sine and cosine harmonics have zero inner product on the symmetric interval.

∫₋Lᴸ cos(mπx/L) cos(nπx/L) dx = 0,  m ≠ n
∫₋Lᴸ sin(mπx/L) sin(nπx/L) dx = 0,  m ≠ n
∫₋Lᴸ sin(mπx/L) cos(nπx/L) dx = 0

Here, (m) and (n) are positive integers. When (m=n), each squared sine or cosine integral equals (L).

• Connection with complex exponentials: Euler’s exponential identities combine sine and cosine into one harmonic.

e^(iθ) = cos θ + i sin θ
cos θ = [e^(iθ) + e^(-iθ)]/2
sin θ = [e^(iθ) - e^(-iθ)]/(2i)

Here, (i^2=-1) and (\theta) is a real angle. Consequently, a Fourier series may also be expressed in complex form using integer-frequency terms (e^{in\pi x/L}).

• Engineering interpretation: Each pair (a_n,b_n) describes the contribution of the (n)-th harmonic. Its amplitude and phase are

Aₙ = √(aₙ² + bₙ²)
φₙ = atan2(bₙ, aₙ)
aₙ cos θ + bₙ sin θ = Aₙ cos(θ - φₙ)

Here, (A_n) is harmonic amplitude, (\phi_n) is phase angle, and (\operatorname{atan2}) selects the correct quadrant.

II. Fourier Expansion — Existence and Convergence

Fourier coefficients can be calculated for many integrable functions, but the value to which the resulting series converges is governed by the local behavior of the periodic extension.

A. conditions for a Fourier expansion and functions having points of discontinuity

The standard Dirichlet conditions provide practical sufficient conditions for pointwise convergence of a Fourier series.

• Dirichlet conditions: Over one period, it is sufficient that (f):
  • is absolutely integrable;
  • has only finitely many finite discontinuities;
  • has only finitely many maxima and minima;
  • is piecewise smooth, or at least piecewise continuously differentiable.

∫₋Lᴸ |f(x)| dx < ∞

These conditions are sufficient rather than necessary: some functions outside this class still possess convergent Fourier expansions.

• Continuous points: If (f) is continuous at (x=x_0), its Fourier series converges there to the function value.

S(x₀) = f(x₀)

Here, (S(x_0)) denotes the sum of the Fourier series at (x_0).

• Jump discontinuities: If finite one-sided limits exist, the series converges to their arithmetic mean.

S(x₀) = [f(x₀⁻) + f(x₀⁺)]/2

Here, (f(x_0^-)) and (f(x_0^+)) are the left- and right-hand limits. The assigned value (f(x_0)) itself does not affect the Fourier coefficients because changing a function at finitely many points does not change its integrals.

• Endpoint behavior: The Fourier series treats the given interval as one period of a periodic function. At (x=-L) and (x=L), it therefore converges to

S(±L) = [f(L⁻) + f(-L⁺)]/2

Thus, even a function continuous on the closed interval may acquire an endpoint jump when its periodic copies are joined.

• Gibbs phenomenon: Near a jump, finite partial sums develop oscillatory overshoots and undershoots. As more terms are included, the oscillations become narrower, but the maximum overshoot approaches approximately (9\%) of the jump magnitude rather than disappearing.

• Mean-square convergence: If (f) is square-integrable, Fourier partial sums converge to (f) in the energy sense.

lim(N→∞) ∫₋Lᴸ |f(x) - S_N(x)|² dx = 0

Here, (S_N) is the partial sum through harmonic (N). This form of convergence is especially important for signal power, vibration energy, and numerical approximation.

III. Interval Transformation — Rescaling the Domain

A Fourier expansion on any finite interval can be obtained by translating and scaling that interval to the standard symmetric interval.

A. change of interval

Changing the interval changes the fundamental frequency and therefore the arguments of all sine and cosine terms.

• Arbitrary interval: For an interval ([\alpha,\beta]), define its midpoint (c) and half-length (L) by

c = (α + β)/2
L = (β - α)/2

Here, (\alpha) and (\beta) are the endpoints. The periodic extension has period (2L=\beta-\alpha).

• Shifted Fourier series: Using the centered variable (x-c), the expansion becomes

f(x) ~ a₀/2
       + Σₙ₌₁^∞ [aₙ cos(nπ(x-c)/L) + bₙ sin(nπ(x-c)/L)]

The coefficients are found by integrating from (\alpha) to (\beta):

a₀ = (1/L) ∫ᵅᵝ f(x) dx

aₙ = (1/L) ∫ᵅᵝ f(x) cos(nπ(x-c)/L) dx

bₙ = (1/L) ∫ᵅᵝ f(x) sin(nπ(x-c)/L) dx

• Normalized coordinate: The substitution

t = π(x-c)/L

maps (x\in[\alpha,\beta]) to (t\in[-\pi,\pi]). Its differential relation is (dx=(L/\pi)\,dt), allowing standard (2\pi)-periodic formulas to be used.

• Choice of full-period interval: Coefficients may be integrated over any consecutive interval of one complete period, such as ([-L,L]), ([0,2L]), or ([q,q+2L]). Here, (q) is any real starting point.

• Frequency effect: The fundamental angular frequency is

ω₀ = π/L = 2π/T

Here, (T=2L) is the period. Increasing the interval length lowers the fundamental frequency and packs the harmonics more closely.

IV. Symmetry — Reduction of Fourier Coefficients

Symmetry determines whether cosine or sine terms vanish, substantially reducing the required integration.

A. even and odd functions

Even functions generate cosine series, whereas odd functions generate sine series on a symmetric interval.

1. Even functions:
   • Definition: An even function satisfies

f(-x) = f(x)

• Coefficient reduction: Because an even function multiplied by sine is odd, every sine coefficient vanishes.

a₀ = (2/L) ∫₀ᴸ f(x) dx
aₙ = (2/L) ∫₀ᴸ f(x) cos(nπx/L) dx
bₙ = 0

• Concrete example: For (f(x)=|x|) on ([-\pi,\pi]), symmetry gives

|x| ~ π/2 - (4/π) Σₙ odd cos(nx)/n²

 The summation is over positive odd integers \(n=1,3,5,\ldots\).

2. Odd functions:
   • Definition: An odd function satisfies

f(-x) = -f(x)

• Coefficient reduction: An odd function multiplied by cosine remains odd, so the constant and cosine coefficients vanish.

a₀ = 0
aₙ = 0
bₙ = (2/L) ∫₀ᴸ f(x) sin(nπx/L) dx

• Parity rules: On ([-L,L]), the integral of an odd integrand is zero, while the integral of an even integrand is twice its integral on ([0,L]). Products obey: even × even = even, odd × odd = even, and even × odd = odd.

• Symmetric decomposition: Any function defined symmetrically can be separated into even and odd parts.

fₑ(x) = [f(x) + f(-x)]/2
fₒ(x) = [f(x) - f(-x)]/2
f(x) = fₑ(x) + fₒ(x)

Here, (f_e) contributes only cosine terms and (f_o) contributes only sine terms.

V. Half-Interval Representation — Sine and Cosine Extensions

When a function is specified only on (0<x<L), it can be extended to (-L<x<0) with chosen symmetry to produce a half-range expansion.

A. half range series

A half-range series represents the same positive-half function either by cosines through even extension or by sines through odd extension.

1. Half-range cosine series:
   • Even extension: Define the negative-half values by (F(-x)=F(x)). The resulting period is (2L).
   • Series and coefficients:

f(x) ~ a₀/2 + Σₙ₌₁^∞ aₙ cos(nπx/L)

a₀ = (2/L) ∫₀ᴸ f(x) dx
aₙ = (2/L) ∫₀ᴸ f(x) cos(nπx/L) dx

• Endpoint meaning: The even extension joins smoothly at (x=0) when appropriate, but its periodic continuation may jump at (x=\pm L).

2. Half-range sine series:
   • Odd extension: Define the negative-half values by (F(-x)=-F(x)).
   • Series and coefficients:

f(x) ~ Σₙ₌₁^∞ bₙ sin(nπx/L)

bₙ = (2/L) ∫₀ᴸ f(x) sin(nπx/L) dx

• Endpoint meaning: Every sine partial sum is zero at (x=0) and (x=L). If the original endpoint limits are nonzero, the periodic odd extension has jumps there and the series converges to the corresponding midpoint values.

• Worked comparison: For (f(x)=x) on (0<x<L), the two valid extensions produce different series outside that interval.

Cosine series:
x ~ L/2 - (4L/π²) Σₙ odd cos(nπx/L)/n²

Sine series:
x ~ (2L/π) Σₙ₌₁^∞ [(-1)^(n+1)/n] sin(nπx/L)

Both represent (x) for interior points (0<x<L), but the cosine series represents an even triangular-type extension, while the sine series represents an odd sawtooth-type extension.

• Engineering use: The choice reflects boundary conditions. Cosine bases naturally model zero-derivative or insulated boundaries, whereas sine bases naturally model zero-value or fixed boundaries in heat flow, wave motion, and vibration problems.