Questions: Fourier Series: Definition and Coefficients
3 questions to test your understanding
Score: 0 / 3
Question 1 Multiple Choice
The Fourier coefficient a₁ for f on [-L, L] is computed as which of the following?
A(1/L) ∫₋ₗᴸ f(x) dx
B(1/L) ∫₋ₗᴸ f(x) cos(πx/L) dx
C(2/L) ∫₋ₗᴸ f(x) cos(πx/L) dx
D(2/L) ∫₋ₗᴸ f(x) sin(πx/L) dx
The general formula is aₙ = (1/L) ∫₋ₗᴸ f(x) cos(nπx/L) dx for n ≥ 1, but the factor convention produces (1/L) only when the a₀ term is written as a₀/2. The correct formula with the standard normalization is aₙ = (1/L) ∫₋ₗᴸ f(x) cos(nπx/L) dx. Option C uses (2/L), which corresponds to the convention where the series constant term is a₀ (not a₀/2). The key insight is that orthogonality of the cosine functions under integration isolates each coefficient.
Question 2 True / False
A Fourier series typically converges to f(x) at most point where f is defined.
TTrue
FFalse
Answer: False
At a jump discontinuity, the Fourier series converges to the average of the left- and right-hand limits, not to f(x) itself. Additionally, near a jump discontinuity the partial sums exhibit the Gibbs phenomenon — an overshoot of about 9% that does not disappear as more terms are added. Pointwise convergence to f(x) requires additional conditions (e.g., f is continuous and piecewise smooth).
Question 3 Short Answer
Why do sines and cosines form a useful basis for representing arbitrary periodic functions?
Think about your answer, then reveal below.
Model answer: The sine and cosine functions on [-L, L] are orthogonal: the integral of the product of any two distinct basis functions over the interval is zero. This orthogonality means we can isolate each coefficient by multiplying both sides by a single basis function and integrating — all other terms vanish.
Orthogonality is the key structural property. Just as perpendicular vectors in a plane have zero dot product (allowing you to find components independently), orthogonal functions have zero inner product (the integral of their product). This lets you compute each Fourier coefficient aₙ or bₙ independently of all the others — a property that would fail if the basis functions were not orthogonal.