4.7.3 · D5Partial Differential Equations
Question bank — Fourier series — motivation from periodic functions
The objects we quiz you on (definitions first)
Everything below refers to a function that repeats. We measure that repeat with a length.
Two more words we lean on:
- A function is ==periodic with period == if for every ; the smallest such positive is the fundamental period.
- Two waves are orthogonal over one period if the integral of their product over that period is — the function analogue of two arrows meeting at a right angle. The picture below shows exactly this. See Orthogonality of Functions.
Look at the shaded area: for two different waves the positive humps and negative dips cancel to zero net area (orthogonal); for a wave times itself the product is never negative, so a real positive area survives — that surviving area is what the coefficient formula reads off.
True or false — justify
TRUE or FALSE: Every periodic function has a convergent Fourier series equal to it at every point.
FALSE — you need the Dirichlet conditions (piecewise-smooth is the usual sufficient case), and even then, at a jump the series converges to the midpoint , not to at that point.
TRUE or FALSE: If is even then every .
TRUE — an even times the odd is odd, and an odd integrand over the symmetric interval integrates to . See Even and Odd Functions.
TRUE or FALSE: If all , then must be even.
FALSE for a raw function, but TRUE for its periodic extension in effect — vanishing says the series (hence the reconstructed periodic signal) is even. A given on a half-interval can be chosen even (cosine series) or odd (sine series); the coefficients follow the extension you pick.
TRUE or FALSE: The constant term equals the average value of over one period.
TRUE — , and dividing that integral by the interval length (the true average) gives exactly .
TRUE or FALSE: Adding more Fourier terms always makes the maximum error near a jump smaller.
FALSE — near a jump the overshoot stays at about of the jump height forever; more terms only squeeze it closer to the jump, not shrink its height. This is the Gibbs Phenomenon.
TRUE or FALSE: and are orthogonal over because they have different frequencies.
TRUE — different integer multiples of the base frequency give ; product-to-sum turns it into of nonzero integer multiples, which integrate over full cycles to .
TRUE or FALSE: and with are orthogonal, but two equal ones are not.
TRUE — the case leaves , which is a positive number (), i.e. the wave has nonzero "length squared"; that surviving term is exactly what isolates the coefficient.
TRUE or FALSE: A pure cosine has the Fourier series consisting of just itself.
TRUE — orthogonality means its only nonzero coefficient is ; the recipe returns the wave you started with, a sanity check on the whole machinery.
TRUE or FALSE: For a piecewise-smooth with no jumps, the Fourier partial sums converge uniformly (same error bound everywhere at once).
TRUE — a continuous, piecewise-smooth periodic has no Gibbs spikes, so the partial sums close in evenly across the whole interval; it is precisely a jump that breaks uniform convergence and forces only pointwise convergence.
Spot the error
Spot the error: "The average of over is , so ."
The average formula is right, but that average equals , not . The coefficient is (same as every ); halving it gives the average.
Spot the error: "The series is an infinite tangled sum of sines and cosines, so I can't pull out a single coefficient."
Orthogonality untangles it: multiply by one wave and integrate, and every term dies except the matching one. The infinite sum collapses to a single survivor.
Spot the error: "For a function of period , I'll use and coefficient ."
That is only correct when . For general period you must use inside the wave and out front of the integral — the has to appear in both places.
Spot the error: " is odd, so ; therefore its Fourier series is zero."
Odd only kills the cosine coefficients ; the sine coefficients carry the whole function. Zeroing is a shortcut, not a vanishing act.
Spot the error: "At the jump of a square wave, the value should appear, so the series gives there."
The series gives the midpoint of the left and right limits, , regardless of how you defined — the series can only see averaged behaviour at a jump.
Spot the error: ", same as the other diagonal cases."
The case is special: the integrand is the constant , so the integral is , not . That doubled value is precisely why the constant term is written .
Spot the error: "Since sines and cosines never mix, I can compute by integrating against ."
You isolate by integrating against the matching cosine; against a sine you would recover instead. Match the wave to the coefficient you want.
Spot the error: "The complex-exponential Fourier series is a different theory from the sine/cosine one."
It is the same series repackaged: Euler's lets you fold each pair into a single complex (with for real ). One theory, two notations.
Why questions
Why do PDE solvers even need Fourier series?
Separation of Variables on the Heat Equation or Wave Equation produces sine/cosine modes; matching an arbitrary initial shape forces you to write as a sum of exactly those modes — that expansion is the Fourier series.
Why is the constant term written as rather than just ?
So the single formula works even at ; the leftover factor of (from the being at ) is absorbed by dividing the constant term by .
Why does orthogonality feel like the dot product ?
Both say "these building blocks don't overlap," so to read off "how much of one you have" you project onto it (dot for vectors, integrate the product for functions) and every other block contributes zero.
Why do only odd harmonics appear in the square wave (period , so )?
Being odd, ; because , the bracket is for even and for odd , so all even harmonics vanish and only survive.
Why does an odd integrand over integrate to ?
The left half is the exact negative mirror of the right half, so their signed areas cancel — this is the engine behind every even/odd shortcut.
Why can a jaggy, non-smooth function be built from perfectly smooth waves?
Each finite partial sum is smooth, but the infinite sum is a limit; sharp corners emerge only in the limit where infinitely many high-frequency waves conspire — near a corner they never fully settle, which is the source of Gibbs overshoot.
Why does convergence at a point not require to be continuous there?
The coefficients are integrals, which are blind to the value at a single point; the series reconstructs an averaged picture, so it converges to the midpoint at a jump without ever needing to be defined there.
Why is the Fourier Transform the natural next step after Fourier series?
A series handles a periodic function (a discrete set of frequencies); as the period grows to infinity the allowed frequencies crowd into a continuum, and the sum becomes an integral — that limit is the Fourier transform.
Edge cases
Edge case: What is the Fourier series of a constant function ?
Just with every other coefficient zero — a constant is its own average and needs no waves.
Edge case: What does the series give exactly at a jump discontinuity?
The midpoint of the left and right limits, independent of any value you assign to at that point.
Edge case: If is neither even nor odd, which coefficients survive?
Both and in general — no symmetry shortcut applies, so you compute every integral. You can still split into its even part (cosines) and odd part (sines) if you want.
Edge case: Two different piecewise functions that agree except at finitely many points — same Fourier series?
Yes — integrals ignore isolated points, so changing at finitely many places leaves every coefficient unchanged and produces the identical series.
Edge case: As you take more and more terms of the square wave, what happens to the overshoot spike right at the corner?
It narrows and hugs the jump ever more tightly but keeps its height near of the jump — it never dies, only migrates toward the discontinuity (Gibbs Phenomenon).
Edge case: Pointwise vs uniform convergence — what is the distinction for these series?
Pointwise means at each fixed the partial sums eventually get close to the target, but the number of terms needed can vary with ; uniform means one term-count works everywhere at once. Jumps (Gibbs) allow only pointwise; a continuous piecewise-smooth gives uniform.
Edge case: For real , how do the complex coefficients relate to ?
for , , and ; the pair of real coefficients and the conjugate pair of complex ones carry identical information.
Edge case: The zero function — what are its coefficients?
Every and is ; the empty sum reproduces , confirming the recipe is consistent at the degenerate extreme.
Recall One-line summary to carry away
Orthogonality isolates each coefficient, symmetry halves the work, and the "" sign is honest everywhere except at jumps, where the series politely reports the midpoint.