Fourier series
Named by 14 essays across 6 fields — each of them below, with the objects they name alongside it.
A curve with a corner at every point
Continuity means a curve can be drawn without lifting the pen. Differentiability means it has a tangent. The first was assumed to nearly imply the second until 1872, when Weierstrass exhibited a curve that is continuous everywhere and has a tangent nowhere — and it is a sum of cosines.
The shape described from outside
A convex shape can be given by its boundary or by the family of lines that touch it, and the second description turns the constant-width condition into one line of arithmetic — after which the perimeter falls out, and curves with no corners at all can simply be written down.
Nearly the most means nearly round
A shape that holds almost as much as a circle of the same perimeter must almost be a circle. Bonnesen made that exact: the ring between a convex shape's largest inscribed circle and smallest enclosing circle is never wider than √(L² − 4πA)/π. Three quite different shapes holding 99% of the circle's area all have rings under 9.55 wide, and not one of 200 random convex shapes breaks the bound.
Almost every orbit is fair
Double a number and keep the fractional part, and do it again, and again. Where the orbit goes is written in the number's binary digits, and for almost every starting point it spends exactly a quarter of its time in each quarter of the interval. The proof is a short argument about Fourier coefficients being pushed to infinity — and it leaves room for a set of exceptions with no length at all, which includes every fraction and, for all anyone can prove, every number anyone has ever named.
Points on a lattice that see almost nothing
Average a function over the points of a carefully tilted lattice and the error can fall like one over the square of the number of points — far faster than random sampling, and faster than the most evenly spread sequences. The reason is that a lattice rule is blind to only a thin set of frequencies, its dual lattice, and a smooth periodic function has almost nothing there.
An ellipse, not a disc
A Taylor series converges on a disc, and the disc's radius is the distance to the nearest singularity. Ask instead how well polynomials can follow a function on an interval, and the answer is an ellipse with the interval's ends as its foci — the largest one the function is smooth inside.
Closed sets obey the continuum hypothesis
Whether some set of reals is bigger than the whole numbers and smaller than the line is a question the axioms cannot answer. For closed sets it has an answer, found in 1883: strip away the isolated points, again and again, and what is left is either nothing or a perfect set, which is as large as the line — so every closed set is countable or of the line's size. The same answer holds for Borel and analytic sets, and fails to be provable exactly one step further up.
The sums of primitive roots are always whole
Take the roots of unity of order exactly q, raise each to the power n and add them. The answer is always a whole number, it depends on n only through what n shares with q, and as n varies the sums behave like the sines and cosines of a Fourier series — so well that Ramanujan could rebuild the sum of the divisors of any number from them, and prove that their weighted total is nought, a fact that at n = 1 is the prime number theorem.
How smooth the disguise is
An unlocked circle map is a rigid rotation in different coordinates, and whether those coordinates are smooth is decided by arithmetic. Solving for the change of coordinates means dividing, harmonic by harmonic, by numbers that come arbitrarily close to nought — and the golden rotation number keeps them far enough away that the result is not merely smooth but analytic, while a number lying close to a fraction makes it buckle.
Multiplying makes the digit one common
Multiply a few random numbers together and the product starts with 1 about 30 per cent of the time and with 9 under 5 per cent — Benford's law, which no factor contains. The logarithm of a product is a sum, the central limit theorem spreads that sum across many powers of ten, and once it is spread its fractional part is uniform. The approach is geometric, at a rate fixed by a single number for each kind of factor; sums never get there, and the powers of two get there with no randomness at all.
Close in height and nowhere close in slope
Uniform convergence carries continuity to the limit and carries integrals to the limit. It carries slopes nowhere. The functions sin(n²x)/n flatten to nothing while their slopes grow without bound; smooth curves converge uniformly to a corner; a Fourier series converges and its differentiated series diverges. What does carry a slope is uniform convergence of the slopes themselves — and the reason is that integration averages wiggles away while differentiation multiplies them.
![Riemann's function on one period, with its one kind of smooth point. R(x) = Σ sin(n²x)/n² on [0, 2π]; R(π) = 0 with derivative −1/2; R(π/2) = 1.2336, a cusp.](/f/riemann-s-function-on-one-period-with-its-one-kind-of-smooth-a763cec7.svg)
The slope a Gauss sum leaves behind
Riemann is said to have offered sin x + sin(4x)/4 + sin(9x)/9 + … as a continuous function with no derivative anywhere. It has one after all, at π and at every π times an odd number over an odd number, and the slope there is always exactly −1/2. Computed with a million terms, the function's behaviour at every fraction is read off a single number — a quadratic Gauss sum — and the slope appears exactly where that number is zero.
A wave known only at its ticks
Read a sine wave nine cycles a second at ten instants a second and the readings describe a wave of one cycle a second instead. Read a wave with nothing faster than 3.7 cycles a second at more than 7.4 instants a second and the readings determine it exactly. Both facts are the same fold of the frequency axis, and it is why wagon wheels turn backwards on film.
A grid that packs the copies
Sampling a pattern on a grid makes copies of its frequencies, one for every point of a lattice, and the pattern can be read back from the samples only if the copies do not collide. For a pattern whose frequencies fill a disc the square grid wastes room between its copies. Searching every lattice of samples for the fewest that work finds a nearly hexagonal one every time — 124 samples where the square grid needs 169 — and the saving is the hexagonal packing's over the square, 13.4% in the limit. In three dimensions it is 29%, and in twenty-four the cubic grid needs 16,777,216 times as many samples as the best.
Named alongside it
The objects these essays reach for when they reach for this one.
LatticeAliasingContinuityConvergenceConvex hullConvexityCounterexampleDerivativeDifferentiabilityEllipseFrequencyMeasure zero