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Concept

Interpolation

Finding a function of a chosen kind that passes exactly through given data points. For polynomials there is exactly one of degree below the number of points, and how good it is depends far more on where the points were placed than on how many there are.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

A degree-2 polynomial over GF(11), and the 7 values sent. A grid of the finite field with the polynomial's value at each point marked, and the transmitted symbols picked out.

A polynomial through the gaps

Write the message as the coefficients of a polynomial and send its values instead. Any k of them determine the polynomial, so it does not matter which ones are lost — and it does not matter how many, as long as k survive.

computation · Error-correcting codes 11 points, equally spaced. 1/(1 + x²) and the polynomial of degree 10 through 11 of its points, spaced evenly across the interval. The worst error is 2.48e-1, at x = -2.350.

The points that ruin the fit

A polynomial through eleven points of a gentle curve should be a good approximation to it, and adding more points should make it better. On evenly spaced points it makes it worse, without limit, and the reason is not the polynomial but where the points were put.

analysis · Taylor series A polygon of 4 points averaged down to its curve at t = 0.40. A control polygon of 4 points, the 3 rounds of weighted averaging at t = 0.40 drawn as nested polylines, the single point they end at, and the whole curve those points trace. The weights on the control points are 0.216, 0.432, 0.288, 0.064.

Averaging down the triangle

Change one word in the rule that builds Pascal's triangle — take a share of each entry above instead of adding them — and the triangle stops counting and starts averaging. The same rule then draws smooth curves from polygons and approximates every continuous function by polynomials, at a rate that no amount of smoothness can improve.

discrete · Pascals triangle The best degree-3 polynomial to eˣ: its error touches its largest size 5 times. The error curve of the best uniform polynomial approximation of degree 3 to eˣ on the interval from −1 to 1. It reaches its maximum size 5.528 × 10⁻³ at 5 points, alternately above and below, at x = -1.000, -0.682, 0.050, 0.732, 1.000.

The error that keeps coming back to its worst

Judge a polynomial by its largest error on an interval and there is exactly one best one of each degree. It is recognised without comparing it to anything else — its error rises to the same largest size, alternately above and below, one more time than there are coefficients.

analysis · Taylor series Where interpolation through 11 nodes magnifies an error. Lebesgue functions of equispaced, Chebyshev and optimal nodes of degree 10; constants 29.900, 2.489, 2.052.

The best nodes have no formula

Interpolating through n points magnifies any error in the data by at most the Lebesgue constant of the points. Chebyshev's points keep it near (2/π)·log n; stretching them to the ends of the interval brings it within two hundredths of the best possible; and the best possible points, characterised in 1978 by having every bump of the error curve the same height, have never been given a formula.

analysis · Taylor series The factorials, and three smooth curves through all of them. Γ(x+1), Γ(x+1)(1 + 0.5 sin² πx) and the degree-six interpolating polynomial through 0!…6!, on a log scale over 0 ≤ x ≤ 5.4; at x = 1/2: 0.88623, 1.32934, -3.58301.

Only one curve through the factorials bends the right way

Infinitely many smooth curves pass through 1, 1, 2, 6, 24, 120, and some of them even obey the factorial's own rule, f(x + 1) = (x + 1)·f(x), at every x. Ask that the logarithm of the curve bend upwards everywhere and exactly one survives — Euler's integral. A wiggle of any size breaks the condition somewhere, the smaller the wiggle the further out, and the proof that nothing else survives is a squeeze that computes √π along the way.

analysis · The integral

Named alongside it

The objects these essays reach for when they reach for this one.

ApproximationPolynomialError analysisPolynomial approximationUniform convergenceAnalytic functionBinomial coefficientBinomial distributionBoundChebyshev nodesChebyshev polynomialConditioning

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