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 2 essays across 2 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

Named alongside it

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

Analytic function Approximation Convergence Erasure Error analysis Error-correcting code Finite field Minimum distance Polynomial Polynomial approximation Radius of convergence Reed solomon

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