Concept

Lebesgue constant

The largest factor by which interpolation at a given set of points can magnify an error in the data, taken over the whole interval. It grows exponentially for equally spaced points and only like the logarithm of their number for Chebyshev's.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The Dirichlet kernel, a spike with ripples that do not die. The Dirichlet kernel for N = 4, 12 on the interval from −π to π: a central spike of height 2N + 1 and side ripples whose total area in absolute value grows with N.

The ripples that make a series run away

Adding up the first N terms of a Fourier series is the same as averaging the function against one fixed wiggly curve. Its area is always one, but the area of its absolute value grows like the logarithm of N, without limit — and that single number is enough to force a continuous function, with no jump and no corner anywhere, whose Fourier series diverges at a point. Averaging the partial sums removes the negative ripples, and with them the whole problem.

analysis · Fourier 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

Named alongside it

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

Cesaro summation Chebyshev nodes Conditioning Continuity Convergence Counterexample Dirichlet kernel Equioscillation Fourier analysis Growth rate Interpolation Polynomial approximation

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