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Concept

Dirichlet kernel

The function whose average against a signal gives the signal's Fourier series cut off at a fixed frequency. Its oscillating tails, which never shrink in total size, are why partial Fourier sums can overshoot and fail to converge.

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 Rebuilding a ball at its own centre. Fourier inversion of the indicator of the unit segment, disc and ball, keeping frequencies below R, evaluated at the centre: at R = 8 the values are 0.9873, 0.9206, 0.9873; in three dimensions at R = 7.75 it is 1.6369.

A ball rebuilt everywhere but its centre

Rebuild a solid ball from its frequencies, keeping those below a cut-off, and read the result at the centre, where the ball is perfectly smooth. In the plane the answer settles on 1. In space it swings between 0.36 and 1.64 for ever, because the whole surface of the ball is the same distance from the centre and its jump arrives there in step.

analysis · Fourier series

Named alongside it

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

Bessel functionCesaro summationContinuityConvergenceCounterexampleDimensionFourier analysisFourier seriesFourier transformGibbs' phenomenonGrowth rateLebesgue constant

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