T-13 · Functional analysis

Functions as vectors.

Give functions an inner product and they acquire lengths, angles and perpendicular directions. Fourier coefficients become coordinates, and Parseval's identity is Pythagoras with infinitely many legs. Gram–Schmidt, run in exact fractions, produces the Legendre polynomials. Among all the ways to measure length, the parallelogram law singles out p = 2.

Harmonics N = 9

Coordinates in a space of functions

With ⟨f, g⟩ = (1/π)∫f g dx, the functions 1/√2, cos nx, sin nx are perpendicular unit vectors. A Fourier coefficient is the length of f's shadow on one of them.

aₙ	0
bₙ	4/(nπ) for odd n, else 0
closed form vs ⟨f, ·⟩	residual … n = 1, 2, 3, by quadrature

√(aₙ² + bₙ²) for n = 1, 2, …: each harmonic's leg

Parseval: Pythagoras with infinitely many legs

The error f − Sₙ is perpendicular to everything in Sₙ, so the squared lengths add. As N grows the error's share goes to zero, and ‖f‖² is the sum of the squares of all its coordinates.

‖f‖² = 2	2 residual …
‖S₉‖²	1.9192095736
‖f − S₉‖²	0.0807904264
‖S‖² + ‖f − S‖² = ‖f‖²	residual …

Gibbs

The partial sums converge in the mean but not uniformly: next to the jump they overshoot by about 9% of the jump, and the overshoot moves closer to the jump without ever shrinking.

peak of S₉	1.18232821 at x = π/10
(2/π)·Si(π)	1.17897974
overshoot	9.1164% of the jump