Wirtinger's inequality

Wirtinger's inequality bounds the squared integral norm of a mean-zero periodic function by the squared integral norm of its derivative. Its sharp constant is determined by the period.

Fourier series: A representation of a periodic function as a sum of sinusoidal modes. The inequality follows by bounding each nonconstant Fourier mode using its frequency.
L² norm: The square root of the integral of a function's squared absolute value. Wirtinger's inequality compares the L² sizes of a function and its derivative.
Calculus of variations: The study of extrema of functionals defined on spaces of functions. The inequality supplies coercive bounds for energies of periodic variations.
Poincaré inequality on an interval: A bound on a function's deviation from its mean over a bounded interval, often with boundary conditions. Unlike the periodic case, interval versions depend on endpoint conditions and interval geometry.
Poincaré inequality: An inequality bounding a function's distance from its mean by the size of its gradient. Wirtinger's inequality is the one-dimensional periodic Poincaré inequality with a sharp constant.
Sobolev space: A function space whose elements and weak derivatives satisfy specified integrability conditions. The inequality extends naturally from smooth periodic functions to Sobolev functions.
Sturm–Liouville theory: The theory of eigenvalue problems for linear differential operators with boundary conditions. Its periodic eigenvalues explain the derivative bound and its equality modes.
Hardy inequality: An integral inequality controlling a function by a weighted integral of its derivative. Hardy inequalities use singular weights rather than periodic frequencies to produce derivative bounds.
Periodic boundary condition: A boundary condition requiring a function's values to agree at opposite ends of an interval. Periodicity quantizes Fourier frequencies and fixes the inequality's constant.
Fundamental frequency: The lowest positive frequency in a periodic signal's harmonic spectrum. It determines the first nonconstant mode and hence the sharp period-dependent bound.

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