Inden for matematik og signalbehandling er Hilberttransformationen en lineær operator, der tager funktionen og producerer . Operatoren er givet ved en foldning med funktionen :

the improper integral being understood in the principal value sense. The Hilbert transform has a particularly simple representation in the frequency domain: it imparts a phase shift of 90° to every Fourier component of a function. For example, the Hilbert transform of , where ω > 0, is .

The Hilbert transform is important in signal processing, where it derives the analytic representation of a real-valued signal u(t). Specifically, the Hilbert transform of u is its harmonic conjugate v, a function of the real variable t such that the complex-valued function u + iv admits an extension to the complex upper half-plane satisfying the Cauchy–Riemann equations. The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions.

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