Hilbert transform

The Hilbert transform maps a function to its Cauchy principal-value convolution with the kernel 1/(πx). It shifts Fourier phases by a quarter-cycle, with opposite signs at positive and negative frequencies.

Cauchy principal value: A generalized integral defined by excluding a symmetric neighborhood of a singularity and taking a limit. The kernel is singular at zero, so this limiting prescription defines the transform.
Singular integral: An integral operator whose kernel is singular or insufficiently integrable at some point. The reciprocal kernel places the Hilbert transform in this broad operator family.
Analytic signal: A complex signal whose Fourier spectrum vanishes at negative frequencies. Combining a real signal with its Hilbert transform constructs its analytic signal.
Hardy space: A space of analytic functions whose boundary values satisfy specified integrability bounds. The transform connects real and imaginary boundary components of functions in these spaces.
Hardy–Littlewood maximal operator: An operator assigning to each point the largest local average of a function over intervals or balls. It is a positive averaging operator, unlike the Hilbert transform’s oscillatory signed kernel.
Fourier transform: A mathematical transform that decomposes a function into its frequency components. In frequency space, the Hilbert transform is multiplication by a simple sign-dependent phase factor.
Lᵖ space: A function space defined by integrability of the pth power of a function. For 1 < p < ∞, the transform extends to a bounded operator on these spaces.
Instantaneous amplitude and phase: Time-varying amplitude and phase obtained from the complex analytic representation of a signal. The transform supplies the quadrature component used to define these signal attributes.
Cauchy integral formula: A complex-analysis formula expressing an analytic function’s interior values through its boundary values. Boundary limits of Cauchy integrals produce operators closely related to the Hilbert transform.
Riesz potential: A fractional integration operator with a locally integrable power-law kernel. It smooths functions through fractional integration, whereas the Hilbert transform preserves scale and shifts phase.

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