 Hilbert transform

With the sign conventionthe Hilbert transform is a singular integral operator. For a Schwartz function, the Cauchy principal value removes equal small intervals on both sides of $y=x$. Subtracting $f(x)$ locally makes the remaining quotient bounded there, while rapid decay controls infinity. Reversing the denominator reverses the operator sign.

For the angular-frequency Fourier transform, its Fourier multiplier is $-i\mathrm{sgn}\xi$, as derived under Hilbert-transform Fourier multiplier. Plancherel theorem therefore extends it to an isometry on $L^2(\mathbb{R})$, and $H^2=-I$ on that space because the squared multiplier is minus one almost everywhere. This frequency-dependent phase shift connects real and imaginary boundary parts of suitable holomorphic functions; Hilbert transform of the sinc function supplies an explicit example.

The transform need not preserve boundedness. For the indicator function of $[-1,1]$, direct integration givesIt diverges logarithmically near the endpoints and has tail $2/(\pi x)+O(x^{-3})$. This example also explains why a compactly supported input can produce a long-range response. The finite Hilbert transform instead restricts the integration interval and has its own endpoint and null-space issues.

The Hilbert transform is a mathematical operation that takes a real-valued function and produces a related complex-valued function. It is widely used in signal processing, communication theory, and various fields of applied mathematics. The transform is particularly useful for analyzing signals and extracting their phase and amplitude characteristics.