The finite Hilbert transform is a singular integral operator whose integration is restricted to a finite interval. The displayed convention uses the denominator ; reversing that denominator reverses the sign. The Cauchy principal value deletes equal intervals on the two sides of the interior point . Endpoint behavior matters: the integrable function belongs to the null space for this interval. This nontrivial null space is one reason that an inversion formula needs a normalization or a stated endpoint class.
For , the displayed expression solves , where and is the finite Hilbert transform with denominator . To derive it, put at infinity, take to be the Cauchy transform of , and set . The Sokhotski–Plemelj formula gives and the displayed jump . The endpoint behavior is at most inverse square root, and at infinity. Cauchy reconstruction from a jump across a finite interval then gives and hence . Adding any constant multiple of gives another solution in the integrable endpoint class.
Multiplying an auxiliary sign-changing factor by a growing analytic factor can invalidate this reconstruction. For example, using instead of with produces the constant function and zero jump. The boundary sum alone is not enough: the entire remainder in reconstructing from its jump must also be controlled.
With convention , the analytic function has removable singularity at zero and decays on large upper semicircles. Its boundary real part is and imaginary part is . The upper-half-plane contour indented above gives . Taking imaginary parts after reversing the denominator proves the displayed transform, with value zero at by continuous extension.
For a unit vector , the directional Hilbert transform integrates along lines parallel to . Its Fourier multiplier is under the usual transform convention. Applied to a rectangle whose long axis is , it reduces to the one-dimensional logarithmic tail of an interval indicator function, separately on each transverse coordinate.
For a Schwartz function , the Hilbert transform satisfies at both infinities. Splitting the inverse Fourier transform at the sign multiplier's jump and integrating each half-line by parts extracts the zero-frequency value ; the remainder vanishes by the Riemann-Lebesgue lemma. A nonzero zeroth moment therefore obstructs Schwartz decay.
For and , the Fourier transform multiplier is . The Heaviside function transform and the double-transform reflection identity give the sign. Plancherel theorem proves the isometry, and multiplication by proves commutation with differentiation. The map need not preserve the Schwartz space.
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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.