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.