Hilbert Transform (original) (raw)

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology

Alphabetical Index New in MathWorld

The Hilbert transform (and its inverse) are the integral transform

where the Cauchy principal value is taken in each of the integrals. The Hilbert transform is an improper integral.

They will be implemented in a future version of the Wolfram Language as HilbertTransform[f, x, _y_] and InverseHilbertTransform[g,y, _x_], respectively.

In the following table, Pi(x) is the rectangle function,sinc(x) is the sinc function, delta(x) is the delta function, AdjustmentBox[I, BoxMargins -> {{0.13913, -0.13913}, {-0.5, 0.5}}]I(x) and AdjustmentBox[I, BoxMargins -> {{0.101266, -0.101266}, {0.375, -0.375}}, BoxBaselineShift -> -0.375]AdjustmentBox[I, BoxMargins -> {{0, 0}, {-0.25, 0.25}}, BoxBaselineShift -> 0.25](x) are impulse symbols, and _1F_1(a;b;x) is a confluent hypergeometric function of the first kind.

See also

Abel Transform, Fourier Transform, Improper Integral, Integral Transform, Titchmarsh Theorem, Inverse Hilbert Transform, Wiener-Lee Transform

Explore with Wolfram|Alpha

References

Bracewell, R. "The Hilbert Transform." The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 267-272, 1999.Papoulis, A. "Hilbert Transforms." The Fourier Integral and Its Applications. New York: McGraw-Hill, pp. 198-201, 1962.

Referenced on Wolfram|Alpha

Hilbert Transform

Cite this as:

Weisstein, Eric W. "Hilbert Transform." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HilbertTransform.html

Subject classifications