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Hilbert Transform

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The Hilbert transform is defined, together with the corresponding inverse Hilbert transform, by the integral transform pair

g(y)=H[f(x)]
(1)
=1/piPVint_(-infty)^infty(f(x)dx)/(x-y)
(2)
f(x)=H^(-1)[g(y)]
(3)
=-1/piPVint_(-infty)^infty(g(y)dy)/(y-x)
(4)

(Bracewell 1999), where the Cauchy principal value is taken in each of the integrals.

The Hilbert transform is therefore an improper integral.

The opposite convention, with denominator y-x in the forward transform, is also common (NIST DLMF, eqn. 1.14.41; King 2009, vol. 2, p. 6). The Wolfram Language functions HilbertTransform[f, x, y] and InverseHilbertTransform[g, y, x] also use this opposite sign convention, so their results are the negatives of the corresponding transforms shown below (though the normalization by 1/pi is the same).

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.

f(x)g(y)
sinxcosy
cosx-siny
(sinx)/x(cosy-1)/y
Pi(x)1/piln|(y-1/2)/(y+1/2)|
1/(1+x^2)-y/(1+y^2)
sinc^'(x)(1-cosy-ysiny)/(y^2)
delta(x)-1/(piy)
AdjustmentBox[I, BoxMargins -> {{0.13913, -0.13913}, {-0.5, 0.5}}]I(x)y/(pi(1/4-y^2))
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)-1/(2pi(1/4-y^2))
e^(-x^2)-e^(-y^2)erfi(y)

See also

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

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References

Bracewell, R. "The Hilbert Transform." The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 267-272, 1999.King, F. W. Hilbert Transforms, Vol. 2. Cambridge, England: Cambridge University Press, p. 6, 2009.NIST Digital Library of Mathematical Functions. "Hilbert Transform." §1.14(v). https://dlmf.nist.gov/1.14.v.Papoulis, A. "Hilbert Transforms." The Fourier Integral and Its Applications. New York: McGraw-Hill, pp. 198-201, 1962.

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Hilbert Transform

Cite this as:

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

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