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Hilbert Transform Along a Vector Field

The Hilbert transform along a vector field v:R^2->S^1 integrates along the straight line whose direction is frozen at the output point. Its short hard truncation is

H_(v,a)^epsilonf(x)=int_(epsilon<|t|<a)f(x-tv(x))(dt)/t,

where 0<epsilon<a. For a Schwartz function, the symmetric Cauchy principal value exists at every point.

See also

Hilbert Transform, Lipschitz Constant, Lipschitz Function, Schwartz Function, Vector Field

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References

Lacey, M. and Li, X. On a Conjecture of E. M. Stein on the Hilbert Transform on Vector Fields. Providence, RI: Amer. Math. Soc., 2010. https://doi.org/10.1090/S0065-9266-10-00572-7.

Cite this as:

Weisstein, Eric W. "Hilbert Transform Along a Vector Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HilbertTransformAlongaVectorField.html

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