Riesz transform

In the mathematical theory of harmonic analysis, the Riesz transforms are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension "d" > 1. They are a type of singular integral operator, meaning that they are given by a convolution of one function with another function having a singularity at the origin. Specifically, the Riesz transforms of a complex-valued function ƒ on R^"d" are defined byfor "j" = 1,2,...,"d". The constant "c"_"d" is a dimensional normalization given by

:$c\_d\; =\; frac\{1\}\{piomega\_\{d-1\; =\; frac\{Gamma\; [(d+1)/2]\; \}\{pi^\{(d+1)/2.$

where ω_"d"−1 is the volume of the ("d" − 1)-ball.

Somewhat informally, the Riesz transforms of ƒ give the first partial derivatives of a solution of the Poisson equation on R^d:$Delta\; u\; =\; f$since the Riesz transform "R"_"j" is obtained by differentiating the Newtonian potential under the sign of the integral:

:$R\_jf(x)\; =\; Cint\_\{mathbf\{R\}^d\}frac\{partial\}\{partial\; x\_j\}frac\{f(t)\}(mathcal\{F\}f)(x)$

(up to an overall positive constant depending on the normalization of the Fourier transform). In this form, the Riesz transforms are seen to be generalizations of the Hilbert transform. The kernel is a distribution which is homogeneous of degree zero. A particular consequence of this last observation is that the Riesz transform defines a bounded linear operator from "L"²(R^"d") to itself. [Strictly speaking, the definition (EquationNote|1) may only make sense for Schwartz function "f". Boundedness on a dense subspace of "L"² implies that each Riesz transform admits a continuous linear extension to all of "L"².]

This homogeneity property can also be stated more directly without the aid of the Fourier transform. If σ_"s" is the dilation on R^"d" by the scalar "s", that is σ_"s""x" = "sx", then σ_"s" defines an action on functions via pullback:

:$sigma\_s^*\; f\; =\; fcircsigma\_s.$

The Riesz transforms commute with σ_"s":

:$sigma\_s^*\; (R\_jf)\; =\; R\_j(sigma\_x^*f).$

Similarly, the Riesz transforms commute with translations. Let τ_"a" be the translation on R^{"d" along the vector "a"; that is, τ_"a"("x") = "x" + "a". Then}

:$au\_a^*\; (R\_jf)\; =\; R\_j(\; au\_a^*f).$

For the final property, it is convenient to regard the Riesz transforms as a single vectorial entity "R"ƒ = ("R"₁ƒ,…,"R"_"d"ƒ). Consider a rotation ρ in R^"d". The rotation acts on spatial variables, and thus on functions via pullback. But it also can act on the spatial vector "R"ƒ. The final transformation property asserts that the Riesz transform is equivariant with respect to these two actions; that is,

:$ho^*\; R\_j\; [(\; ho^\{-1\})^*f]\; =\; sum\_\{k=1\}^d\; ho\_\{jk\}\; R\_kf.$

These three properties in fact characterize the Riesz transform in the following sense. Let "T"=("T"_"1",…,"T"_"d") be a "d"-tuple of bounded linear operators from "L"²(R^{"d") to "L"2(R"d") such that}

* "T" commutes with all dilations and translations.
* "T" is equivariant with respect to rotations.

Then, for some constant "c", "T" = "cR".

ee also

* Poisson kernel
* Riesz potential

References

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