Integral transforms | Harmonic analysis | Singular integrals | Potential theory

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 Rd are defined by for j = 1,2,...,d. The constant cd is a dimensional normalization given by where ωd−1 is the volume of the unit (d − 1)-ball. The limit is written in various ways, often as a principal value, or as a convolution with the tempered distribution The Riesz transforms arises in the study of differentiability properties of harmonic potentials in potential theory and harmonic analysis. In particular, they arise in the proof of the Calderón-Zygmund inequality . (Wikipedia).

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Poisson kernel | Riesz potential | Potential theory | Operator (mathematics) | Partial derivative | Cauchy principal value | Homogeneous function | Rotation | Mathematics | Distribution (mathematics) | Hilbert transform | Singular integral | Euclidean space | Volume of an n-ball | Hessian matrix | Convolution | Fourier transform | Harmonic analysis | Pullback (differential geometry)