Lesson plan
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 by {{NumBlk|:| R_jf(x) = c_d\lim_{\epsilon\to 0}\int_{\mathbf{R}^d\backslash B_\epsilon(x)}\frac{(x_j-t_j)f(t)}{|x-t|^{d+1}}\,dt | }} for j = 1,2,...,d. The constant c d is a dimensional normalization given by : c_d = \frac{1}{\pi\omega_{d-1}} = \frac{\Gamma[(d+1)/2]}{\pi^{(d+1)/2}}. where ω d−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 : K(x) = \frac{1}{\pi\omega_{d-1}} \, p.v. \frac{x_j}{|x|^{d+1}}. 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 .
Article summary
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 by {{NumBlk|:| R_jf(x) = c_d\lim_{\epsilon\to 0}\int_{\mathbf{R}^d\backslash B_\epsilon(x)}\frac{(x_j-t_j)f(t)}{|x-t|^{d+1}}\,dt | }} for j = 1,2,...,d. The constant c d is a dimensional normalization given by : c_d = \frac{1}{\pi\omega_{d-1}} = \frac{\Gamma[(d+1)/2]}{\pi^{(d+1)/2}}. where ω d−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 : K(x) = \frac{1}{\pi\omega_{d-1}} \, p.v. \frac{x_j}{|x|^{d+1}}. 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 .
Key claims
No claims recorded.
Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
No sources recorded.
Key claims and bibliography are generated from the same tracked claims and sources as the article page.