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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

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  • 리스 변환 (ko)
  • リース変換 (ja)
  • Riesz transform (en)
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  • 조화해석학에서 리스 변환은 힐베르트 변환을 1보다 큰 차원의 유클리드 공간으로 확장해 일반화한 것이다. 힐베르트 변환과 같이 한 함수를 원점에 특이점이 있는 다른 함수와 컨볼루션함으로써 이루어진다. d차원 실수공간 Rd 에 정의된 복소함수 f에 대한 리스 변환은 다음과 같이 정의된다. 여기서 j = 1,2,...,d이다. 상수 cd는 다음과 같이 정의된 차원 정규화값이다: 여기서 ωd−1는 단위 (d − 1)-구의 부피이다. (ko)
  • 数学の調和解析の分野におけるリース変換（リースへんかん、英: Riesz transform）とは、次元 d > 1 のユークリッド空間へのヒルベルト変換の一般化の族である。ある函数と、原点に特異性を持つ別の函数の畳み込みであることから、ある種のと見なすことが出来る。より正確に言うと、Rd 上の複素数値函数 ƒ のリース変換は、j = 1,2,...,d に対して次式で定義される。 (1) ここで定数 cd は次元の正規化 であり、ωd−1 は (d − 1)-次元球の体積を表す。上式の極限は様々な方法で書き表すことが出来、しばしば主値や、緩増加超函数（tempered distribution） との畳み込みとして書き表される。リース変換は、ポテンシャル論や調和解析における調和ポテンシャルの微分可能性の研究に現れる。特に、カルデロン＝ジグムントの不等式の証明に現れる。 (ja)
  • 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 (en)
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  • 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 . (en)
  • 조화해석학에서 리스 변환은 힐베르트 변환을 1보다 큰 차원의 유클리드 공간으로 확장해 일반화한 것이다. 힐베르트 변환과 같이 한 함수를 원점에 특이점이 있는 다른 함수와 컨볼루션함으로써 이루어진다. d차원 실수공간 Rd 에 정의된 복소함수 f에 대한 리스 변환은 다음과 같이 정의된다. 여기서 j = 1,2,...,d이다. 상수 cd는 다음과 같이 정의된 차원 정규화값이다: 여기서 ωd−1는 단위 (d − 1)-구의 부피이다. (ko)
  • 数学の調和解析の分野におけるリース変換（リースへんかん、英: Riesz transform）とは、次元 d > 1 のユークリッド空間へのヒルベルト変換の一般化の族である。ある函数と、原点に特異性を持つ別の函数の畳み込みであることから、ある種のと見なすことが出来る。より正確に言うと、Rd 上の複素数値函数 ƒ のリース変換は、j = 1,2,...,d に対して次式で定義される。 (1) ここで定数 cd は次元の正規化 であり、ωd−1 は (d − 1)-次元球の体積を表す。上式の極限は様々な方法で書き表すことが出来、しばしば主値や、緩増加超函数（tempered distribution） との畳み込みとして書き表される。リース変換は、ポテンシャル論や調和解析における調和ポテンシャルの微分可能性の研究に現れる。特に、カルデロン＝ジグムントの不等式の証明に現れる。 (ja)
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