In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
theory of
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
, the Riesz transforms are a family of generalizations of the
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...
to
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
s of dimension ''d'' > 1. They are a type of
singular integral In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator
: T(f)(x) = \int K(x,y)f(y) \, dy,
who ...
operator, meaning that they are given by a
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
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
for ''j'' = 1,2,...,''d''. The constant ''c''
''d'' is a dimensional normalization given by
:
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
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positive ...
, or as a
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
with the
tempered distribution
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
:
The Riesz transforms arises in the study of differentiability properties of harmonic potentials in
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...
and
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
. In particular, they arise in the proof of the Calderón-Zygmund inequality .
Multiplier properties
The Riesz transforms are given by a
Fourier multiplier. Indeed, the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of ''R''
''j''ƒ is given by
:
In this form, the Riesz transforms are seen to be generalizations of the
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...
. The kernel is a
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
which is
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
of degree zero. A particular consequence of this last observation is that the Riesz transform defines a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
from ''L''
2(R
''d'') to itself.
[Strictly speaking, the definition () may only make sense for ]Schwartz function
In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables on ...
''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
Dilation (or dilatation) may refer to:
Physiology or medicine
* Cervical dilation, the widening of the cervix in childbirth, miscarriage etc.
* Coronary dilation, or coronary reflex
* Dilation and curettage, the opening of the cervix and surgic ...
on R
''d'' by the scalar ''s'', that is σ
''s''''x'' = ''sx'', then σ
''s'' defines an action on functions via
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: in ...
:
:
The Riesz transforms commute with σ
''s'':
:
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
:
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
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...
with respect to these two actions; that is,
:
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''.
Relationship with the Laplacian
Somewhat imprecisely, the Riesz transforms of
give the first
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
s of a solution of the equation
:
where Δ is the Laplacian. Thus the Riesz transform of
can be written as:
:
In particular, one should also have
:
so that the Riesz transforms give a way of recovering information about the entire
Hessian
A Hessian is an inhabitant of the German state of Hesse.
Hessian may also refer to:
Named from the toponym
*Hessian (soldier), eighteenth-century German regiments in service with the British Empire
**Hessian (boot), a style of boot
**Hessian f ...
of a function from knowledge of only its Laplacian.
This is now made more precise. Suppose that
is a
Schwartz function
In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables on ...
. Then indeed by the explicit form of the Fourier multiplier, one has
:
The identity is not generally true in the sense of
distributions. For instance, if ''
'' is a
tempered distribution
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
such that
, then one can only conclude that
:
for some polynomial
.
See also
*
Hilbert Transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...
*
Poisson kernel
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriv ...
*
Riesz potential In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space. They generalize to ...
References
* .
* .
* .
* {{citation, first1=N., last1=Arcozzi, title=Riesz Transform on spheres and compact Lie groups, publisher=Springer, publication-place=New York, year=1998, doi=10.1007/BF02384766 , s2cid=119919955 , issn=0004-2080.
Harmonic analysis
Integral transforms
Potential theory
Singular integrals