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Concept
Singular integral
Résumé
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
whose kernel function K : Rn×Rn → R is singular along the diagonal x = y. Specifically, the singularity is such that |K(x, y)| is of size |x − y|−n asymptotically as |x − y| → 0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over |y − x| > ε as ε → 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on Lp(Rn).
Hilbert transform
The archetypal singular integral operator is the Hilbert transform H. It is given by convolution against the kernel K(x) = 1/(πx) for x in R. More precisely,
The most straightforward higher dimension analogues of these are the Riesz transforms, which replace K(x) = 1/x with
where i = 1, ..., n and is the i-th component of x in Rn. All of these operators are bounded on Lp and satisfy weak-type (1, 1) estimates.
Singular integral operators of convolution type
A singular integral of convolution type is an operator T defined by convolution with a kernel K that is locally integrable on Rn{0}, in the sense that
Suppose that the kernel satisfies:
The size condition on the Fourier transform of K
The smoothness condition: for some C > 0,
Then it can be shown that T is bounded on Lp(Rn) and satisfies a weak-type (1, 1) estimate.
Property 1. is needed to ensure that convolution () with the tempered distribution p.v. K given by the principal value integral
is a well-defined Fourier multiplier on L2. Neither of the properties 1. or 2. is necessarily easy to verify, and a variety of sufficient conditions exist. Typically in applications, one also has a cancellation condition
which is quite easy to check. It is automatic, for instance, if K is an odd function. If, in addition, one assumes 2. and the following size condition
then it can be shown that 1.
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Personnes associées (5)
Unités associées (1)
Concepts associés (5)
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.
Transformation de Hilbert
En mathématiques et en traitement du signal, la transformation de Hilbert, ici notée , d'une fonction de la variable réelle est une transformation linéaire qui permet d'étendre un signal réel dans le domaine complexe, de sorte qu'il vérifie les équations de Cauchy-Riemann. La transformation de Hilbert tient son nom en honneur du mathématicien David Hilbert, mais fut principalement développée par le mathématicien anglais G. H. Hardy.
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 several variables the Riemann–Liouville integrals of one variable. If 0 < α < n, then the Riesz potential Iαf of a locally integrable function f on Rn is the function defined by where the constant is given by This singular integral is well-defined provided f decays sufficiently rapidly at infinity, specifically if f ∈ Lp(Rn) with 1 ≤ p < n/α.
Cours associés (3)
MATH-405: Harmonic analysis
An introduction to methods of harmonic analysis.
Covers convergence of Fourier series, Hilbert transform, Calderon-Zygmund theory, Fourier restriction, and applications to PDE.
MATH-206: Analysis IV
En son coeur, c'est un cours d'analyse fonctionnelle pour les physiciens et traite les bases de théorie de mesure, des espaces des fonctions et opérateurs linéaires.
MICRO-211: Analog circuits and systems
This course introduces the analysis and design of linear analog circuits based on operational amplifiers. A Laplace early approach is chosen to treat important concepts such as time and frequency resp