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Concept
Singular integral
Summary
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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Related concepts (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.
Hilbert transform
In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). The Hilbert transform is given by the Cauchy principal value of the convolution with the function (see ). The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° ( radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see ).
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.