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).
In mathematics, singularintegrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly...
Regular: An integral equation is called regular if the integrals used are all proper integrals. Singular or weakly singular: An integral equation is called...
In mathematics, singularintegral operators of convolution type are the singularintegral operators that arise on Rn and Tn through convolution by distributions;...
improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval...
In mathematics, singularintegral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The...
general nature, it is a singularintegral operator, defined by convolution with a function having a mathematical singularity at the origin, the Newtonian...
mathematics and signal processing, the Hilbert transform is a specific singularintegral that takes a function, u(t) of a real variable and produces another...
derivation of the Grunsky inequalities using reproducing kernels and singularintegral operators in geometric function theory; a more recent related approach...
there is a singularity at 0 and the antiderivative becomes infinite there. If the integral above were to be used to compute a definite integral between −1...
one to consider powers of D. The operators arising are examples of singularintegral operators; and the generalisation of the classical theory to higher...
Integral theory as developed by Ken Wilber is a synthetic metatheory aiming to unify a broad spectrum of Western theories and models and Eastern meditative...
^{n/2}2^{\alpha }{\frac {\Gamma (\alpha /2)}{\Gamma ((n-\alpha )/2)}}.} This singularintegral is well-defined provided f decays sufficiently rapidly at infinity...
linear elasticity, singularintegrals and numerical analysis: he is best known for the introduction of the symbol of a singularintegral operator, which...
the most significant were the results he obtained with Calderón on singularintegral operators. George G. Lorentz called it Zygmund's crowning achievement...
mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear...
{\displaystyle p\in [1,\infty )} . The Laplacian can also be viewed as a singularintegral operator which is defined as the following limit taken in X {\displaystyle...
kernel. The Hilbert transform H is the integral transform given by the Cauchy principal value of the singularintegral H f ( t ) = 1 π ∫ − ∞ ∞ f ( x ) d x...
type of singularintegral operator, meaning that they are given by a convolution of one function with another function having a singularity at the origin...
bounded on these weighted Lp spaces. In fact, any Calderón-Zygmund singularintegral operator is also bounded on these spaces. Let us describe a simpler...
basic problems of the mathematical theory of elasticity" (1933) and "SingularIntegral Equations" (1947). During World War II Muskhelishvili was responsible...
such that F′ = f on the interval. Provided the integrand and integral have no singularities on the path of integration, by the fundamental theorem of calculus...
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a rescaling followed...
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f ( x ) = e − x 2 {\displaystyle f(x)=e^{-x^{2}}}...
In probability, a singular distribution is a probability distribution concentrated on a set of Lebesgue measure zero, where the probability of each point...
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