Singular Integral

singular integral

[′siŋ·gyə·lər ¦int·ə·grəl əv ə ‚dif·ə′ren·chəl i′kwā·zhən]
(mathematics)

Singular Integral

 

(1) A method of representing functions. A singular integral is an integral of the form

which, when certain conditions on ∫ are fulfilled, converges to its generating function f(x) as n → ∞. The function Kn is called the kernel of the singular integral. For example,

and

are the singular integrals of Dirichlet and Vallée Poussin, respectively. The systematic investigation of singular integrals was begun by H. Lebesgue in 1909. Singular integrals attracted attention in connection with the problem of representing and approximating functions of various classes by simpler functions, such as smooth functions and polynomials.

(2) An improper integral. [23–1231–]

References in periodicals archive ?
Satisfying boundary conditions (5) by the functions (13)-(14) on the faces of cracks with end zones, we get a system of singular integral equations with respect to the unknown functions [g.
Then, we use the Toeplitz matrix method and product Nystrom method, as the best two methods, to obtain the solution of the singular integral equation numerically.
p](D), p > 2, is a solution of the singular integral equation
Shamir, Calderon's reproducing formula and singular integral operators on a real line.
of Memphis, Tennessee) quantitatively study the basic approximation properties of the general Picard, Gauss-Weierstrass, and Poisson-Cauchy singular integral operators over the real line, which are not positive linear operators.
Calderon (1920-1998) was an important Argentine mathematician known for his work on the theory of partial differential equations and singular integral operators.
The q-Fourier transform and its inverse are certain singular integral equations, somewhat similar to the case of the classical Hilbert transform; see [24], [39] for an account of the theory of singular integral equations.
In [30] the Haar wavelet method is applied to solving different types of linear integral equations (Fredholm, Volterra, integro-differential, weakly singular integral equations), also the eigenvalue problem is solved.
This small volume on linear singular integral equations examines solutions to complex problems in partial differential equations.
He covers Volterra integral equations, Fredholm integral equations, nonlinear integral equations, the singular integral equation, integro-differential equations, symmetric kernals and orthogonal systems of functions, and a range of applications.
w] can be found as a solution of the singular integral equation