Singular integral

An integral

$$ I _ {n} ( f, x) = \int\limits _ { a } ^ { b } f ( t) \Phi _ {n} ( t, x) dt $$

with a singularity at the point $ x $, defined for a function $ f $ integrable on $ [ a, b] $, whose kernel $ \Phi _ {n} $ satisfies the following conditions: For any $ \delta > 0 $ and an arbitrary interval $ [ \alpha , \beta ] \subset [ a, b] $,

$$ \tag{1 } \lim\limits _ {n \rightarrow \infty } \ \int\limits _ {[ a, b] \cap [ x - \delta , x + \delta ] } \Phi _ {n} ( t, x) dt = 1, $$

$$ \tag{2 } \lim\limits _ {n \rightarrow \infty } \int\limits _ {[ \alpha , \beta ] - [ x - \delta , x + \delta ] } \Phi _ {n} ( t, x) dt = 0 $$

and

$$ \tag{3 } { \mathop{\rm ess} \sup } _ {t \in [ a, x - \delta ] \cup [ x + \delta , b] } \ | \Phi _ {n} ( t, x) | \leq \ \Phi _ {x} ( \delta ) < \infty , $$

where $ \Phi _ {x} ( \delta ) $ depends only on $ \delta $ and $ x $ and not on $ n $. If (1), (2) and (3) are fulfilled uniformly on an $ x $- set $ E \subset [ a, b] $, then the integral $ I _ {n} ( f, x) $ is said to be uniformly singular on $ E $. Most attention has been paid to properties of so-called positive kernels $ ( \Phi _ {n} ( t, x) \geq 0) $; Dirichlet kernels (cf. Dirichlet kernel)

$$ D _ {n} ( t, x) = \ \frac{\sin { \frac{2n + 1 }{2} } ( t - x) }{2 \sin { \frac{t - x }{2} } } , $$

Fejér kernels (cf. Fejér integral)

$$ F _ {n} ( t, x) = \ \frac{\sin ^ {2} { \frac{n + 1 }{2} } ( t - x) }{2 ( n + 1) \sin ^ {2} { \frac{t - x }{2} } } , $$

Poisson–Abel kernels (cf. Poisson integral)

$$ P _ {r} ( t, x) = \ \frac{1 - r ^ {2} }{2 [ 1 - 2r \cos ( t - x) + r ^ {2} ] } ,\ \ 0 \leq r < 1, $$

and kernels induced by various methods of summation of orthogonal expansions in orthonormal polynomials.

The concept of a "singular integral" was introduced by H. Lebesgue [1], who pointed out its importance in the investigation of questions of convergence. Thus, the investigation of the convergence of singular integrals bears on questions of convergence and summability of trigonometric Fourier series; Fourier series in orthogonal polynomials, and also expansions in general orthogonal systems.

Lebesgue established a criterion for the convergence of singular integrals in the case of continuous functions $ f $ of bounded variation. D.K. Faddeev [2] established necessary and sufficient conditions for the convergence of singular integrals at a Lebesgue point in the case of summable functions $ f $. Since the conditions of Lebesgue and Faddeev are difficult to verify for concrete singular integrals, a whole series of papers was devoted to the search for effective sufficient conditions for the convergence of singular integrals, both at isolated points and uniformly. For a singular integral to converge at a continuity point, it is sufficient that the operator $ I _ {n} ( f, x) $ is bounded in norm, i.e. the sequence of integrals

$$ \int\limits _ { a } ^ { b } | \Phi _ {n} ( t, x) | dt ,\ n= 1, 2 \dots $$

must be bounded, and for convergence at a Lebesgue point it is necessary that there exists a so-called "hump-backed majoranthump-backed majorant" for the kernel $ \Phi _ {n} ( t, x) $, that is, an integrable function $ \Psi _ {n} ( t, x) \geq 0 $ which monotonically increases on $ [ a, x) $ and monotonically decreases on $ ( x, b) $, such that for almost-all $ t \in [ a, b] $,

$$ | \Phi _ {n} ( t, x) | \leq \Psi _ {n} ( t, x), $$

where

$$ \int\limits _ { a } ^ { b } \Psi _ {n} ( t, x) dt = O ( 1). $$

References

[1]	H. Lebesgue, "Sur les intégrales singulières" Ann. Fac. Sci. Univ. Toulouse , 1 (1909) pp. 25–117
[2]	D.K. Faddeev, "On the representation of mean-square integrable functions by singular integrals" Mat. Sb. , 1 (1936) pp. 351–368 (In Russian) (French abstract)
[3]	P.P. Korovkin, "Linear operators and approximation theory" , Hindushtan Publ. Comp. (1960) (Translated from Russian)
[4]	I.P. Natanson, "Functions of a real variable" , 1–2 , F. Ungar (1955–1961) (Translated from Russian)
[5]	G. Alexits, "Convergence problems of orthogonal series" , Pergamon (1961) (Translated from Russian)
[6]	A.V. Efimov, "On linear summation methods for Fourier series" Izv. Akad. Nauk. SSSR Ser. Mat. , 24 : 5 (1960) pp. 743–756 (In Russian)
[7]	S.A. Telyakovskii, "Integrability conditions for trigonometric series and their application to the study of linear summation methods for Fourier series" Izv. Akad. Nauk. SSSR Ser. Mat. , 28 : 6 (1964) pp. 1209–1236 (In Russian)

Comments

A basic example of a singular integral is the Hilbert singular integral (cf. also Hilbert transform).

References

[a1]	E.M. Stein, "Singular integrals and differentiability properties of functions" , Princeton Univ. Press (1979)