View source for Singular integral

An integral

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with a singularity at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s0855702.png" />, defined for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s0855703.png" /> integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s0855704.png" />, whose kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s0855705.png" /> satisfies the following conditions: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s0855706.png" /> and an arbitrary interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s0855707.png" />,

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and

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557011.png" /> depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557013.png" /> and not on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557014.png" />. If (1), (2) and (3) are fulfilled uniformly on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557015.png" />-set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557016.png" />, then the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557017.png" /> is said to be uniformly singular on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557018.png" />. Most attention has been paid to properties of so-called positive kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557019.png" />; Dirichlet kernels (cf. [[Dirichlet kernel|Dirichlet kernel]])

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Fejér kernels (cf. [[Fejér singular integral|Fejér integral]])

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Poisson–Abel kernels (cf. [[Poisson integral|Poisson integral]])

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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 [[#References|[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]]; [[Fourier series in orthogonal polynomials|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557023.png" /> of bounded variation. D.K. Faddeev [[#References|[2]]] established necessary and sufficient conditions for the convergence of singular integrals at a [[Lebesgue point|Lebesgue point]] in the case of summable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557024.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557025.png" /> is bounded in norm, i.e. the sequence of integrals

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557027.png" />, that is, an integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557028.png" /> which monotonically increases on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557029.png" /> and monotonically decreases on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557030.png" />, such that for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085570/s08557031.png" />,

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where

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====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Lebesgue,   "Sur les intégrales singulières"  ''Ann. Fac. Sci. Univ. Toulouse'' , '''1'''  (1909)  pp. 25–117</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.P. Korovkin,   "Linear operators and approximation theory" , Hindushtan Publ. Comp.  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.P. Natanson,   "Functions of a real variable" , '''1–2''' , F. Ungar  (1955–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G. Alexits,   "Convergence problems of orthogonal series" , Pergamon  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.V. Efimov,   "On linear summation methods for Fourier series"  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''24''' :  5  (1960)  pp. 743–756  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  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)</TD></TR></table>



====Comments====
A basic example of a singular integral is the [[Hilbert singular integral|Hilbert singular integral]] (cf. also [[Hilbert transform|Hilbert transform]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.M. Stein,   "Singular integrals and differentiability properties of functions" , Princeton Univ. Press  (1979)</TD></TR></table>