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Functions in harmonic analysis mathematics
In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly
Singular_integral
Equations with an unknown function under an integral sign
Regular: An integral equation is called regular if the integrals used are all proper integrals. Singular or weakly singular: An integral equation is called
Integral_equation
Mathematical concept
In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rn and Tn through convolution by distributions;
Singular integral operators of convolution type
Singular_integral_operators_of_convolution_type
Area of mathematical analysis
modern harmonic analysis also studies maximal functions, singular integrals, oscillatory integrals, Fourier multipliers, Littlewood–Paley theory, and spectral
Harmonic_analysis
{\displaystyle (a,b,c)} this way leads to what are called singular integrals. Usually, most integrals fall into three categories defined above, but it may
First-order partial differential equation
First-order_partial_differential_equation
In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The
Singular integral operators on closed curves
Singular_integral_operators_on_closed_curves
Method for assigning values to integrals
improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval
Cauchy_principal_value
Argentine mathematician
and his mentor, the analyst Antoni Zygmund, developed the theory of singular integral operators. This created the "Chicago School of (hard) Analysis" (sometimes
Alberto_Calderón
Australian and American mathematician (born 1975)
101–121. Coifman, R. R.; Meyer, Yves On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc. 212 (1975), 315–331. Coifman
Terence_Tao
Number, approximately 3.14
kernel. The Hilbert transform H is the integral transform given by the Cauchy principal value of the singular integral H f ( t ) = 1 π ∫ − ∞ ∞ f ( x ) d x
Pi
Integral transform and linear operator
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
Hilbert_transform
bounded on these weighted Lp spaces. In fact, any Calderón-Zygmund singular integral operator is also bounded on these spaces. Let us describe a simpler
Muckenhoupt_weights
Polish mathematician (1900–1992)
the most significant were the results he obtained with Calderón on singular integral operators. George G. Lorentz called it Zygmund's crowning achievement
Antoni_Zygmund
Method of evaluating certain integrals along paths in the complex plane
a contour integral between fixed endpoints is not governed by the precise shape of the contour, but by its winding around the singularities of the integrand
Contour_integration
Soviet mathematician
linear elasticity, singular integrals and numerical analysis: he is best known for the introduction of the symbol of a singular integral operator, which
Solomon_Mikhlin
Type of singular integral operator
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
Riesz_transform
Georgian mathematician (1891–1976)
basic problems of the mathematical theory of elasticity" (1933) and "Singular Integral Equations" (1947). During World War II Muskhelishvili was responsible
Nikoloz_Muskhelishvili
Operation in mathematical calculus
integral is the continuous analog of a sum, and is used to calculate areas, volumes, and their generalizations. The process of computing an integral,
Integral
Class of integral and differential operator
mathematics, in the field of harmonic analysis, an oscillatory integral operator is an integral operator of the form T λ u ( x ) = ∫ R n e i λ S ( x , y )
Oscillatory_integral_operator
Analysis theorem
a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calderón and Antoni Zygmund
Calderón–Zygmund_lemma
Green's function for Laplacian
general nature, it is a singular integral operator, defined by convolution with a function having a mathematical singularity at the origin, the Newtonian
Newtonian_potential
Nonlocal mathematical operator
{\displaystyle p\in [1,\infty )} . The Laplacian can also be viewed as a singular integral operator which is defined as the following limit taken in X {\displaystyle
Fractional_Laplacian
Differential operator in mathematics
above. Equivalently, the fractional Laplacian can be defined by a singular integral: ( − Δ ) α / 2 f ( x ) = c n , α PV ∫ R n f ( x ) − f ( y ) | x −
Laplace_operator
Branch of mathematical analysis
one to consider powers of D. The operators arising are examples of singular integral operators; and the generalisation of the classical theory to higher
Fractional_calculus
Matrix used in complex analysis
derivation of the Grunsky inequalities using reproducing kernels and singular integral operators in geometric function theory; a more recent related approach
Grunsky_matrix
French mathematician (1889–1943)
working in potential theory, partial differential equations, singular integrals and singular integral equations: he is mainly known for his solution of the regular
Georges_Giraud
understanding, for example, the differentiability properties of functions, singular integrals and partial differential equations. They often provide a deeper and
Maximal_function
Complex analysis theorem
\nu )-i{\mathcal {P}}{\Big (}{\frac {1}{\omega \pm \nu }}{\Big )}} Singular integral operators on closed curves (account of the Sokhotski–Plemelj theorem
Sokhotski–Plemelj_theorem
Potential in mathematics
^{n/2}2^{\alpha }{\frac {\Gamma (\alpha /2)}{\Gamma ((n-\alpha )/2)}}.} This singular integral is well-defined provided f decays sufficiently rapidly at infinity
Riesz_potential
Definite integral of a scalar or vector field along a path
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
Line_integral
Mathematical problems related to differential equations
Clancey, K.; Gohberg, I. (1981), Factorization of matrix functions and singular integral operators, Oper. Theory: Advances and Appl., vol. 3, Basel-Boston-Stuttgart:
Riemann–Hilbert_problem
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
Lists_of_integrals
Algebraic structure used in topology
} . Differentials of a periodic function have the property that their integral over a whole period is zero: by the fundamental theorem of calculus, ∫
Cohomology
Mathematical potential
Fractional Schrödinger equation Yukawa potential Stein, Elias (1970). Singular integrals and differentiability properties of functions. Princeton University
Bessel_potential
Generalized function whose value is zero everywhere except at zero
ISBN 978-0-387-97655-6. Kracht, Manfred; Kreyszig, Erwin (1989), "On singular integral operators and generalizations", in Rassias, Themistocles M. (ed.)
Dirac_delta_function
Framework for integrating diverse theories
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
Integral_theory
Special function defined by an integral
denotes the natural logarithm. The function 1/(ln t) has a singularity at t = 1, and the integral for x > 1 is interpreted as a Cauchy principal value, li
Logarithmic_integral_function
Matrix decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a scaling, followed
Singular_value_decomposition
American mathematician
multilinear singular integral operators", Acta Mathematica 159(1–2): 51–80. 1990: "A T(b) theorem with remarks on analytic capacity and the Cauchy integral", Colloquium
F._Michael_Christ
Attribute of a mathematical function
number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be
Residue_(complex_analysis)
Class of operator mapping
u {\displaystyle Au} at y {\displaystyle y} . An example of a singular integral operator is the fractional Laplacian ( − Δ ) s f ( x ) = c d , s ∫
Nonlocal_operator
One of six awards by the Wolf Foundation
Fourier integral operators to linear partial differential equations. 1989 Alberto Calderón Argentina for his groundbreaking work on singular integral operators
Wolf_Prize_in_Mathematics
Type of differential operator
differential inequalities with m ≤ 0, it can be shown that the kernel is a singular integral kernel. Differential algebra for a definition of pseudo-differential
Pseudo-differential_operator
Russian mathematician (1937–2008)
the Hua Luogeng problem of finding the exponent of convergency of the integral: ϑ 0 = ∫ − ∞ + ∞ ⋯ ∫ − ∞ + ∞ | ∫ 0 1 e 2 π i ( α n x n + ⋯ + α 1 x ) d
Anatoly_Karatsuba
Calculation of strain energy release rate
show that this integral is zero when the boundary Γ {\displaystyle \Gamma } is closed and encloses a region that contains no singularities and is simply
J-integral
Provides integral formulas for all derivatives of a holomorphic function
2). To find the integral of g ( z ) {\displaystyle g(z)} around the contour C {\displaystyle C} , we need to know the singularities of g ( z ) {\displaystyle
Cauchy's_integral_formula
Azerbaijani mathematician (1907–1981)
Sciences from 1962). His area of contributions embraced nonlinear singular integral equations, differential equations, potential theory and functional
Ashraf_Huseynov
French mathematician
exceptionelle. David is known for his research on Hardy spaces and on singular integral equations using the methods of Alberto Calderón. In 1998 David solved
Guy_David_(mathematician)
Integral of the Gaussian function, equal to sqrt(π)
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}}}
Gaussian_integral
Generalization of the Riemann integral
{1}{x^{3}}}\right).} This function has a singularity at 0, and is not Lebesgue-integrable. However, it seems natural to calculate its integral except over the interval
Henstock–Kurzweil_integral
Concept of complex analysis
the singularity of c {\displaystyle c} due to nature of isolated singularities. This may be used for calculation in cases where the integral can be
Residue_theorem
Type of operator in Fourier analysis
conjecture. Calderón–Zygmund lemma Marcinkiewicz theorem Singular integrals Singular integral operators of convolution type Duoandikoetxea 2001, Section
Multiplier_(Fourier_analysis)
boundary is just a Lipschitz curve was constructed by Calderón using singular integral operators and generalized by Stein (1970). It is sufficient to construct
Sobolev spaces for planar domains
Sobolev_spaces_for_planar_domains
Formulation of quantum mechanics
normalization, although singular potentials require careful treatment. Since the states obey the Schrödinger equation, the path integral must reproduce the
Path_integral_formulation
American mathematician (b. 1949)
Fourier analysis, in particular convergence, multipliers, divergence, singular integrals and Hardy spaces earned him a Fields Medal at the International Congress
Charles_Fefferman
Concept in mathematical analysis
improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context
Improper_integral
Indian mathematician
when he died in 1982. Prabhakar function T. R. Prabhakar (1971). "A singular integral equation with a generalized Mittag–Leffler function in the kernel"
Tilak_Raj_Prabhakar
Russian mathematician
"The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis". Algebra
Sergei_Treil
Azerbaijani mathematician
for polyharmonic equations, proposed abstract generalizations of singular integral operators and made some other contributions. In 1955, Khalilov became
Zahid_Khalilov
Special function defined by an integral
{\displaystyle x} , but the integral has to be understood in terms of the Cauchy principal value due to the singularity of the integrand at zero. For
Exponential_integral
Mathematical theory by discovered by Józef Marcinkiewicz
Zygmund, and was absent from his original works on the theory of singular integral operators. Later Zygmund (1956) realized that Marcinkiewicz's result
Marcinkiewicz interpolation theorem
Marcinkiewicz_interpolation_theorem
the Hα, β, γ function space and proved some theorems for nonlinear singular integral equations with Cauchy kernel within that space Heydar Huseynov — philosopher
List of Azerbaijani scientists and philosophers
List_of_Azerbaijani_scientists_and_philosophers
Vietnamese-American mathematician
PMC 534412. PMID 16593402. Phong D. H., Stein E. M. (1986). "Hilbert integrals, singular integrals, and Radon transforms I". Acta Mathematica. 157: 99–157. doi:10
Duong_Hong_Phong
Triangle inequality in Lp spaces
H. (1953). Geometrie der Zahlen. Chelsea.. Stein, Elias (1970). Singular integrals and differentiability properties of functions. Princeton University
Minkowski_inequality
Mathematical element
step in resolution of singularities since it gives a process for resolving singularities of codimension 1. For example, the integral closure of C [ x , y
Integral_element
Greek mathematician
Missouri. Grafakos' research interests include Fourier analysis, singular integrals, and Calderón–Zygmund theory. He is well known for his contributions
Loukas_Grafakos
Study of geometric properties of sets through measure theory
manifolds, Carnot groups, Heisenberg groups, etc. Connections to singular integrals, Fourier transform, Frostman measures, harmonic measures, etc Currents
Geometric_measure_theory
American mathematician (1931–2018)
complications of lymphoma in 2018, aged 87. Stein, Elias (1970). Singular Integrals and Differentiability Properties of Functions. Princeton University
Elias_M._Stein
Theorem in complex analysis
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard
Cauchy's_integral_theorem
Distribution concentrated on a set of measure zero
A singular distribution or singular continuous distribution is a probability distribution concentrated on a set of Lebesgue measure zero, for which the
Singular_distribution
Chinese-American mathematician
Fields Mathematics Institutions University of Michigan Thesis Nonlinear Singular Integrals and Analytic Dependence (1990) Doctoral advisor Ronald Coifman
Sijue_Wu
Polish mathematician (1931–2012)
title of doctor (with her dissertation titled On systems of strongly-singular integral equations, under the supervision of Witold Pogorzelski), and in 1964
Danuta_Przeworska-Rolewicz
Concept in mathematics
also a basic example in mathematical analysis, in connection with singular integral operators. Conjugate harmonic functions (and the transform between
Harmonic_conjugate
Partial differential equation
equation on C and relies on the Lp theory of the Beurling transform, a singular integral operator defined on Lp(C) for all 1 < p < ∞. The same method applies
Beltrami_equation
Operator equation in the style of Fredholm theory
in the integral is called the kernel. Such equations can be analyzed and solved by means of Laplace transform techniques. For a weakly singular kernel
Volterra_integral_equation
Technique in analytic number theory
have radius of convergence 1, so it has singularities on the unit circle – thus one cannot take the contour integral over the unit circle. The circle method
Hardy–Ramanujan–Littlewood circle method
Hardy–Ramanujan–Littlewood_circle_method
Mathematics award
spectrum of problems ranging from character sums in number theory to singular integral operators in Euclidean spaces". Mihaela Ignatova (2020), "in recognition
Sadosky_Prize
Theorem in functional analysis
Alexander A.; Nikolski, Nikolai K. (eds.). Systems, Approximation, Singular Integral Operators, and Related Topics. Operator Theory: Advances and Applications
Grothendieck_inequality
Type of vector space in math
Concepts and Contexts (3rd ed.), Thomson/Brooks/Cole. Stein, E (1970), Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press
Hilbert_space
Azerbaijani mathematician
partial differential equations on Lie groups Singular integrals, maximal functions, and other integral operators, generated by Bessel differential operators
Vagif_Guliyev
Summability method in physics
"Complex powers of an elliptic operator", in Calderón, Alberto P. (ed.), Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Proceedings of Symposia
Zeta_function_regularization
boundary value problems, including moving boundary value problems, singular integrals and classic harmonic analysis. In particular Clifford analysis has
Clifford_analysis
Problem of solving a partial differential equation subject to prescribed boundary values
Mathematical Society, ISBN 978-0-8218-4910-1 Stein, Elias M. (1970), Singular Integrals and Differentiability Properties of Functions, Princeton University
Dirichlet_problem
Awarded every year by the American Mathematical Society
Mathematical Society. ISBN 9780821812808. Stein, Elias M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical
Leroy_P._Steele_Prize
Estonian mathematician (1938–2024)
1007/978-94-010-2715-1. Vainikko, Gennadi (1993). Multidimensional Weakly Singular Integral Equations. Lecture Notes in Mathematics. Vol. 1549. Berlin and Heidelberg:
Gennadi_Vainikko
Function spaces generalizing finite-dimensional p norm spaces
which has broad applications to harmonic analysis and the study of singular integrals. As before, consider a measure space ( S , Σ , μ ) . {\displaystyle
Lp_space
Romanian-American mathematician (1935–2025)
limit theorem, transference principles, square functions and other singular integral techniques are now part of the daily arsenal of people working in
Alexandra_Bellow
{\displaystyle \textstyle n} -dimensional manifold M {\displaystyle M} , an integral manifold is an immersed (not necessarily embedded) submanifold whose tangent
Integrability conditions for differential systems
Integrability_conditions_for_differential_systems
Branch of mathematics studying functions of a complex variable
depend on the details of the contour, only how it winds around the singularities of the function. Being able to move a given contour to a more suitable
Complex_analysis
Soviet and Israeli mathematician and chemical engineer
07701. A masterpiece in the multidimensional theory of singular integrals and singular integral equations summarizing all the results from the beginning
Aizik_Volpert
Statement about integration on manifolds
from de Rham cohomology to singular cohomology. On the level of forms, this means: closed forms, i.e., dω = 0, have zero integral over boundaries, i.e. over
Generalized_Stokes_theorem
Prabhakar fractional integral have been extensively studied in the literature. Tilak Raj Prabhakar (1971). "A singular integral equation with a generalized
Prabhakar_function
American mathematician
University of Chicago, focusing her dissertation on the study of Singular Integrals (1956), basing her research on Zygmund's work on harmonic analysis
Vivienne_Esta_Morley
Special mathematical function
closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein
Polylogarithm
Bangladeshi Canadian mathematician and writer (1932–2015)
Ph.D. in 1965 with a thesis on the kinetic theory of plasma using singular integral equation techniques. After his PhD, he became an assistant professor
Mizan_Rahman
Theorem in complex analysis
Complex Analysis (1987), 3rd Ed., McGraw Hill, New York. Elias Stein, Singular integrals and differentiability properties of functions (1970), Princeton University
Fatou's_theorem
Hungarian mathematician
groups, in particular he researched orthogonal systems of functions, singular integrals, analytic functions, differential equations, set theory, function
Alfréd_Haar
1966 result in mathematical analysis
MR 0199632. Sjölin, Per (1971). "Convergence almost everywhere of certain singular integrals and multiple Fourier series". Arkiv för Matematik. 9 (1–2): 65–90
Carleson's_theorem
Unsolved problem in geometry
geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. In simple terms, the Hodge
Hodge_conjecture
Natural number
Arabic numeral. Linguistically, in English, "one" is a determiner for singular nouns and a gender-neutral pronoun. In mathematics, 1 is the multiplicative
1
SINGULAR INTEGRAL
SINGULAR INTEGRAL
Girl/Female
Muslim
Unique, Singular, Exclusive
Boy/Male
Afghan, Arabic, Danish, French, Kashmiri, Muslim, Pashtun, Sindhi
Singular; Unique; Alone; Exclusively; Unequalled; Exceptional; Peerless
Girl/Female
Indian
Unique, Singular
Girl/Female
Arabic, Gujarati, Indian, Kannada, Kashmiri, Muslim, Sindhi
Unique; Singular; Sole; Exclusive
Girl/Female
Indian
Unique, Singular, Exclusive
Biblical
lot, singular of Purim (lots, as in Cleromancy [casting of lots])
Girl/Female
Arabic, Muslim
Unique; Singular
Girl/Female
Indian
Unique, Singular, Exclusive
Girl/Female
Arabic, Muslim
Unique; Singular; Single
Girl/Female
Muslim
Unique, Singular
Surname or Lastname
English
English : from Middle English sengler, syngler ‘singular’ (Old French se(i)ngler), perhaps a nickname for a solitary person.German : topographic name for a valley dweller, from a diminutive of Middle High German senke ‘valley’ + the suffix -er, denoting an inhabitant.German : habitational name for someone from Singeln near Waldshut.German : variant of Sing 1.
Girl/Female
Celtic
Mythical daughter of Lyr.
Girl/Female
Arabic, Muslim
Singular; Unparalleled; Alone; Unique
Girl/Female
Arabic, Muslim
Present; Gift; Singular of Nihel
Girl/Female
Indian
Unique, Singular, Exclusive
Girl/Female
Muslim
Unique, Singular, Exclusive
Girl/Female
Arabic, Muslim
Present; Gift; Singular of Nihel
Girl/Female
Arabic, Muslim
Wish; Desire; Purpose; Use; Aim; Singular of Marib
Girl/Female
Muslim
Unique, Singular, Exclusive
Boy/Male
Muslim/Islamic
Singular exclusive, unequalled
SINGULAR INTEGRAL
SINGULAR INTEGRAL
Girl/Female
Muslim
Like deer
Girl/Female
Hindu, Indian
Name of a Nakhatra out of 27 Nakhatras
Boy/Male
Latin
F: Ameaning bringer of joy. In the Divine Comedy, Beatrice was Dante's guide through Paradise,...
Girl/Female
Hebrew Scottish
Rich. God beholds. The daughter of Shylock in Shakespeare's play 'The Merchant of Venice'.
Boy/Male
Muslim/Islamic
Success
Female
Spanish
Spanish name ARACELI means "altar of the sky."
Boy/Male
Hindu, Indian
Most Power Full
Girl/Female
Indian
Having golden garlands
Boy/Male
Hindu, Indian, Punjabi, Sikh
One who Realizes the Home Within
Male
Egyptian
, the deity of the soul of the sun.
SINGULAR INTEGRAL
SINGULAR INTEGRAL
SINGULAR INTEGRAL
SINGULAR INTEGRAL
SINGULAR INTEGRAL
a.
Being alone; belonging to, or being, that of which there is but one; unique.
n.
An individual instance; a particular.
adv.
So as to express one, or the singular number.
n.
Singular; wonderful; extraordinary.
adv.
Strangely; oddly; as, to behave singularly.
a.
Each; individual; as, to convey several parcels of land, all and singular.
a.
Relating to an angle or to angles; having an angle or angles; forming an angle or corner; sharp-cornered; pointed; as, an angular figure.
a.
Rather queer; somewhat singular.
a.
Fig.: Lean; lank; raw-boned; ungraceful; sharp and stiff in character; as, remarkably angular in his habits and appearance; an angular female.
adv.
In a singular manner; in a manner, or to a degree, not common to others; extraordinarily; as, to be singularly exact in one's statements; singularly considerate of others.
n.
Anything singular, rare, or curious.
a.
Standing by itself; out of the ordinary course; unusual; uncommon; strange; as, a singular phenomenon.
a.
Measured by an angle; as, angular distance.
a.
Of or pertaining to the people of an island; narrow; circumscribed; illiberal; contracted; as, insular habits, opinions, or prejudices.
a.
Denoting one person or thing; as, the singular number; -- opposed to dual and plural.
n.
Any one of numerous species of brachiopod shells belonging to the genus Lingula, and related genera. See Brachiopoda, and Illustration in Appendix.
n.
The singular number, or the number denoting one person or thing; a word in the singular number.
n.
See Kickshaws, the correct singular.
a.
Distinguished as existing in a very high degree; rarely equaled; eminent; extraordinary; exceptional; as, a man of singular gravity or attainments.
a.
Of or pertaining to an island; of the nature, or possessing the characteristics, of an island; as, an insular climate, fauna, etc.