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Hardy–Littlewood Maximal Operator
In mathematics, the Hardy–Littlewood maximal operator M is a significant non-linear operator used in real analysis and harmonic analysis
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Mathematics
Mathematics
Mathematics
(from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity,[1] structure,[2] space,[1] and change.[3][4][5] It has no generally accepted definition.[6][7] Mathematicians seek out patterns[8][9] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist
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Elias M. Stein
Elias Menachem Stein (born January 13, 1931) is a mathematician. He is a leading figure in the field of harmonic analysis. He is a professor emeritus of Mathematics
Mathematics
at Princeton University.Contents1 Biography 2 Personal life 3 Bibliography 4 Notes 5 References 6 External linksBiography[edit] Stein was born to Elkan Stein and Chana Goldman, Ashkenazi Jews
Ashkenazi Jews
from Belgium.[1] After the German invasion in 1940, the Stein family fled to the United States, first arriving in New York City.[1] He graduated from Stuyvesant High School
Stuyvesant High School
in 1949,[1] where he was classmates with future Fields Medalist
Fields Medalist
Paul Cohen,[2] before moving on to the University of Chicago
University of Chicago
for college. In 1955, Stein earned a Ph.D
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Special
Special
Special
or specials may refer to:Contents1 Music 2 Film and television 3 Other uses 4 See alsoMusic[edit] Special
Special
(album), a 1992
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Gerald Teschl
Gerald Teschl
Gerald Teschl
(born May 12, 1970 in Graz) is an Austrian mathematical physicist and Professor
Professor
of Mathematics. He is working in the area of mathematical physics; in particular direct and inverse spectral theory with application to completely integrable partial differential equations (soliton equations). Career[edit] After studying physics at the Graz
Graz
University of Technology (diploma thesis 1993), he continued with a PhD
PhD
in mathematics at the University of Missouri – Columbia. The title of his thesis supervised by Fritz Gesztesy was Spectral Theory for Jacobi Operators (1995). After a postdoctoral position at the Rheinisch-Westfälischen Technische Hochschule Aachen (1996/97), he moved to Vienna, where he received his Habilitation at the University of Vienna in May 1998
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John B. Garnett
John Brady Garnett (born December 15, 1940) is an American mathematician at the University of California, Los Angeles, known for his work in harmonic analysis. He received his Ph.D. at the University of Washington in 1966, under the supervision of Irving Glicksberg. He received the Steele Prize for Mathematical Exposition in 2003 for his book, Bounded Analytic Functions.[1] As of June 2011, he has supervised the dissertations of 25 students [2] including Peter Jones. In 2012 he became a fellow of the American Mathematical Society.[3] Publications[edit]Analytic Capacity and Measure. Springer-Verlag. 1972. ISBN 3-540-06073-1.  Bounded Analytic Functions. Academic Press. 1981. ISBN 0-122-76150-2. [4] with Donald E. Marshall: Harmonic Measures. Cambridge University Press. 2005. ISBN 0-521-47018-8. [5]References[edit]^ "Archived copy" (PDF). Archived from the original (PDF) on December 26, 2010
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Dyadic Cubes
In mathematics, the dyadic cubes are a collection of cubes in Rn of different sizes or scales such that the set of cubes of each scale partition Rn and each cube in one scale may be written as a union of cubes of a smaller scale. These are frequently used in mathematics (particularly harmonic analysis) as a way of discretizing objects in order to make computations or analysis easier
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Elias Stein
Elias Menachem Stein (born January 13, 1931) is a mathematician. He is a leading figure in the field of harmonic analysis. He is a professor emeritus of Mathematics
Mathematics
at Princeton University.Contents1 Biography 2 Personal life 3 Bibliography 4 Notes 5 References 6 External linksBiography[edit] Stein was born to Elkan Stein and Chana Goldman, Ashkenazi Jews
Ashkenazi Jews
from Belgium.[1] After the German invasion in 1940, the Stein family fled to the United States, first arriving in New York City.[1] He graduated from Stuyvesant High School
Stuyvesant High School
in 1949,[1] where he was classmates with future Fields Medalist
Fields Medalist
Paul Cohen,[2] before moving on to the University of Chicago
University of Chicago
for college. In 1955, Stein earned a Ph.D
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Approximation Of The Identity
In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element.Contents1 Definition 2 C*-algebras 3 Convolution
Convolution
algebras 4 Rings 5 See alsoDefinition[edit] A right approximate identity in a Banach algebra A is a net e λ : λ ∈ Λ displaystyle ,e_ lambda colon lambda in Lambda , such that for every element a of A, lim λ ∈ Λ ‖ a e λ − a ‖ = 0. displaystyle lim _ lambda in Lambda lVert ae_ lambda -arVert =0
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Fatou's Theorem
In complex analysis, Fatou's theorem, named after Pierre Fatou, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk.Contents1 Motivation and statement of theorem 2 Discussion 3 See also 4 ReferencesMotivation and statement of theorem[edit] If we have a holomorphic function f displaystyle f defined on the open unit disk D = z : z < 1 displaystyle mathbb D = z:z<1 , it is reasonable to ask under what conditions we can extend this function to the boundary of the unit disk. To do this, we can look at what the function looks like on each circle inside the disk centered at 0, each with some radius r displaystyle r
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Rademacher's Theorem
In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If U is an open subset of Rn and  f : U → Rm  is Lipschitz continuous, then f  is differentiable almost everywhere in U; that is, the points in U at which f  is not differentiable form a set of Lebesgue measure
Lebesgue measure
zero. Generalizations[edit] There is a version of Rademacher's theorem that holds for Lipschitz functions from a Euclidean space
Euclidean space
into an arbitrary metric space in terms of metric differentials instead of the usual derivative. See also[edit]Alexandrov theoremReferences[edit]Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, 153, Berlin–Heidelberg–New York: Springer-Verlag, pp. xiv+676, ISBN 978-3-540-60656-7, MR 0257325, Zbl 0176.00801 
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Lebesgue Differentiation Theorem
In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for Henri Lebesgue.Contents1 Statement 2 Proof 3 Discussion of proof 4 Discussion 5 See also 6 ReferencesStatement[edit] For a Lebesgue integrable
Lebesgue integrable
real or complex-valued function f on Rn, the indefinite integral is a set function which maps a measurable set A  to the Lebesgue integral of f ⋅ 1 A displaystyle fcdot mathbf 1 _ A , where 1 A displaystyle mathbf 1 _ A denotes the characteristic function of the set A
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Inner Regular Measure
In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets.Contents1 Definition 2 Examples 3 References 4 See alsoDefinition[edit] Let (X, T) be a Hausdorff topological space and let Σ be a σ-algebra on X that contains the topology T (so that every open set is a measurable set, and Σ is at least as fine as the Borel σ-algebra on X). Then a measure μ on the measurable space (X, Σ) is called inner regular if, for every set A in Σ, μ ( A ) = sup μ ( K ) ∣ compact  K ⊆ A . displaystyle mu (A)=sup mu (K)mid text compact Ksubseteq A
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Vitali Covering Lemma
In mathematics, the Vitali covering lemma
Vitali covering lemma
is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem
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Essential Supremum
In mathematics, the concepts of essential supremum and essential infimum are related to the notions of supremum and infimum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for all elements in a set, but rather almost everywhere, i.e., except on a set of measure zero.Contents1 Definition 2 Examples 3 Properties 4 See also 5 Notes 6 ReferencesDefinition[edit] Let f : X → R be a real valued function defined on a set X. A real number a is called an upper bound for f if f(x) ≤ a for all x in X, i.e., if the set f − 1 ( a , ∞ ) = x ∈ X : f ( x ) > a displaystyle f^ -1 (a,infty )= xin X:f(x)>a is empty
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Marcinkiewicz Theorem
In mathematics, the Marcinkiewicz interpolation theorem, discovered by Józef Marcinkiewicz (1939), is a result bounding the norms of non-linear operators acting on Lp spaces. Marcinkiewicz' theorem is similar to the Riesz–Thorin theorem about linear operators, but also applies to non-linear operators.Contents1 Preliminaries 2 Formulation 3 Applications and examples 4 History 5 See also 6 ReferencesPreliminaries[edit] Let f be a measurable function with real or complex values, defined on a measure space (X, F, ω). The distribution function of f is defined by λ f ( t ) = ω x ∈ X ∣ f ( x ) > t . displaystyle lambda _ f (t)=omega left xin Xmid f(x)>tright
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