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Hardy–Littlewood Maximal Operator In mathematics, the Hardy–Littlewood maximal operator M is a significant nonlinear operator used in real analysis and harmonic analysis [...More...]  "Hardy–Littlewood Maximal Operator" on: Wikipedia Yahoo Parouse 

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 [...More...]  "Mathematics" on: Wikipedia Yahoo Parouse 

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 [...More...]  "Elias M. Stein" on: Wikipedia Yahoo Parouse 

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 [...More...]  "Special" on: Wikipedia Yahoo Parouse 

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 RheinischWestfälischen Technische Hochschule Aachen (1996/97), he moved to Vienna, where he received his Habilitation at the University of Vienna in May 1998 [...More...]  "Gerald Teschl" on: Wikipedia Yahoo Parouse 

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. SpringerVerlag. 1972. ISBN 3540060731. Bounded Analytic Functions. Academic Press. 1981. ISBN 0122761502. [4] with Donald E. Marshall: Harmonic Measures. Cambridge University Press. 2005. ISBN 0521470188. [5]References[edit]^ "Archived copy" (PDF). Archived from the original (PDF) on December 26, 2010 [...More...]  "John B. Garnett" on: Wikipedia Yahoo Parouse 

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 [...More...]  "Dyadic Cubes" on: Wikipedia Yahoo Parouse 

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 [...More...]  "Elias Stein" on: Wikipedia Yahoo Parouse 

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 [...More...]  "Approximation Of The Identity" on: Wikipedia Yahoo Parouse 

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 [...More...]  "Fatou's Theorem" on: Wikipedia Yahoo Parouse 

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: SpringerVerlag, pp. xiv+676, ISBN 9783540606567, MR 0257325, Zbl 0176.00801 [...More...]  "Rademacher's Theorem" on: Wikipedia Yahoo Parouse 

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 complexvalued 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 [...More...]  "Lebesgue Differentiation Theorem" on: Wikipedia Yahoo Parouse 

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 σalgebra 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 σalgebra 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 [...More...]  "Inner Regular Measure" on: Wikipedia Yahoo Parouse 

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 [...More...]  "Vitali Covering Lemma" on: Wikipedia Yahoo Parouse 

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 [...More...]  "Essential Supremum" on: Wikipedia Yahoo Parouse 

Marcinkiewicz Theorem In mathematics, the Marcinkiewicz interpolation theorem, discovered by Józef Marcinkiewicz (1939), is a result bounding the norms of nonlinear operators acting on Lp spaces. Marcinkiewicz' theorem is similar to the Riesz–Thorin theorem about linear operators, but also applies to nonlinear 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 [...More...]  "Marcinkiewicz Theorem" on: Wikipedia Yahoo Parouse 