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Hardy–Littlewood Maximal Function
In mathematics, the Hardy–Littlewood maximal operator ''M'' is a significant non-linear operator used in real analysis and harmonic analysis. Definition The operator takes a locally integrable function ''f'' : R''d'' → C and returns another function ''Mf''. For any point ''x'' ∈ R''d'', the function ''Mf'' returns the maximum of a set of reals, namely the set of average values of ''f'' for all the balls ''B''(''x'', ''r'') of any radius ''r'' at ''x''. Formally, : Mf(x)=\sup_ \frac\int_ , f(y), \, dy where , ''E'', denotes the ''d''-dimensional Lebesgue measure of a subset ''E'' ⊂ R''d''. The averages are jointly continuous in ''x'' and ''r'', therefore the maximal function ''Mf'', being the supremum over ''r'' > 0, is measurable. It is not obvious that ''Mf'' is finite almost everywhere. This is a corollary of the Hardy–Littlewood maximal inequality. Hardy–Littlewood maximal inequality This theorem of G. H. Hardy and J. E. Littlewood states that ''M' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Definition 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, :\mu (A) = \sup \. This property is sometimes referred to in words as "approximation from within by compact sets." Some authors use the term tight as a synonym for inner regular. This use of the term is closely related to tightness of a family of measures, since a finite measure ''μ'' is inner regular if and only if, for all ''ε'' > 0, there is some compact subset ''K'' of ''X'' such that ''μ''(''X'' \ ''K'') < ''ε''. This is precise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions. Scope Construction of the real numbers The theorems of real analysis rely on the properties of the real number system, which must be established. The real number system consists of an uncountable set (\mathbb), together with two binary operations denoted and , and an order denoted . The operations make the real numbers a field, and, along with the order, an ordered field. The real number system is the unique ''complete ordered field'', in the sense that any other complete ordered field is isomorphic to it. Intuitively, completeness means ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerald Teschl
Gerald Teschl (born 12 May 1970 in Graz) is an Austrian mathematical physicist and professor of mathematics. He works in the area of mathematical physics; in particular direct and inverse spectral theory with application to completely integrable partial differential equations (soliton equations). Career After studying physics at the Graz University of Technology (diploma thesis 1993), he continued with a PhD in mathematics at the University of Missouri. 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. Since then he has been a professor of mathematics there. In 1997 he received the Ludwig Boltzmann Prize from the Austrian Physical Society, 1999 the Prize of the Austrian Mathematical Society. In 2006 he was awarded with the pres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John B
John Bryn Williams (born 1977), known as John B, is an English disc jockey and electronic music producer. He is widely recognised for his eccentric clothing and wild hair and his production of several cutting edge drum and bass tracks. John B ranked number 76 in ''DJ Magazine''s 2010 Top 100 DJs annual poll, announced on 27 October 2010. Career Williams was born on 12 July 1977 in Maidenhead, Berkshire. He started producing music around the age of 14, and now is the head of drum and bass record label Beta Recordings, together with its more specialist drum and bass sub-labels Nu Electro, Tangent, and Chihuahua. He also has releases on Formation Records, Metalheadz and Planet Mu. Williams was ranked 92nd drum and bass DJ on the 2009 ''DJ Magazine'' top 100. Style While his trademark sound has evolved through the years, it generally involves female vocals and trance-like synths (a style which has been dubbed "trance and bass", "trancestep" and "futurestep" by listeners). His m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rising Sun Lemma
In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathema ..., used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zygmund lemma. The lemma is stated as follows:See: * * * * :Suppose ''g'' is a real-valued continuous function on the interval 'a'',''b''and ''S'' is the set of ''x'' in 'a'',''b''such that there exists a ''y''∈(''x'',''b''] with ''g''(''y'') > ''g''(''x''). (Note that ''b'' cannot be in ''S'', though ''a'' may be.) Define ''E'' = ''S'' ∩ (''a'',''b''). :Then ''E'' is an open set, and it may be written as a countable union of disjoint intervals ::E=\bigcup_k (a_k,b_k) :such that ''g''(''a''''k'') = ''g''(''b''''k''), unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyadic Cubes
In mathematics, the dyadic cubes are a collection of cubes in R''n'' of different sizes or scales such that the set of cubes of each scale partition R''n'' 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. For example, to study an arbitrary subset of ''A'' of Euclidean space, one may instead replace it by a union of dyadic cubes of a particular size that cover the set. One can consider this set as a pixelized version of the original set, and as smaller cubes are used one gets a clearer image of the set ''A''. Most notable appearances of dyadic cubes include the Whitney extension theorem and the Calderón–Zygmund lemma. Dyadic cubes in Euclidean space In Euclidean space, dyadic cubes may be constructed as follows: for each integer ''k'' let Δ''k'' be the set of cubes in R''n'' of sidelength 2− ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elias Stein
Elias Menachem Stein (January 13, 1931 – December 23, 2018) was an American mathematician who was a leading figure in the field of harmonic analysis. He was the Albert Baldwin Dod Professor of Mathematics, Emeritus, at Princeton University, where he was a faculty member from 1963 until his death in 2018. Biography Stein was born in Antwerp Belgium, to Elkan Stein and Chana Goldman, Ashkenazi Jews from Belgium.University of St Andrews, Scotland - School of Mathematics and Statistics: "Elias Menachem Stein" by J.J. O'Connor and E F Robertson February 2010 After the German invasion in 1940, the Stein family fl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Definition A right approximate identity in a Banach algebra ''A'' is a net \ such that for every element ''a'' of ''A'', \lim_\lVert ae_\lambda - a \rVert = 0. Similarly, a left approximate identity in a Banach algebra ''A'' is a net \ such that for every element ''a'' of ''A'', \lim_\lVert e_\lambda a - a \rVert = 0. An approximate identity is a net which is both a right approximate identity and a left approximate identity. C*-algebras For C*-algebras, a right (or left) approximate identity consisting of self-adjoint elements is the same as an approximate identity. The net of all positive elements in ''A'' of norm ≤ 1 with its natural order is an approximate identity for any C*-algebra. This is called the canonical approximate identity of a C*-algebra. Ap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Integration Theorem
A fraction is one or more equal parts of something. Fraction may also refer to: * Fraction (chemistry), a quantity of a substance collected by fractionation * Fraction (floating point number), an (ambiguous) term sometimes used to specify a part of a floating point number * Fraction (politics), a subgroup within a parliamentary party * Fraction (radiation therapy), one unit of treatment of the total radiation dose of radiation therapy that is split into multiple treatment sessions * Fraction (religion), the ceremonial act of breaking the bread during Christian Communion People with the surname * Matt Fraction, a comic book author See also * Algebraic fraction, an indicated division in which the divisor, or both dividend and divisor, are algebraic expressions ** Irrational fraction, a type of algebraic fraction * Faction (other) * ''Frazione'', a type of administrative division of an Italian ''commune'' * Free and Independent Fraction, a Romanian political party * Part (d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fatou's Theorem
In mathematics, specifically 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. Motivation and statement of theorem If we have a holomorphic function f defined on the open unit disk \mathbb=\, 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. This defines a new function: :\begin f_r:S^1 \to \Complex \\ f_(e^)=f(re^) \end where :S^1:=\=\, is the unit circle. Then it would be expected that the values of the extension of f onto the circle should be the limit of these functions, and so the question reduces to determining when f_r converges, and in what sense, as r\to 1, and how well defined is this limit. In particular, if the L^p norms of these f_r are well beh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rademacher's Theorem
In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If is an open subset of and is Lipschitz continuous, then is differentiable almost everywhere in ; that is, the points in at which is ''not'' differentiable form a set of Lebesgue measure zero. Differentiability here refers to infinitesimal approximability by a linear map, which in particular asserts the existence of the coordinate-wise partial derivatives. Sketch of proof The one-dimensional case of Rademacher's theorem is a standard result in introductory texts on measure-theoretic analysis. In this context, it is natural to prove the more general statement that any single-variable function of bounded variation is differentiable almost everywhere. (This one-dimensional generalization of Rademacher's theorem fails to extend to higher dimensions.) One of the standard proofs of the general Rademacher theorem was found by Charles Morrey. In the following, let denote a Lipschitz-c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
