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Maximal Function
Maximal functions appear in many forms in harmonic analysis (an area of mathematics). One of the most important of these is the Hardy–Littlewood maximal function. They play an important role in understanding, for example, the differentiability properties of functions, singular integrals and partial differential equations. They often provide a deeper and more simplified approach to understanding problems in these areas than other methods. The Hardy–Littlewood maximal function In their original paper, G.H. Hardy and J.E. Littlewood explained their maximal inequality in the language of cricket averages. Given a function ''f'' defined on R''n'', the uncentred Hardy–Littlewood maximal function ''Mf'' of ''f'' is defined as :(Mf)(x) = \sup_ \frac \int_B , f, at each ''x'' in R''n''. Here, the supremum is taken over balls ''B'' in R''n'' which contain the point ''x'' and , ''B'', denotes the measure of ''B'' (in this case a multiple of the radius of the ball raised to the powe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis). In the past two centuries, it has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience. The term "harmonics" originated as the Ancient Greek word ''harmonikos'', meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are Multiple (mathematics), integer multiples of one another, as are the frequencies of the Harmonic series (music), harmonics of music notes, but the term has been generalized beyond its original meaning. The classical Fourier transform on R''n'' is still an area of ongoing research, ... [...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] |
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] |
Cricket
Cricket is a bat-and-ball game played between two teams of eleven players on a field at the centre of which is a pitch with a wicket at each end, each comprising two bails balanced on three stumps. The batting side scores runs by striking the ball bowled at one of the wickets with the bat and then running between the wickets, while the bowling and fielding side tries to prevent this (by preventing the ball from leaving the field, and getting the ball to either wicket) and dismiss each batter (so they are "out"). Means of dismissal include being bowled, when the ball hits the stumps and dislodges the bails, and by the fielding side either catching the ball after it is hit by the bat, but before it hits the ground, or hitting a wicket with the ball before a batter can cross the crease in front of the wicket. When ten batters have been dismissed, the innings ends and the teams swap roles. The game is adjudicated by two umpires, aided by a third umpire and match referee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Inequality
In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function (mathematics), function of a random variable is greater than or equal to some positive Constant (mathematics), constant. It is named after the Russian mathematician Andrey Markov, although it appeared earlier in the work of Pafnuty Chebyshev (Markov's teacher), and many sources, especially in Mathematical analysis, analysis, refer to it as Chebyshev's inequality (sometimes, calling it the first Chebyshev inequality, while referring to Chebyshev's inequality as the second Chebyshev inequality) or Irénée-Jules Bienaymé, Bienaymé's inequality. Markov's inequality (and other similar inequalities) relate probabilities to expected value, expectations, and provide (frequently loose but still useful) bounds for the cumulative distribution function of a random variable. Statement If is a nonnegative random variable and , then the probability that is at least is at most th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vitali Covering Lemma
In mathematics, the 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. The covering theorem is credited to the Italian mathematician Giuseppe Vitali.. The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset ''E'' of R''d'' by a disjoint family extracted from a ''Vitali covering'' of ''E''. Vitali covering lemma There are two basic version of the lemma, a finite version and an infinite version. Both lemmas can be proved in the general setting of a metric space, typically these results are applied to the special case of the Euclidean space \mathbb^d. In both theorems we will use the following notation: if B = B(x,r) is a ball and c \in \mathbb, we will write cB for the ball B(x,cr). Finite version Theorem (Finite Covering Lemma). Let B_, \dots, B_ be any finite coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marcinkiewicz Interpolation Theorem
In mathematics, the Marcinkiewicz interpolation theorem, discovered by , is a result bounding the norms of non-linear operators acting on ''L''p spaces. Marcinkiewicz' theorem is similar to the Riesz–Thorin theorem about linear operators, but also applies to non-linear operators. Preliminaries 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 :\lambda_f(t) = \omega\left\. Then ''f'' is called weak L^1 if there exists a constant ''C'' such that the distribution function of ''f'' satisfies the following inequality for all ''t'' > 0: :\lambda_f(t)\leq \frac. The smallest constant ''C'' in the inequality above is called the weak L^1 norm and is usually denoted by \, f\, _ or \, f\, _. Similarly the space is usually denoted by ''L''1,''w'' or ''L''1,∞. (Note: This terminology is a bit misleading since the weak norm does not satisfy the triangl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Statement For a Lebesgue integrable real or complex-valued function ''f'' on R''n'', the indefinite integral is a set function which maps a measurable set ''A'' to the Lebesgue integral of f \cdot \mathbf_A, where \mathbf_ denotes the characteristic function of the set ''A''. It is usually written A \mapsto \int_ f\ \mathrm\lambda, with ''λ'' the ''n''–dimensional Lebesgue measure. The ''derivative'' of this integral at ''x'' is defined to be \lim_ \frac \int_f \, \mathrm\lambda, where , ''B'', denotes the volume (i.e., the Lebesgue measure) of a ball ''B'' centered at ''x'', and ''B'' → ''x'' means that the diameter of ''B'' tends to 0. The ''Lebesgue differentiation theorem'' states th ... [...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] |
Singular Integrals
In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator : T(f)(x) = \int K(x,y)f(y) \, dy, whose kernel function ''K'' : R''n''×R''n'' → R is singular along the diagonal ''x'' = ''y''. Specifically, the singularity is such that , ''K''(''x'', ''y''), is of size , ''x'' − ''y'', −''n'' asymptotically as , ''x'' − ''y'', → 0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over , ''y'' − ''x'', > ε as ε → 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on ''L''''p''(R''n''). The Hilbert transform The archetypal singular integral operator is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Delta Function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. The current understanding of the unit impulse is as a linear functional that maps every continuous function (e.g., f(x)) to its value at zero of its domain (f(0)), or as the weak limit of a sequence of bump functions (e.g., \delta(x) = \lim_ \frace^), which are zero over most of the real line, with a tall spike at the origin. Bump functions are thus sometimes called "approximate" or "nascent" delta distributions. The delta function was introduced by physicist Paul Dirac as a tool for the normalization of state vectors. It also has uses in probability theory and signal processing. Its validity was disputed until Laurent Schwartz developed the theory of distributions where it is defined as a linear form acting on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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