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Marcinkiewicz Theorem
In mathematics, particularly in functional analysis, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Transform
In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, of a real variable and produces another function of a real variable . The Hilbert transform is given by the Cauchy principal value of the convolution with the function 1/(\pi t) (see ). The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° (/2 radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see ). The Hilbert transform is important in signal processing, where it is a component of the Analytic signal, analytic representation of a real-valued signal . The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions. Definition The Hilbert transform of can be thought of as the convolution of with the function , known as the Cauchy ker ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Guido Weiss
Guido Leopold Weiss (29 December 1928–25 December 2021 in St. Louis) was an American mathematician, working in analysis, especially Fourier analysis and harmonic analysis. Childhood Weiss was born in Trieste Italy into a Jewish family. His parents, Edoardo and Vonda Weiss, were both psychiatrists. Weiss was forced out of school at the age of 9, upon the passage of Italy's Italian Racial Laws, which forbade all Jewish children from attending public school. He attended a Jewish school in Rome until the end of 1939 when his father was sponsored by members of the Menninger family to emigrate to America. The family settled in Topeka, Kansas. Career Weiss studied at the University of Chicago, where he received in 1951 his master's degree and in 1956 under Antoni Zygmund his PhD with thesis ''On certain classes of function spaces and on the interpolation of sublinear operators''. At DePaul University he became an instructor in 1955, an assistant professor in 1956, and in 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Allen Hunt
Richard Allen Hunt (16 June 1937 – 22 March 2009) was an American mathematician. He graduated from Washington University in St. Louis in 1965 with a dissertation entitled ''Operators acting on Lorentz Spaces''. An important result of states that the Fourier expansion of a function in ''L''''p'', ''p'' > 1, converges almost everywhere. The case ''p=2'' is due to Lennart Carleson, and for this reason the general result is called the Carleson-Hunt theorem. Hunt was the 1969 recipient of the Salem Prize. He was a faculty member at Purdue University from 1969 to 2000, when he retired as professor emeritus. See also * Convergence of Fourier series In mathematics, the question of whether the Fourier series of a given periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily gi ... References * * 20th-century American mathematicians 21st-century ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Integral Operator
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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antoni Zygmund
Antoni Zygmund (December 26, 1900 – May 30, 1992) was a Polish-American mathematician. He worked mostly in the area of mathematical analysis, including harmonic analysis, and he is considered one of the greatest analysts of the 20th century. Zygmund was responsible for creating the Chicago school of mathematical analysis together with his doctoral student Alberto Calderón, for which he was awarded the National Medal of Science in 1986. Biography Born in Warsaw, Zygmund obtained his Ph.D. from the University of Warsaw (1923) and was a professor at Stefan Batory University at Wilno from 1930 to 1939, when World War II broke out and Poland was occupied. In 1940 he managed to emigrate to the United States, where he became a professor at Mount Holyoke College in South Hadley, Massachusetts. In 1945–1947 he was a professor at the University of Pennsylvania, and from 1947, until his retirement, at the University of Chicago. He was a member of several scientific societies. Fro ... [...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 versions 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 \geq 0 , we will write cB for the ball B(x,cr). Finite version Infinite version The following proof is based on . Remarks *In the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sublinear Operator
In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. Unlike seminorms, a sublinear function does not have to be nonnegative-valued and also does not have to be absolutely homogeneous. Seminorms are themselves abstractions of the more well known notion of norms, where a seminorm has all the defining properties of a norm that it is not required to map non-zero vectors to non-zero values. In functional analysis the name Banach functional is sometimes used, reflecting that they are most commonly used when applying a general formulation of the Hahn–Banach theorem. The notion of a sublinear function was introduced by Stefan Banach when he proved his version of the Hahn-Banach theorem. There is also a different notion in computer science, described below, that also goes by the name "subli ... [...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 \to \mathbb C and returns another function Mf : \R^d \to , \infty/math>, where Mf (x) is the supremum of the average of f among all possible balls centered on x. Formally, : Mf(x)=\sup_ \frac\int_ , f(y), \, dy , where , ''E'', denotes the ''d''-dimensional Lebesgue measure of a subset ''E'' ⊂ R''d'', and B(x,\,r) is the ball of radius, r>0, centered at the point x\in\mathbb^d. Since f is locally integrable, the averages are jointly continuous in ''x'' and ''r'', so the maximal function ''Mf'', being the supremum over ''r'' > 0, is measurable. A nontrivial corollary of the Hardy–Littlewood maximal inequality states that Mf is finite almost everywhere for functions in L^1 . Hardy–Littlewood maximal inequality This theorem of G. H. Hardy and J. E. L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ahn 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parseval's Theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh. Although the term "Parseval's theorem" is often used to describe the unitarity of ''any'' Fourier transform, especially in physics, the most general form of this property is more properly called the Plancherel theorem. Statement of Parseval's theorem Suppose that A(x) and B(x) are two complex-valued functions on \mathbb of period 2 \pi that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series :A(x)=\sum_^\infty a_ne^ and :B(x)=\sum_^\infty b_ne^ respecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Fourier Transform
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency#Frequency_of_waves, frequency and phase (waves), phase information about a wave then we may reconstruct the original wave precisely. The theorem says that if we have a function f:\R \to \Complex satisfying certain conditions, and we use the Fourier transform#Other conventions, convention for the Fourier transform that :(\mathcalf)(\xi):=\int_ e^ \, f(y)\,dy, then :f(x)=\int_ e^ \, (\mathcalf)(\xi)\,d\xi. In other words, the theorem says that :f(x)=\iint_ e^ \, f(y)\,dy\,d\xi. This last equation is called the Fourier integral theorem. Another way to state the theorem is that if R is the flip operator i.e. (Rf)(x) := f(-x), then :\mathcal^=\mathcalR=R\mathcal. The theorem holds if both f and its Fourier transform are absolutely integrable function, ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
