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Wiener–Lévy Theorem
Wiener–Lévy theorem is a theorem in Fourier analysis, which states that a function of an absolutely convergent Fourier series has an absolutely convergent Fourier series under some conditions. The theorem was named after Norbert Wiener and Paul Lévy. Norbert Wiener first proved Wiener's 1/''f'' theorem, see Wiener's theorem. It states that if has absolutely convergent Fourier series and is never zero, then its inverse also has an absolutely convergent Fourier series. Wiener–Levy theorem Paul Levy generalized Wiener's result, showing that Let F(\theta ) = \sum\limits_^\infty c_k e^, \quad\theta \in ,2\pi /math> be an absolutely convergent Fourier series with : \, F\, = \sum\limits_^\infty , c_k, has an absolutely convergent Fourier series. The proof can be found in the Zygmund's classic book ''Trigonometric Series''. Example Let H(\theta )=\ln(\theta ) and F(\theta ) = \sum\limits_^\infty p_k e^,(\sum\limits_^\infty p_k = 1 ) is characteristic function of discr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term ''Fourier ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norbert Wiener
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher in stochastic and mathematical noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems. Wiener is considered the originator of cybernetics, the science of communication as it relates to living things and machines, with implications for engineering, systems control, computer science, biology, neuroscience, philosophy, and the organization of society. Norbert Wiener is credited as being one of the first to theorize that all intelligent behavior was the result of feedback mechanisms, that could possibly be simulated by machines and was an important early step towards the development of modern artificial intelligence. Biography Youth Wiener was born in Columbia, Missouri, the first ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Lévy (mathematician)
Paul Pierre Lévy (15 September 1886 – 15 December 1971) was a French mathematician who was active especially in probability theory, introducing fundamental concepts such as local time, stable distributions and characteristic functions. Lévy processes, Lévy flights, Lévy measures, Lévy's constant, the Lévy distribution, the Lévy area, the Lévy arcsine law, and the fractal Lévy C curve are named after him. Biography Lévy was born in Paris to a Jewish family which already included several mathematicians. His father Lucien Lévy was an examiner at the École Polytechnique. Lévy attended the École Polytechnique and published his first paper in 1905, at the age of nineteen, while still an undergraduate, in which he introduced the Lévy–Steinitz theorem. His teacher and advisor was Jacques Hadamard. After graduation, he spent a year in military service and then studied for three years at the École des Mines, where he became a professor in 1913. During Worl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener's 1/f Theorem
In mathematics, the Wiener algebra, named after Norbert Wiener and usually denoted by , is the space of absolutely convergent Fourier series. Here denotes the circle group. Banach algebra structure The norm of a function is given by :\, f\, =\sum_^\infty , \hat(n), ,\, where :\hat(n)= \frac\int_^\pi f(t)e^ \, dt is the th Fourier coefficient of . The Wiener algebra is closed under pointwise multiplication of functions. Indeed, : \begin f(t)g(t) & = \sum_ \hat(m)e^\,\cdot\,\sum_ \hat(n)e^ \\ & = \sum_ \hat(m)\hat(n)e^ \\ & = \sum_ \left\e^ ,\qquad f,g\in A(\mathbb); \end therefore : \, f g\, = \sum_ \left, \sum_ \hat(n-m)\hat(m) \ \leq \sum_ , \hat(m), \sum_n , \hat(n), = \, f\, \, \, g\, .\, Thus the Wiener algebra is a commutative unitary Banach algebra. Also, is isomorphic to the Banach algebra , with the isomorphism given by the Fourier transform. Properties The sum of an absolutely convergent Fourier series is continuous, so :A(\mathbb)\subset C(\mathbb) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compositio Mathematica
''Compositio Mathematica'' is a monthly peer-reviewed mathematics journal established by L.E.J. Brouwer in 1935. It is owned by the Foundation Compositio Mathematica, and since 2004 it has been published on behalf of the Foundation by the London Mathematical Society in partnership with Cambridge University Press. According to the ''Journal Citation Reports'', the journal has a 2020 2-year impact factor of 1.456 and a 2020 5-year impact factor of 1.696. The editors-in-chief are Jochen Heinloth, Bruno Klingler, Lenny Taelman, and Éric Vasserot. Early history The journal was established by L. E. J. Brouwer in response to his dismissal from ''Mathematische Annalen'' in 1928. An announcement of the new journal was made in a 1934 issue of the ''American Mathematical Monthly''. In 1940 the publication of the journal was suspended due to the German occupation of the Netherlands Despite Dutch neutrality, Nazi Germany invaded the Netherlands on 10 May 1940 as part of Fall Gelb (Case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Series
In mathematics, a trigonometric series is a infinite series of the form : \frac+\displaystyle\sum_^(A_ \cos + B_ \sin), an infinite version of a trigonometric polynomial. It is called the Fourier series of the integrable function f if the terms A_ and B_ have the form: :A_=\frac\displaystyle\int^_0\! f(x) \cos \,dx\qquad (n=0,1,2,3 \dots) :B_=\frac\displaystyle\int^_0\! f(x) \sin\, dx\qquad (n=1,2,3, \dots) The zeros of a trigonometric series The uniqueness and the zeros of trigonometric series was an active area of research in 19th century Europe. First, Georg Cantor proved that if a trigonometric series is convergent to a function f(x) on the interval , 2\pi/math>, which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero. Later Cantor proved that even if the set ''S'' on which f is nonzero is infinite, but the derived set ''S of ''S'' is finite, then the coefficients are all zero. In fact, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Function (probability Theory)
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables. In addition to univariate distributions, characteristic functions can be defined for vector- or matrix-valued random variables, and can also be extended to more generic cases. The characteristic function always exists when treated as a function of a real-valued argument, unlike the moment-generating function. There are relations between the behavior of the characteristic function of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Poisson Distribution
In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable. The result can be either a continuous or a discrete distribution. Definition Suppose that :N\sim\operatorname(\lambda), i.e., ''N'' is a random variable whose distribution is a Poisson distribution with expected value λ, and that :X_1, X_2, X_3, \dots are identically distributed random variables that are mutually independent and also independent of ''N''. Then the probability distribution of the sum of N i.i.d. random variables :Y = \sum_^N X_n is a compound Poisson distribution. In the case ''N'' = 0, then this is a sum of 0 terms, so the value of ''Y'' is 0. Hence the conditional distribution of ''Y'' given that ''N'' = 0 is a degenerate distribution. The compound Poisson distribution is obtained by marginalising the j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero-inflated Model
In statistics, a zero-inflated model is a statistical model based on a zero-inflated probability distribution, i.e. a distribution that allows for frequent zero-valued observations. Zero-inflated Poisson One well-known zero-inflated model is Diane Lambert's zero-inflated Poisson model, which concerns a random event containing excess zero-count data in unit time. For example, the number of insurance claims within a population for a certain type of risk would be zero-inflated by those people who have not taken out insurance against the risk and thus are unable to claim. The zero-inflated Poisson (ZIP) model mixes two zero generating processes. The first process generates zeros. The second process is governed by a Poisson distribution that generates counts, some of which may be zero. The mixture distribution is described as follows: : \Pr (Y = 0) = \pi + (1 - \pi) e^ :\Pr (Y = y_i) = (1 - \pi) \frac ,\qquad y_i = 1,2,3,... where the outcome variable y_i has any non-negative in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener's Theorem (other)
{{mathematical disambiguation ...
Wiener's theorem is any of several theorems named after Norbert Wiener: *Paley–Wiener theorem * Wiener's 1/''ƒ'' theorem about functions with absolutely convergent Fourier series. *Wiener–Ikehara theorem *Wiener–Khinchin theorem *Wiener's tauberian theorem *Wiener–Wintner theorem See also *Wiener's lemma In mathematics, Wiener's lemma is a well-known identity which relates the asymptotic behaviour of the Fourier coefficients of a Borel measure on the Circle group, circle to its atomic part. This result admits an analogous statement for measures on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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