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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] |
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] |
Paley–Wiener Theorem
In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (1894–1964). The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz. These theorems heavily rely on the triangle inequality (to interchange the absolute value and integration). Holomorphic Fourier transforms The classical Paley–Wiener theorems make use of the holomorphic Fourier transform on classes of square-integrable functions supported on the real line. Formally, the idea is to take the integral defining the (inverse) Fourier transform :f(\zeta) = \int_^\infty F(x)e^\,dx and allow ''ζ'' to be a complex number in the upper half-plane. One may then expect to differentiate under the integral in orde ... [...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] |
Wiener–Ikehara Theorem
The Wiener–Ikehara theorem is a Tauberian theorem introduced by . It follows from Wiener's Tauberian theorem, and can be used to prove the prime number theorem (Chandrasekharan, 1969). Statement Let ''A''(''x'') be a non-negative, monotonic nondecreasing function of ''x'', defined for 0 ≤ ''x'' 1 to the function ''ƒ''(''s'') and that, for some non-negative number ''c'', :f(s) - \frac has an extension as a continuous function for ℜ(''s'') ≥ 1. Then the limit as ''x'' goes to infinity of ''e''−''x'' ''A''(''x'') is equal to c. One Particular Application An important number-theoretic application of the theorem is to Dirichlet series of the form :\sum_^\infty a(n) n^ where ''a''(''n'') is non-negative. If the series converges to an analytic function in :\Re(s) \ge b with a simple pole of residue ''c'' at ''s'' = ''b'', then :\sum_a(n) \sim \frac X^b. Applying this to the logarithmic derivative of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener–Khinchin Theorem
In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary random process has a spectral decomposition given by the power spectrum of that process. History Norbert Wiener proved this theorem for the case of a deterministic function in 1930; Aleksandr Khinchin later formulated an analogous result for stationary stochastic processes and published that probabilistic analogue in 1934. Albert Einstein explained, without proofs, the idea in a brief two-page memo in 1914. The case of a continuous-time process For continuous time, the Wiener–Khinchin theorem says that if x is a wide-sense stochastic process whose autocorrelation function (sometimes called autocovariance) defined in terms of statistical expected value, r_(\tau) = \mathbb\big (t)^*x(t - \tau)\big/math> (the asterisk denotes complex co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener's Tauberian Theorem
In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in or can be approximated by linear combinations of translations of a given function.see . Informally, if the Fourier transform of a function vanishes on a certain set , the Fourier transform of any linear combination of translations of also vanishes on . Therefore, the linear combinations of translations of can not approximate a function whose Fourier transform does not vanish on . Wiener's theorems make this precise, stating that linear combinations of translations of are dense if and only if the zero set of the Fourier transform of is empty (in the case of ) or of Lebesgue measure zero (in the case of ). Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the L1 group ring L1(R) of the group R of real numbers is the d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener–Wintner Theorem
In mathematics, the Wiener–Wintner theorem, named after Norbert Wiener and Aurel Wintner, is a strengthening of the ergodic theorem, proved by . Statement Suppose that ''τ'' is a measure-preserving transformation of a measure space ''S'' with finite measure. If ''f'' is a real-valued integrable function on ''S'' then the Wiener–Wintner theorem states that there is a measure 0 set ''E'' such that the average : \lim_\frac\sum_^\ell e^ f(\tau^j P) exists for all real λ and for all ''P'' not in ''E''. The special case for ''λ'' = 0 is essentially the Birkhoff ergodic theorem Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ..., from which the existence of a suitable measure 0 set ''E'' for any fixed ''λ'', or any countable set of values ''λ'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
