Editing Norbert Wiener (section)

==Work==
{{blockquote|Information is information, not matter or energy.|Norbert Wiener|''[[Cybernetics: Or Control and Communication in the Animal and the Machine]]''}}

Wiener was an early studier of [[stochastic processes|stochastic]] and mathematical [[Noise (disambiguation)#Noise in mathematics|noise]] processes, contributing work relevant to [[electronic engineering]], [[electronic communication]], and [[control system]]s. It was Wiener's idea to model a signal as if it were an exotic type of noise, giving it a sound mathematical basis. The example often given to students is that English text could be modeled as a random string of letters and spaces, where each letter of the alphabet (and the space) has an assigned probability. But Wiener dealt with analog signals, where such a simple example doesn't exist. Wiener's early work on information theory and signal processing was limited to analog signals, and was largely forgotten with the development of the digital theory.<ref>''John Von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death'', Steve Joshua Heims, MIT Press, 1980</ref>

Wiener is one of the key originators of [[cybernetics]], a formalization of the notion of [[feedback]], with many implications for [[engineering]], [[systems control]], [[computer science]], [[biology]], [[philosophy]], and the organization of [[society]]. His work with cybernetics influenced [[Gregory Bateson]] and [[Margaret Mead]], and through them, [[anthropology]], [[sociology]], and [[education]].<ref>{{cite journal |last=Heims |first=Steve P. |date=April 1977 |title=Gregory Bateson and the mathematicians: From interdisciplinary interaction to societal functions |journal=Journal of the History of the Behavioral Sciences |volume=13 |issue=2 |pages=141–159 |pmid=325068 |doi=10.1002/1520-6696(197704)13:2<141::AID-JHBS2300130205>3.0.CO;2-G }}</ref>

[[File:Wiener process 3d.png|thumb|230px|In the mathematical field of probability, the "[[Wiener sausage]]" is a neighborhood of the trace of a [[Brownian motion]] up to a time  {{mvar|t}},  given by taking all points within a fixed distance of Brownian motion. It can be visualized as a cylinder of fixed radius the centerline of which is Brownian motion.]]

===Wiener equation===
A simple mathematical representation of [[Brownian motion]], the [[Wiener equation]], named after Wiener, assumes the current [[velocity]] of a [[fluid]] particle fluctuates randomly.

===Wiener filter===
For signal processing, the [[Wiener filter]] is a [[filter (signal processing)|filter]] proposed by Wiener during the 1940s and published in 1942 as a classified document. Its purpose is to reduce the amount of [[noise]] present in a signal by comparison with an estimate of the desired noiseless signal. Wiener developed the filter at the Radiation Laboratory at MIT to predict the position of German bombers from radar reflections. What emerged was a mathematical theory of great generality – a theory for predicting the future as best one can on the basis of incomplete information about the past. It was a statistical theory that included applications that did not, strictly speaking, predict the future, but only tried to remove noise. It made use of Wiener's earlier work on [[integral equations]] and [[Fourier transforms]].<ref>''John Von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death'', Steve Joshua Heims, MIT Press, 1980, p.183</ref>
<ref>{{cite book |first=Norbert |last=Wiener |author-link=Norbert Wiener |year=1949 |orig-year=1942 |title=Extrapolation, Interpolation, and Smoothing of Stationary Time Series |publisher=MIT Press}}
: Originally published as a classified document in 1942</ref>

=== Nonlinear control theory ===
Wiener studied [[polynomial chaos]], a key piece of which is the Hermite-Laguerre expansion. This was developed in detail in ''Nonlinear Problems in Random Theory''.

Wiener applied Hermite-Laguerre expansion to nonlinear system identification and control. Specifically, a nonlinear system can be identified by inputting a white noise process and computing the Hermite-Laguerre expansion of its output. The identified system can then be controlled.<ref>{{cite journal |last=Brick |first=Donald B. |date=March 1968 |title=On the applicability of Wiener's canonical expansions |journal=IEEE Transactions on Systems Science and Cybernetics |volume=4 |issue=1 |pages=29–38 |doi=10.1109/TSSC.1968.300185 |issn=0536-1567 |url=https://ieeexplore.ieee.org/document/4082114 }}</ref><ref>{{Cite journal |last1=Harris |first1=G.H. |last2=Lapidus |first2=Leon |date=1967-06-01 |title=Identification of Nonlinear Systems |journal=Industrial & Engineering Chemistry |language=en |volume=59 |issue=6 |pages=66–81 |doi=10.1021/ie50690a012 |issn=0019-7866 |url=https://pubs.acs.org/doi/abs/10.1021/ie50690a012 }}</ref>
[[File:Norbert Wiener.webp|thumb|Norbert Wiener in [[Massachusetts Institute of Technology|MIT]], 1963]]

===In mathematics ===
Wiener took a great interest in the mathematical theory of [[Brownian motion]] (named after [[Robert Brown (Scottish botanist from Montrose)|Robert Brown]]) proving many results now widely known, such as the non-differentiability of the paths. Consequently, the one-dimensional version of Brownian motion was named the [[Wiener process]]. It is the best known of the [[Lévy process]]es, [[càdlàg]] stochastic processes with stationary statistically [[independent increments]], and occurs frequently in pure and applied mathematics, physics and economics (e.g. on the stock-market).

[[Wiener's tauberian theorem]], a 1932 result of Wiener, developed [[Tauberian theorem]]s in [[summability theory]], on the face of it a chapter of [[real analysis]], by showing that most of the known results could be encapsulated in a principle taken from [[harmonic analysis]]. In its present formulation, the theorem of Wiener does not have any obvious association with Tauberian theorems, which deal with [[infinite series]]; the translation from results formulated for integrals, or using the language of [[functional analysis]] and [[Banach algebra]]s, is however a relatively routine process.

The [[Paley–Wiener theorem]] relates growth properties of [[entire function]]s on '''C'''<sup>n</sup> and Fourier transformation of Schwartz distributions of compact support.

The [[Wiener–Khinchin theorem]], (also known as the ''Wiener – Khintchine theorem'' and the ''Khinchin – Kolmogorov theorem''), states that the power spectral density of a wide-sense-stationary random process is the Fourier transform of the corresponding autocorrelation function.

An [[abstract Wiener space]] is a mathematical object in [[measure theory]], used to construct a "decent", strictly positive and locally finite measure on an infinite-dimensional vector space. Wiener's original construction only applied to the space of real-valued continuous paths on the unit interval, known as [[classical Wiener space]]. Leonard Gross provided the generalization to the case of a general [[separable space|separable]] [[Banach space]].

The notion of a Banach space itself was discovered independently by both Wiener and [[Stefan Banach]] at around the same time.<ref>{{cite journal |last1=Wiener |first1=Norbert |year=1923 |title=Note on a paper of M. Banach |journal=Fund. Math. |volume=4 |pages=136–143 |doi=10.4064/fm-4-1-136-143 |doi-access=free }}
:See
{{cite book |last1=Albiac |first1=F. |last2=Kalton |first2=N. |author2-link=Nigel Kalton |year=2006 |title=Topics in Banach Space Theory |publisher=Springer |location=New York, NY |isbn=978-0-387-28141-4 |page=15 |series=[[Graduate Texts in Mathematics]] |volume=233 }}</ref>