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 the basis of 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 distribution and properties of the distribution, such as the existence of moments and the existence of a density function.
The characteristic function of a uniform U(–1,1) random variable. This function is real-valued because it corresponds to a random variable that is symmetric around the origin; however characteristic functions may generally be complex-valued.
Introduction
The characteristic function provides an alternative way for describing a random variable. Similar to the cumulative distribution function,
![F_{X}(x)=\operatorname {E} \left[\mathbf {1} _{\{X\leq x\}}\right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/60d71e50c71db8ecbb3c1c4301ee7a449cf637fd)
( where 1{X ≤ x} is the indicator function — it is equal to 1 when X ≤ x, and zero otherwise), which completely determines behavior and properties of the probability distribution of the random variable X, the characteristic function,
![\varphi _{X}(t)=\operatorname {E} \left[e^{itX}\right],](https://wikimedia.org/api/rest_v1/media/math/render/svg/220ee899de62e4b930fbe4ac3a92c2e9544b2b8d)
also completely determines behavior and properties of the probability distribution of the random variable X. The two approaches are equivalent in the sense that knowing one of the functions it is always possible to find the other, yet they both provide different insight for understanding the features of the random variable. However, in particular cases, there can be differences in whether these functions can be represented as expressions involving simple standard functions.
If a random variable admits a density function, then the characteristic function is its dual, in the sense that each of them is a Fourier transform of the other. If a random variable has a moment-generating function, then the domain of the characteristic function can be extended to the complex plane, and
Note however that the characteristic function of a distribution always exists, even when the probability density function or moment-generating function do not.
The characteristic function approach is particularly useful in analysis of linear combinations of independent random variables: a classical proof of the Central Limit Theorem uses characteristic functions and Lévy’s continuity theorem. Another important application is to the theory of the decomposability of random variables.
Definition
For a scalar random variable X the characteristic function is defined as the expected value of eitX, where i is the imaginary unit, and t ∈ R is the argument of the characteristic function:
![{\displaystyle {\begin{cases}\varphi _{X}\!:\mathbb {R} \to \mathbb {C} \\\varphi _{X}(t)=\operatorname {E} \left[e^{itX}\right]=\int _{\mathbb {R} }e^{itx}\,dF_{X}(x)=\int _{\mathbb {R} }e^{itx}f_{X}(x)\,dx=\int _{0}^{1}e^{itQ_{X}(p)}\,dp\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e8285f8052fa7b4d20bb73b82bd8be429e32d5c)
Here FX is the cumulative distribution function of X, and the integral is of the Riemann–Stieltjes kind. If random variable X has a probability density function fX, then the characteristic function is its Fourier transform with sign reversal in the complex exponential,[2][3] and the last formula in parentheses is valid. QX(p) is the inverse cumulative distribution function of X also called the quantile function of X.[4]
It should be noted though, that this convention for the constants appearing in the definition of the characteristic function differs from the usual convention for the Fourier transform.[5] For example, some authors[6] define φX(t) = Ee−2πitX, which is essentially a change of parameter. Other notation may be encountered in the literature: 
。一時想起了 M♪o
聲聲慢‧李清照
尋尋覓覓，冷冷清清，淒淒慘慘戚戚。
乍暖還寒時候，最難將息。
三杯兩盞淡酒，怎敵他晚來風急。
雁過也，正傷心，卻是舊時相識。
滿地黃花堆積，憔悴損，如今有誰堪摘？
守著窗兒，獨自怎生得黑！
梧桐更兼細雨，到黃昏點點滴滴。
這次第，怎一個愁字了得！
白金義︰無課。
誰家五月飛柳絮？大雪紛紛後暴雨！
願訴 
幸運草，西方
傅立葉又非朱麗葉？因何能解芳菲語？如今已《失候》！豈是個『讀』字能善了！更羞提︰
故國失土苦無路
學問道上讀書聲
天文物理極其數
社會人文難化成
卻只有一聯︰
風聲雨聲讀書聲，聲聲入耳；
家事國事天下事，事事關心。
問幾時方可聞地賴？有幸得天籟的耶？？
── 摘自《M♪o 之學習筆記本《子》開關︰【白金義】風聲雨聲》
或許此刻暫且不必太記掛傅立葉的了吧？？
‧ 勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧傅立葉
何不就將
調和分析
調和分析（Harmonic analysis）也稱為諧波分析，是數學中的一個分枝，是由基本波的疊加來表示其他函數或是信號，並且研究及擴展傅立葉級數及傅立葉變換（也是傅立葉分析的擴展）。自十九世紀以來，調和分析已用在許多的領域中，像是信號處理、量子力學 、潮汐理論及神經科學。
Rn以下的經典傅立葉變換目前仍然是一個正在研究的領域，特別是將傅立葉變換應用在一些較廣義的概念下，例如緩增廣義函數（tempered distribution）。例如若在某一分布f上加上一些條件，也會試圖將此條件轉換到f的傅立葉變換上。培力-威納定理即為此例。培力-威納定理指出若f是一個緊支撐下的非零分布（這裡包括緊支撐下的函數），則其傅立葉變換一定不會是緊支撐。這是調和分析下不確定性原理的一個基本形式。
調和分析中的調和（harmonic，或稱為諧波）起源自古希臘文harmonikos，意思是「有音樂上的技巧」[1]。在物理的特徵值問題中，開始用harmonic一詞表示某些特定的波，其頻率是其他波頻率的整數倍，就像泛音列的頻率是第一泛音的整數倍一様，後來這個詞也漸漸擴展，超過原來的意思。
傅立葉級數也常用希爾伯特空間的方式來進行研究，因此調和分析和泛函分析也有一些關係。
光的調和分析。圖中是紅光和其他波長光線的相互作用。若波長差為λ/2（半波長），紅光完全的和二次諧波紫光同相位。圖中所有其他的光和紅光的波長差都小 於λ/2，因此合成波中會出現諧波振盪 。若波長差是λ/14，每14個週期都會出現一次振盪。若波長差是λ/8，每8個週期都會出現一次振盪。振盪在λ /4波長差時最為頻繁，每4個週期出現一次振盪，若波長差為λ/3，每7個週期出現一次振盪。若波長差為λ/2.5，每13個週期出現一次振盪。
以及
算符
在物理學裏，算符（operator），又稱算子，作用於物理系統的狀態空間，使得物理系統從某種狀態變換為另外一種狀態。這變換可能相當複雜，需要用很多方程式來表明，假若能夠使用算符來代表，可以更為簡單扼要地表達論述。
對於很多案例，假若作用的對象有所迥異，算符的物理行為也會不同；但是，對於有些案例，算符的物理行為具有一般性，這時，就可以將論題抽象化，專注於研究算符的物理行為，不必顧慮到狀態的獨特性。這方法比較適用於一些像對稱性或守恆定律的論題。因此，在經典力學裏，算符是很有用的工具。在量子力學裏，算符為理論表述不可或缺的要素。
對於更深奧的理論研究，可能會遇到很艱難的數學問題，算符理論（operator theory）能夠提供高功能的架構，使得數學推導更為簡潔精緻、易讀易懂，更能展現出內中物理涵意。
一般而言，在經典力學裏的算符大多作用於函數，這些函數的參數為各種各樣的物理量，算符將某函數映射為另一種函數。這種算符稱為「函數算符」。在量子力學裏的算符稱為「量子算符」，作用的對象是量子態。量子算符將某量子態映射為另一種量子態。
天地留與其人來者耶！！