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| In [[mathematics]], an '''exponential sum''' may be a finite [[Fourier series]] (i.e. a [[trigonometric polynomial]]), or other finite sum formed using the [[exponential function]], usually expressed by means of the function
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| :<math>e(x) = \exp(2\pi ix).\,</math>
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| Therefore a typical exponential sum may take the form
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| :<math>\sum e(x_n),</math>
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| summed over a finite sequence of [[real number]]s ''x''<sub>''n''</sub>.
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| ==Formulation== | |
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| If we allow some real coefficients ''a''<sub>''n''</sub>, to get the form
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| :<math>\sum a_n e(x_n)</math>
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| it is the same as allowing exponents that are [[complex number]]s. Both forms are certainly useful in applications. A large part of twentieth century [[analytic number theory]] was devoted to finding good estimates for these sums, a trend started by basic work of [[Hermann Weyl]] in [[diophantine approximation]].
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| ==Estimates==
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| The main thrust of the subject is that a sum
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| :<math>S=\sum e(x_n)</math> | |
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| is ''trivially'' estimated by the number ''N'' of terms. That is, the [[absolute value]]
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| :<math>|S| \le N\,</math>
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| by the [[triangle inequality]], since each summand has absolute value 1. In applications one would like to do better. That involves proving some cancellation takes place, or in other words that this sum of complex numbers on the [[unit circle]] is not of numbers all with the same [[Parameter|argument]]. The best that is reasonable to hope for is an estimate of the form
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| :<math>|S|= O(\sqrt{N})\,</math>
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| which signifies, up to the implied constant in the [[big O notation]], that the sum resembles a [[random walk]] in two dimensions.
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| Such an estimate can be considered ideal; it is unattainable in many of the major problems, and estimates
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| :<math>|S|= o(N)\,</math>
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| have to be used, where the o(''N'') function represents only a ''small saving'' on the trivial estimate. A typical 'small saving' may be a factor of log(''N''), for example. Even such a minor-seeming result in the right direction has to be referred all the way back to the structure of the initial sequence ''x''<sub>''n''</sub>, to show a degree of [[randomness]]. The techniques involved are ingenious and subtle.
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| A variant of 'Weyl differencing' investigated by Weyl involving a generating exponential sum
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| <math> G(\tau)= \sum e^{iaf(x)+ia\tau n} </math>
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| Was previously studied by Weyl himself, he developed a method to express the sum as the value <math> G(0)</math>, where 'G' can be defined via a linear differential equation similar to [[Dyson equation]] obtained via summation by parts.
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| ==History==
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| If the sum is of the form
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| :<math> S(x)= e^{ia f(x) } </math>
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| where ''ƒ'' is a smooth function, you could use the [[Euler–Maclaurin formula]] to convert the series into an integral, plus some corrections involving derivatives of ''S''(''x''), then for large values of ''a'' you could use "stationary phase" method to calculate the integral and give an approximate evaluation of the sum. Major advances in the subject were ''[[Van der Corput's method]]'' (c. 1920), related to the [[principle of stationary phase]], and the later ''[[Vinogradov method]]'' (c.1930).
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| The [[large sieve method]] (c.1960), the work of many researchers, is a relatively transparent general principle; but no one method has general application.
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| ==Types of exponential sum==
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| Many types of sums are used in formulating particular problems; applications require usually a reduction to some known type, often by ingenious manipulations. [[Partial summation]] can be used to remove coefficients ''a''<sub>''n''</sub>, in many cases.
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| A basic distinction is between a '''complete exponential sum''', which is typically a sum over all [[residue class]]es ''[[modular arithmetic|modulo]]'' some integer ''N'' (or more general [[finite ring]]), and an '''incomplete exponential sum''' where the range of summation is restricted by some [[inequality (mathematics)|inequality]]. Examples of complete exponential sums are [[Gauss sum]]s and [[Kloosterman sum]]s; these are in some sense [[finite field]] or finite ring analogues of the [[gamma function]] and some sort of [[Bessel function]], respectively, and have many 'structural' properties. An example of an incomplete sum is the partial sum of the quadratic Gauss sum (indeed, the case investigated by [[Carl Friedrich Gauss|Gauss]]). Here there are good estimates for sums over shorter ranges than the whole set of residue classes, because, in geometric terms, the partial sums approximate a [[Cornu spiral]]; this implies massive cancellation.
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| Auxiliary types of sums occur in the theory, for example [[character sum]]s; going back to [[Harold Davenport]]'s thesis. The [[Weil conjectures]] had major applications to complete sums with domain restricted by polynomial conditions (i.e., along an [[algebraic variety]] over a finite field).
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| ==Weyl sums==
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| One of the most general types of exponential sum is the '''Weyl sum''', with exponents 2π''if''(''n'') where ''f'' is a fairly general real-valued [[smooth function]]. These are the sums involved in the distribution of the values
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| :''ƒ''(''n'') modulo 1,
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| according to [[Weyl's equidistribution criterion]]. A basic advance was [[Weyl's inequality]] for such sums, for polynomial ''f''.
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| There is a general theory of [[exponent pair]]s, which formulates estimates. An important case is where ''f'' is logarithmic, in relation with the [[Riemann zeta function]]. See also [[equidistribution theorem]].<ref name=Mont39>Montgomery (1994) p.39</ref>
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| ==Example: the quadratic Gauss sum==
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| Let ''p'' be an odd prime and let <math>\xi = e^{2\pi i / p}</math>. Then
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| the quadratic [[Gauss sum]] is given by | |
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| :<math>\sum_{n=0}^{p-1}\xi^{n^2} =
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| \begin{cases}
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| \sqrt{p}, & p = 1 \mod 4 \\
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| i\sqrt{p}, & p = 3 \mod 4
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| \end{cases}
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| </math>
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| where the square roots are taken to be positive.
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| This is the ideal degree of cancellation one could hope for without any ''a priori'' knowledge of the structure of the sum, since it matches the scaling of a [[random walk]].
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| ==See also==
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| * [[Hua's lemma]]
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| ==References==
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| {{reflist}}
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| * {{cite book | last=Montgomery | first=Hugh L. | authorlink=Hugh Montgomery (mathematician) | title=Ten lectures on the interface between analytic number theory and harmonic analysis | series=Regional Conference Series in Mathematics | volume=84 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1994 | isbn=0-8218-0737-4 | zbl=0814.11001 }}
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| * {{cite book | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=[[Springer-Verlag]] | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 }}
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| ==Further reading==
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| * {{cite book | zbl=0754.11022 | last=Korobov | first=N.M. | title=Exponential sums and their applications | others=Translated from the Russian by Yu. N. Shakhov | series=Mathematics and Its Applications. Soviet Series. | volume=80 | location=Dordrecht | publisher=Kluwer Academic Publishers | year=1992 | isbn=0-7923-1647-9 }}
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| ==External links==
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| * [http://mathworld.wolfram.com/WeylSum.html A brief introduction to Weyl sums on Mathworld]
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| [[Category:Exponentials]]
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| [[Category:Analytic number theory]]
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To find it in excel, copy-paste this continued plan down into corpuscle B1. A person's again access an majority of time in abnormal in about corpuscle A1, the mass fast in treasures will arise in B1.
Inside your are a parent or possibly gaurdian of any smidgen of one who appreciates taking part in video games, be familiar with multi-player and on-line video video options. These aspects give your kid to interact with many a lot of gamers throughout the . As good as the problem is for your tyke in order to with others, you do not know who many people on the other end is.
Conclusion There are a associated with Apple fans who within the above game all on earth. This generation has just been the JRPG's best; in fact it's resulted in being unanimously its worst. Exclusively at Target: Mission: Impossible 4-Pack DVD Couple with all 4 Mission: Impossible movies). Though it is a special day of grand gifts and gestures, one Valentines Day is likely to blend into another much too easily. clash of clans is one among the quickest rising video gaming as of late.
Clash of Clans is without any doubt a popular sport in order to end up being played on multiple systems, the car . iOS and also android. The overall game is totally intriguing but presently there comes a spot associated with legend, where the character gets trapped because within not enough gems. Should you beloved this post along with you would like to be given more info concerning clash of clans hacks i implore you to go to our own web-page. However, this problem has able to easily becoming resolved.
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Game is infiltrating houses around the world. Some play these games for work, but nevertheless , others play them to achieve enjoyment. This organization is booming and won't disappear anytime soon. Maintain reading for some fantastic tips about gaming.
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