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In harmonic analysis, '''Littlewood–Paley theory''' is a theoretical framework used to extend certain results about ''L''<sup>2</sup> functions to ''L''<sup>''p''</sup> functions for 1<''p''<&infin;. It is typically used as a substitute for orthogonality arguments which only apply to ''L''<sup>''p''</sup> functions when ''p''=2. One implementation involves studying a function by decomposing it in terms of functions with localized frequencies, and using the Littlewood–Paley ''g''-function to compare it with its Poisson integral. The 1-variable case was originated by {{harvs|txt||first=J. E. |last=Littlewood|author1-link=J. E. Littlewood|first2=Raymond
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In addition to permitting players to build their own worlds, Minecraft also allows them to create their own unique stories in the course of their game.<br>Although the initial parts of the game are very simple, the gameplay turn intos more and more complex as the player progresses. Throughout the course of each map for example, players will encounter monsters, loot, food, supplies and other interesting items. Minecraft combines individuals features normally associated with online RPG and combines them open ended gameplay, thus creating a fun and interesting setting for its fans.<br>Unlike other sandbox games however, Minecraft is still in the early phase of createment, and its game mechanics still has a lot of room for improvement. It's also worth mentioning that Minecraft is not particularly unique. There are other games which comply with the sandbox gameplay format.<br>One good instance is Dwarf Fortress, in which the player given the task of overseeing a colony of Dwarves. Like Minecraft, Dwarf Fortress also features a lot of digging and mines. Another good comparison is Terraria, which also side-scroller game with 16-bit graphics. Despite these similarities however, Minecraft remains the most popular sandbox game of them all, which is why more and more individuals are turning out to be interested in playing it.<br>[http://freeminecraftcodes.org/how-to-get-free-minecraft-codes-with-a-minecraft-code-generator how to get free minecraft no surveys] Minecraft is Played<br>Maps in Minecraft are pseudo random, which implies that they can have a surface area that scales much larger than the surface of the entire planet. Each map is also a pastoral setting, wherever the player is given free reign to dig, build and farm to their heart's content.<br>Minecraft usually starts simple. 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Aside from stimulating each player's creativity, Minecraft is also a setting exactly where men and women get to build their own fictional worlds.<br>However, it's also worth mentioning that most players only get access to classic Minecraft, and needless to say, it doesn't have as many features as premium data. Naturally, in sequence to unlock these features, a player will need to pay for them and that's the place gift codes details come in.<br>What's A Minecraft Gift Code?<br>Do you know someone who really enjoys Minecraft? Well, a Minecraft Gift Code may be the best present that you can give to that friend this Christmas. 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You can earn points by doing surveys or by completing offer yous. The more surveys and supplys you complete each day, the more points you get. Once you reach 20 points or more, you routinely get a free Minecraft gift code.<br>Accumulate enough points and you will get your free Minecraft gift code in no time. The site alone will instruct you on how to earn sufficient points to get the free coupon, and you will also be given details on the duration of hold off in between the completion of your tasks and the payout.<br>As you hold out these tasks, you will be provided with information on how many points you will earn for each survey or present. On average, folks earn around 2 to 10 points per hour, so if you're ready to work on these tasks for 2 hours then your a free Minecraft gift code is already as good as yours.
|last2=Paley|author2-link=Raymond Paley|year1=1931|year2=1937|year3=1938}} and developed further by Polish mathematicians [[A. Zygmund]] and [[J. Marcinkiewicz]]  in the 1930s using complex function theory {{harv|Zygmund|2002|loc=chapters XIV, XV}}. [[E. M. Stein]] later extended the theory to higher dimensions using real variable techniques.
 
==The dyadic decomposition of a function==
Littlewood–Paley theory uses a decomposition of a function ''f'' into a sum of functions ''f''<sub>&rho;</sub> with localized frequencies. There are several ways to construct such a decomposition; a typical method is as follows.
 
If ''f'' is a function on '''R''', and &rho; is a measurable set with characteristic function &chi;<sub>&rho;</sub>, then ''f''<sub>&rho;</sub> is defined to be given by
:<math>\hat f_\rho = \chi_\rho\hat f</math>
where the "hat" is used to represent the [[Fourier transform]]. Informally, ''f''<sub>&rho;</sub> is the piece of ''f'' whose frequencies lie in &rho;.
 
If &Delta; is a collection of measurable sets which (up to measure 0) are disjoint and have  union  the real line, then a well behaved function ''f''  can be written as a sum of functions ''f''<sub>&rho;</sub>
for &rho; &isin;&Delta;.
 
When &Delta; consists of the sets of the form
:<math> \rho = [-2^{k+1},-2^k] \cup [2^k,2^{k+1}]</math>.
for ''k'' an integer, this gives a so-called "dyadic decomposition" of ''f'':  &Sigma;<sub>&rho;</sub> ''f''<sub>&rho;</sub>.
 
There are many variations of this construction; for example, the characteristic function of a  set used in the definition of ''f''<sub>&rho;</sub> can be replaced by a smoother function.
 
A key estimate of Littlewood–Paley theory is the Littlewood–Paley theorem, which bounds the size of the functions ''f''<sub>&rho;</sub> in terms of the size of ''f''. There are many versions of this theorem corresponding to the different ways of decomposing ''f''.  
A typical estimate is to bound the ''L''<sup>''p''</sup> norm of (&Sigma;<sub>&rho;</sub> |''f''<sub>&rho;</sub>|<sup>2</sup>)<sup>1/2</sup> by a multiple  of the ''L''<sup>''p''</sup> norm of ''f''.
 
In higher dimensions it is possible to generalize this construction by replacing intervals with rectangles with sides parallel to the coordinate axes. Unfortunately these are rather special sets, which limits the applications to higher dimensions.
 
==The Littlewood–Paley ''g'' function==
The ''g'' function is a non-linear operator on ''L''<sup>''p''</sup>('''R'''<sup>''n''</sup>)  that can be used to control the  ''L''<sup>''p''</sup> norm of a function ''f'' in terms of its [[Poisson integral]].
The Poisson integral ''u''(''x'',''y'') of ''f'' is defined for ''y''>0 by
:<math>u(x,y) = \int_{R^n}P_y(t)f(x-t)dt</math>
where the [[Poisson kernel]] ''P'' is given by
:<math>P_y(x) = \int_{R^n}e^{-2\pi i tx-2\pi |t|y}dt = \frac{\Gamma((n+1)/2)}{\pi^{(n+1)/2}}\frac{y}{(|x|^2+y^2)^{(n+1)/2}}</math>
The Littlewood–Paley ''g'' function ''g''(''f'') is defined by
:<math>g(f)(x) = (\int_0^\infty|\nabla u(x,y)|^2ydy)^{\frac{1}{2}}</math>
 
A basic property of ''g''  is that it approximately preserves norms. More precisely, for 1<''p''<&infin;, the ratio of the ''L''<sup>''p''</sup> norms of ''f'' and ''g''(''f'') is bounded above and below by fixed positive constants depending on ''n'' and ''p'' but not on ''f''.
 
==Applications==
 
One early application of Littlewood–Paley theory was the proof that if ''S''<sub>''n''</sub> are the partial sums of the Fourier series of a periodic  ''L''<sup>''p''</sup> function (''p''>1) and ''n''<sub>''j''</sub> is a sequence satisfying ''n''<sub>''j+1''</sub>/''n''<sub>''j''</sub> > ''q'' for some fixed ''q''>1, then the sequence ''S''<sub>''n''<sub>''j''</sub></sub>  converges almost everywhere. This was later superseded by the [[Carleson's theorem|Carleson–Hunt theorem]] showing that ''S''<sub>''n''</sub> itself converges almost everywhere.
 
Littlewood–Paley theory can also be used to prove the [[Marcinkiewicz multiplier theorem]].
 
==References==
*{{Citation | doi=10.1090/S0002-9904-1978-14464-4 | last1=Coifman | first1=R. R. | last2=Weiss | first2=Guido | title=Book Review: Littlewood-Paley and multiplier theory | url=http://projecteuclid.org/euclid.bams/1183540517 | id={{MathSciNet | id = 1567040}} | year=1978 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=84 | issue=2 | pages=242–250}}
*{{Citation | last1=Edwards | first1=R. E. | last2=Gaudry | first2=G. I. | title=Littlewood-Paley and multiplier theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-07726-8 | id={{MathSciNet | id = 0618663}} | year=1977}}
*{{Citation | last1=Frazier | first1=Michael | last2=Jawerth | first2=Björn | last3=Weiss | first3=Guido | title=Littlewood-Paley theory and the study of function spaces | url=http://www.mrlonline.org/bookstore?arg9=Björn_Jawerth&f=G&fn=105&p=0&pg1=CN&r=1&s1=Jawerth_Bjorn | publisher=Published for the Conference Board of the Mathematical Sciences, Washington, DC | series=CBMS Regional Conference Series in Mathematics | isbn=978-0-8218-0731-6 | id={{MathSciNet | id = 1107300}} | year=1991 | volume=79}}
*{{Citation | last1=Littlewood | first1=J. E. | last2=Paley | first2=R. E. A. C. | title=Theorems on Fourier Series and Power Series  | doi=10.1112/jlms/s1-6.3.230  | year=1931 | journal=J. London Math. Soc. | volume=6 | issue=3 | pages=230–233}}
*{{Citation | last1=Littlewood | first1=J. E. | last2=Paley | first2=R. E. A. C. | title=Theorems on Fourier Series and Power Series (II)  | doi=10.1112/plms/s2-42.1.52  | year=1937 | journal=Proc. London Math. Soc. | volume=42 | issue=1 | pages=52–89}}
*{{Citation | last1=Littlewood | first1=J. E. | last2=Paley | first2=R. E. A. C. | title=Theorems on Fourier Series and Power Series (III)  | doi=10.1112/plms/s2-43.2.105 | year=1938 | journal=Proc. London Math. Soc. | volume=43 | issue=2 | pages=105–126}}
*{{Citation | last1=Stein | first1=Elias M. | author1-link=Elias M. Stein | title=Topics in harmonic analysis related to the Littlewood-Paley theory. | url=http://press.princeton.edu/titles/1977.html | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies, No. 63 | id={{MathSciNet | id = 0252961}} | year=1970}}
*{{Citation | last1=Zygmund | first1=A. | title=Trigonometric series. Vol. I, II | origyear=1935 | url=http://matwbn.icm.edu.pl/kstresc.php?wyd=10&tom=5&jez=en | publisher=[[Cambridge University Press]] | edition=3rd | series=Cambridge Mathematical Library | isbn=978-0-521-89053-3 | id={{MathSciNet | id = 1963498}} | year=2002}}
 
{{DEFAULTSORT:Littlewood-Paley theory}}
[[Category:Fourier analysis]]

Latest revision as of 21:45, 14 December 2014

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Although the initial parts of the game are very simple, the gameplay turn intos more and more complex as the player progresses. Throughout the course of each map for example, players will encounter monsters, loot, food, supplies and other interesting items. Minecraft combines individuals features normally associated with online RPG and combines them open ended gameplay, thus creating a fun and interesting setting for its fans.
Unlike other sandbox games however, Minecraft is still in the early phase of createment, and its game mechanics still has a lot of room for improvement. It's also worth mentioning that Minecraft is not particularly unique. There are other games which comply with the sandbox gameplay format.
One good instance is Dwarf Fortress, in which the player given the task of overseeing a colony of Dwarves. Like Minecraft, Dwarf Fortress also features a lot of digging and mines. Another good comparison is Terraria, which also side-scroller game with 16-bit graphics. Despite these similarities however, Minecraft remains the most popular sandbox game of them all, which is why more and more individuals are turning out to be interested in playing it.
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Maps in Minecraft are pseudo random, which implies that they can have a surface area that scales much larger than the surface of the entire planet. Each map is also a pastoral setting, wherever the player is given free reign to dig, build and farm to their heart's content.
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Aside from developing your own town or castle, it's also worth mentioning that Minecraft features monsters that only come out at night. The player can also go treasure hunting or interact with particular creatures that he or she will encounter along the way. Naturally, players with premium details get access to greater tools and gear.
Another essential aspect of the game are the players themselves. Players who have more experience get access to a broader array of enchantments, most of which are randomly provideed once you finish selected tasks. For example, high level players can get special upgrades for their weapons, but some of these effects are not usually good.
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What's A Minecraft Gift Code?
Do you know someone who really enjoys Minecraft? Well, a Minecraft Gift Code may be the best present that you can give to that friend this Christmas. This is because a buying Minecraft Gift Code is somewhat similar to buying a premium account on Minecraft, except that it is meant to be used as a gift for the buyer's friend or acquaintance.
A Minecraft Gift Code is a long code that allows its recipient to activate all the essential features of their Minecraft accounts, thus enabling them to play the full version alternatively of the classic one.
This is the reason why a lot of Minecraft fans want these kind of codes. Unfortunately, most code generators are just scams, so if you're looking for free Minecraft gift code(s), you should look at other options instead.
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As you hold out these tasks, you will be provided with information on how many points you will earn for each survey or present. On average, folks earn around 2 to 10 points per hour, so if you're ready to work on these tasks for 2 hours then your a free Minecraft gift code is already as good as yours.

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