Mean integrated squared error: Difference between revisions

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In mathematics, the '''Babenko–Beckner inequality''' (after K. Ivan Babenko and [[William E. Beckner]]) is a sharpened form of the [[Hausdorff–Young inequality]] having applications to [[uncertainty principle]]s in the [[Fourier analysis]] of [[Lp space|L<sup>p</sup> spaces]].  The '''(''q'',&nbsp;''p'')-norm''' of the ''n''-dimensional [[Fourier transform]] is defined to be<ref name=Bialynicki>Iwo Bialynicki-Birula. ''Formulation of the uncertainty relations in terms of the Renyi entropies.'' [http://arxiv.org/abs/quant-ph/0608116v2 arXiv:quant-ph/0608116v2]</ref>
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:<math>\|\mathcal F\|_{q,p} = \sup_{f\in L^p(\mathbb R^n)} \frac{\|\mathcal Ff\|_q}{\|f\|_p},\text{ where }1 < p \le 2,\text{ and }\frac 1 p + \frac 1 q = 1.</math>
 
In 1961, Babenko<ref>K.I. Babenko.  ''An ineqality in the theory of Fourier analysis.'' Izv. Akad. Nauk SSSR, Ser. Mat. '''25''' (1961) pp. 531–542 English transl., Amer. Math. Soc. Transl. (2) '''44''', pp. 115–128</ref> found this norm for ''even'' integer values of ''q''. Finally, in 1975,
using [[Hermite functions]] as [[eigenfunction]]s of the Fourier transform, Beckner<ref name=Beckner>W. Beckner, ''Inequalities in Fourier analysis.'' Annals of Mathematics, Vol. 102, No. 6 (1975) pp. 159–182.</ref> proved that the value of this norm for all <math>q \ge 2</math> is
 
:<math>\|\mathcal F\|_{q,p} = \left(p^{1/p}/q^{1/q}\right)^{n/2}.</math>
 
Thus we have the '''Babenko–Beckner inequality''' that
 
:<math>\|\mathcal Ff\|_q \le \left(p^{1/p}/q^{1/q}\right)^{n/2} \|f\|_p.</math>
 
To write this out explicitly, (in the case of one dimension,) if the Fourier transform is normalized so that
 
:<math>g(y) \approx \int_{\mathbb R} e^{-2\pi ixy} f(x)\,dx\text{ and }f(x) \approx \int_{\mathbb R} e^{2\pi ixy} g(y)\,dy,</math>
 
then we have
 
:<math>\left(\int_{\mathbb R} |g(y)|^q \,dy\right)^{1/q} \le \left(p^{1/p}/q^{1/q}\right)^{1/2} \left(\int_{\mathbb R} |f(x)|^p \,dx\right)^{1/p}</math>
 
or more simply
 
:<math>\left(\sqrt q \int_{\mathbb R} |g(y)|^q \,dy\right)^{1/q}
  \le \left(\sqrt p \int_{\mathbb R} |f(x)|^p \,dx\right)^{1/p}.</math>
 
==Main ideas of proof==
Throughout this sketch of a proof, let
:<math>1 < p \le 2, \quad \frac 1 p + \frac 1 q = 1, \quad \text{and} \quad \omega = \sqrt{1-p} = i\sqrt{p-1}.</math>
(Except for ''q'', we will more or less follow the notation of Beckner.)
 
===The two-point lemma===
Let <math>d\nu(x)</math> be the discrete measure with weight <math>1/2</math> at the points <math>x = \pm 1.</math> Then the operator
:<math>C:a+bx \rightarrow a + \omega bx\,</math>
maps <math>L^p(d\nu)</math> to <math>L^q(d\nu)</math> with norm 1; that is,
:<math>\left[\int|a+\omega bx|^q d\nu(x)\right]^{1/q} \le \left[\int|a+bx|^p d\nu(x)\right]^{1/p},</math>
or more explicitly,
:<math>\left[\frac {|a+\omega b|^q + |a-\omega b|^q} 2 \right]^{1/q}
  \le \left[\frac {|a+b|^p + |a-b|^p} 2 \right]^{1/p}</math>
for any complex ''a'', ''b''.  (See Beckner's paper for the proof of his "two-point lemma".)
 
===A sequence of Bernoulli trials===
The measure <math>d\nu</math> that was introduced above is actually a fair [[Bernoulli trial]] with mean 0 and variance 1.  Consider the sum of a sequence of ''n'' such Bernoulli trials, independent and normalized so that the standard deviation remains 1.  We obtain the measure <math>d\nu_n(x)</math> which is the ''n''-fold convolution of <math>d\nu(\sqrt n x)</math> with itself. The next step is to extend the operator ''C'' defined on the two-point space above to an operator defined on the (''n''&nbsp;+&nbsp;1)-point space of <math>d\nu_n(x)</math> with respect to the [[elementary symmetric polynomials]].
 
===Convergence to standard normal distribution===
The sequence <math>d\nu_n(x)</math> converges weakly to the standard [[normal probability distribution]] <math>d\mu(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}\, dx</math> with respect to functions of polynomial growth. In the limit, the extension of the operator ''C'' above in terms of the elementary symmetric polynomials with respect to the measure <math>d\nu_n(x)</math> is expressed as an operator ''T'' in terms of the [[Hermite polynomials]] with respect to the standard normal distribution. These Hermite functions are the eigenfunctions of the Fourier transform, and the (''q'',&nbsp;''p'')-norm of the Fourier transform is obtained as a result after some renormalization.
 
==See also==
*[[Hirschman uncertainty]]
 
==References==
<references/>
 
{{DEFAULTSORT:Babenko-Beckner inequality}}
[[Category:Inequalities]]

Latest revision as of 01:45, 7 September 2014

What"s a criminal defense attorney? She or he is a law firm who organizes a case in order to represent a who is accused of a crime.

These crimes include gender, drug and violent offenses which are things that aren"t approved by society as noticed in this countrys laws.

The work now of the criminal defense lawyer is to fight for the client by protecting the client in open court in a jury of the peers. Sometimes, this person works if the jury provides a award of not guilty. If you believe anything, you will certainly require to check up about cleveland criminal defense. Other times, the jury can say that the accused is guilty.

When the jury does not have a conclusion, the judge has no choice but to call for a retrial. Should there by way of a mistake made in the trial like the selection of the jury, the judge will state that it"s a mistrial and once again, both sides will need to present their case and start over.

There are instances where the criminal defense lawyer and the prosecutor dont need to decide this matter in court. This may happen as both sides have a deal and this is authorized by the judge.

But how does this all happen? First, a crime has to be committed. Best Criminal Defense Attorney Cleveland contains more concerning the inner workings of this viewpoint. Law-enforcement authorities will then have to find out what happen and when they have enough data, a suspect is arrested.

The person who is charged has the right-to remain silent throughout questioning until their lawyer is present. Then the attorney will be provided so even those who dont have money will be properly represented, when the person can"t afford one.

The sixth amendment in the bill of rights declare that this has to be performed where the crime was committed and everyone who"s arrested has the right-to a trial. Be taught additional resources on our related link - Click here: Profile for qualitydrugweb | Feedbooks. After being arraigned, the criminal defense attorney will now conduct an investigation by interviewing witnesses, reviewing police reports and some other data that is associated with the case.

The suspect that"s charged with the crime can only just be convicted if the case presented by the district attorneys office is very persuasive. Which means that a criminal defense lawyer can get a simple verdict by putting holes into the defense of the prosecution. When this happens, your client is freed. Should things go the other away, the criminal defense attorney can make an to the higher court concerning the lower courts certainty.

Can anyone become a criminal defense lawyer? A whole lot of authorities say no because unlike other specializations in the exercise, some find it very intimidating especially when the customer that you"ve to represent is obviously guilty of the crime. But given that you"re that persons attorney, it is your duty to protect the accused to the very best of your ability from begin to finish.

Given that crime of most types and nature occur, some have put up their particular law firms that focus on criminal law. Most who graduate from law assignment work for the federal government and are often referred to as public defenders.

How much you earn like a criminal defense lawyer ranges. This depends primarily while somebody who doesnt make much will have the ability to only provide how much is in their budget who your customer is because if you"re defending a millionaire, you may charge a lot on your legal fees..

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