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In the [[mathematical analysis]], and especially in [[real analysis|real]] and [[harmonic analysis]], a '''Birnbaum–Orlicz space''' is a type of function space which generalizes the [[Lp space|L<sup>p</sup> spaces]]. Like the L<sup>p</sup> spaces, they are [[Banach space]]s.  The spaces are named for [[Władysław Orlicz]] and [[Zygmunt William Birnbaum]], who first defined them in 1931.


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Besides the ''L''<sup>p</sup> spaces, a variety of function spaces arising naturally in analysis are Birnbaum–Orlicz spaces.  One such space ''L''&nbsp;log<sup>+</sup>&nbsp;''L'', which arises in the study of [[Hardy–Littlewood maximal function]]s, consists of measurable functions ''f'' such that the integral
 
:<math>\int_{\mathbb{R}^n} |f(x)|\log^+ |f(x)|\,dx < \infty. </math>
 
Here log<sup>+</sup> is the [[positive part]] of the logarithm. Also included in the class of Birnbaum–Orlicz spaces are many of the most important [[Sobolev space]]s.
 
==Formal definition==
 
Suppose that &mu; is a [[&sigma;-finite measure]] on a set ''X'', and &Phi;&nbsp;:&nbsp;[0,&nbsp;∞)&nbsp;&rarr;&nbsp;[0,&nbsp;∞) is a [[Young function]], i.e., a [[convex function]] such that
 
:<math>\frac{\Phi(x)}{x} \to \infty,\quad\mathrm{as\ \ }x\to \infty,</math>
 
:<math>\frac{\Phi(x)}{x} \to 0,\quad\mathrm{as\ \ }x\to 0.</math>
 
Let <math>L^\dagger_\Phi</math> be the set of measurable functions ''f'' : ''X'' &rarr; '''R''' such that the integral
 
:<math>\int_X \Phi(|f|)\, d\mu</math>
 
is finite, where as usual functions that agree [[almost everywhere]] are identified.
 
This ''may not be'' a vector space (it may fail to be closed under scalar multiplication). The vector space of functions spanned by <math>L^\dagger_\Phi</math> is the Birnbaum–Orlicz space, denoted <math>L_\Phi</math>.
 
To define a norm on <math>L_\Phi</math>, let &Psi; be the Young complement of &Phi;; that is,
:<math>\Psi(x) = \int_0^x (\Phi')^{-1}(t)\, dt.</math>
Note that [[Young's inequality]] holds:
:<math>ab\le \Phi(a) + \Psi(b).</math>
The norm is then given by
:<math>\|f\|_\Phi = \sup\left\{\|fg\|_1\mid \int \Psi\circ |g|\, d\mu \le 1\right\}.</math>
Furthermore, the space <math>L_\Phi</math> is precisely the space of measurable functions for which this norm is finite.
 
An equivalent norm {{harv|Rao|Ren|1991|loc=§3.3}} is defined on L<sub>&Phi;</sub> by
 
:<math>\|f\|'_\Phi = \inf\left\{k\in (0,\infty)\mid\int_X \Phi(|f|/k)\,d\mu\le 1\right\},</math>
 
and likewise L<sub>&Phi;</sub>(&mu;) is the space of all measurable functions for which this norm is finite.
 
===Example===
 
Here is an example where <math>L^\dagger_\Phi</math> is not a vector space and is strictly smaller than <math>L_\Phi</math>.
Suppose that ''X'' is the open unit interval (0,1), &Phi;(''x'')=exp(''x'')–1–''x'', and ''f''(x)=log(x). Then ''af'' is in the space <math>L_\Phi</math> but is only in the set <math>L^\dagger_\Phi</math> if |''a''|<1.
 
==Properties==
 
* Orlicz spaces generalize ''L<sup>p</sup>'' spaces in the sense that if <math>\varphi (t) = | t |^{p}</math>, then <math>\| u \|_{L^{\varphi} (X)} = \| u \|_{L^{p} (X)}</math>, so <math>L^{\varphi} (X) = L^{p} (X)</math>.
* The Orlicz space <math>L^{\varphi} (X)</math> is a [[Banach space]] &mdash; a [[complete space|complete]] [[norm (mathematics)|normed]] [[vector space]].
 
==Relations to Sobolev spaces==
 
Certain [[Sobolev space]]s are embedded in Orlicz spaces: for <math>X \subseteq \mathbb{R}^{n}</math> [[open set|open]] and [[bounded set|bounded]] with [[Lipschitz domain|Lipschitz boundary]] <math>\partial X</math>,
:<math>W_{0}^{1, p} (X) \subseteq L^{\varphi} (X)</math>
for
:<math>\varphi (t) := \exp \left( | t |^{p / (p - 1)} \right) - 1.</math>
 
This is the analytical content of the [[Trudinger inequality]]:  For <math>X \subseteq \mathbb{R}^{n}</math> open and bounded with Lipschitz boundary <math>\partial X</math>, consider the space <math>W_{0}^{k, p} (X)</math>, <math>k p = n</math>. There exist constants <math>C_{1}, C_{2} > 0</math> such that
:<math>\int_{X} \exp \left( \left( \frac{| u(x) |}{C_{1} \| \mathrm{D}^{k} u \|_{L^{p} (X)}} \right)^{p / (p - 1)} \right) \, \mathrm{d} x \leq C_{2} | X |.</math>
 
==Orlicz Norm of a Random Variable==
Similarly, the Orlicz norm of a [[random variable]] characterizes it as follows:
:<math>\|X\|_{\Psi} \triangleq  \inf\left\{k\in (0,\infty)\mid E[ \Psi(|X|/k)] \le 1 \right\}. </math>
This norm is [[Homogeneous_function#Positive_homogeneity|homogeneous]] and is defined only when this set is non-empty.
 
When <math>\Psi(x) = x^p</math>, this coincides with the p-th [[Moment (mathematics)|moment]] of the random variable. Other special cases in the exponential family are taken with respect to the functions <math>\Psi_q(x) = \exp(x^q)-1 </math> (for <math>q \geq 1 </math>). A random variable with finite <math>\Psi_2</math> norm is said to be "sub-Gaussian" and a random variable with finite <math>\Psi_1</math> norm is said to be "sub-exponential". Indeed, the boundedness of the <math>\Psi_p</math> norm characterizes the limiting behavior of the probability density function:
:<math>\|X\|_{\Psi_p} = c \rightarrow \lim_{x \rightarrow \infty} f_X(x) \exp(|x/c|^p) = 0, </math>
so that the tail of this probability density function asymptotically resembles, and is bounded above by <math>\exp(-|x/c|^p)</math>.
 
The <math>\Psi_1</math> norm may be easily computed from a strictly monotonic [[moment-generating function]]. For example, the moment-generating function of a [[Chi-squared distribution|chi-squared]] random variable X with K degrees of freedom is <math>M_X(t) = (1-2t)^{-K/2}</math>, so that the inverse of the <math>\Psi_1</math> norm is related to the functional inverse of the moment-generating function:
:<math>\|X\|_{\Psi_1} ^{-1} = M_X^{-1}(2) = (1-4^{-1/K})/2.</math>
 
==References==
* {{citation|first1=Z. W.|last1=Birnbaum|first2=W.|last2=Orlicz|title=Über die Verallgemeinerung des Begriffes der zueinander Konjugierten Potenzen|journal=Studia Mathematica|volume=3|pages=1–67|year=1931}} [http://matwbn.icm.edu.pl/ksiazki/sm/sm3/sm311.pdf PDF].
* {{citation|first=Iracema|last=Bund|title=Birnbaum–Orlicz spaces of functions on groups|journal=Pacific Mathematics Journal|year=1975|volume=58|issue=2|pages=351–359}}.
* {{citation|first1=Edwin|last1=Hewitt|first2=Karl|last2=Stromberg|title=Real and abstract analysis|publisher=Springer-Verlag}}.
*{{citation | authorlink1 =Mark Krasnosel'skii|last1=Krasnosel'skii|first1=M.A.|last2=Rutickii|first2=Ya.B. | title=Convex Functions and Orlicz Spaces | publisher= P.Noordhoff Ltd | location=Groningen |  year=1961}}
* {{citation|first1=M.M.|last1=Rao|first2=Z.D.|last2=Ren|title=Theory of Orlicz Spaces|publisher=Marcel Dekker|series=Pure and Applied Mathematics|year=1991|isbn=0-8247-8478-2}}.
* {{citation|first=Antoni|last=Zygmund|authorlink=Antoni Zygmund|title=Trigonometric series, Volume 1|edition=3rd|publisher=Cambridge University Press|chapter=Chapter IV: Classes of functions and Fourier series}}.
* {{citation|first1=Michel|last1=Ledoux|first2=Michel|last2=Talagrand|title=Probability in Banach Spaces|publisher=Springer-Verlag}}.
 
{{DEFAULTSORT:Birnbaum-Orlicz space}}
[[Category:Harmonic analysis]]
[[Category:Real analysis]]
[[Category:Banach spaces]]

Latest revision as of 05:31, 12 September 2013

In the mathematical analysis, and especially in real and harmonic analysis, a Birnbaum–Orlicz space is a type of function space which generalizes the Lp spaces. Like the Lp spaces, they are Banach spaces. The spaces are named for Władysław Orlicz and Zygmunt William Birnbaum, who first defined them in 1931.

Besides the Lp spaces, a variety of function spaces arising naturally in analysis are Birnbaum–Orlicz spaces. One such space L log+ L, which arises in the study of Hardy–Littlewood maximal functions, consists of measurable functions f such that the integral

Here log+ is the positive part of the logarithm. Also included in the class of Birnbaum–Orlicz spaces are many of the most important Sobolev spaces.

Formal definition

Suppose that μ is a σ-finite measure on a set X, and Φ : [0, ∞) → [0, ∞) is a Young function, i.e., a convex function such that

Let be the set of measurable functions f : XR such that the integral

is finite, where as usual functions that agree almost everywhere are identified.

This may not be a vector space (it may fail to be closed under scalar multiplication). The vector space of functions spanned by is the Birnbaum–Orlicz space, denoted .

To define a norm on , let Ψ be the Young complement of Φ; that is,

Note that Young's inequality holds:

The norm is then given by

Furthermore, the space is precisely the space of measurable functions for which this norm is finite.

An equivalent norm Template:Harv is defined on LΦ by

and likewise LΦ(μ) is the space of all measurable functions for which this norm is finite.

Example

Here is an example where is not a vector space and is strictly smaller than . Suppose that X is the open unit interval (0,1), Φ(x)=exp(x)–1–x, and f(x)=log(x). Then af is in the space but is only in the set if |a|<1.

Properties

Relations to Sobolev spaces

Certain Sobolev spaces are embedded in Orlicz spaces: for open and bounded with Lipschitz boundary ,

for

This is the analytical content of the Trudinger inequality: For open and bounded with Lipschitz boundary , consider the space , . There exist constants such that

Orlicz Norm of a Random Variable

Similarly, the Orlicz norm of a random variable characterizes it as follows:

This norm is homogeneous and is defined only when this set is non-empty.

When , this coincides with the p-th moment of the random variable. Other special cases in the exponential family are taken with respect to the functions (for ). A random variable with finite norm is said to be "sub-Gaussian" and a random variable with finite norm is said to be "sub-exponential". Indeed, the boundedness of the norm characterizes the limiting behavior of the probability density function:

so that the tail of this probability density function asymptotically resembles, and is bounded above by .

The norm may be easily computed from a strictly monotonic moment-generating function. For example, the moment-generating function of a chi-squared random variable X with K degrees of freedom is , so that the inverse of the norm is related to the functional inverse of the moment-generating function:

References

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    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.