A pear tree-Orlicz space (also Orlicz space ) is a term from the mathematical sub-area of functional analysis and a function space that generalizes the L p spaces . It is named after the Polish mathematicians Zygmunt Wilhelm Birnbaum and Władysław Orlicz .
definition
Orlicz function
Let be a σ-finite measure on a set . A convex function is called an Orlicz function (also Young function ) if the following applies:
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
![{\ displaystyle \ phi \ colon [0, \ infty] \ to [0, \ infty]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da2771cb738ce67d481fe1bd7d16b94053ba0c69)
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and
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.
Orlicz standard
Now be the right inverse function to , that is, it holds . We define the complementary function to as the integral over its right inverse function:
![\ psi](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a)
![\ phi](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4)
![{\ displaystyle \ psi (s) = \ sup \ {t: \ phi (t) \ leq s \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dab140e26b7f39604dff2e34a8ca63ba553c113c)
![\ phi](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4)
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.
The Orlicz norm is then given by:
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.
Pear Tree Orlicz Room
The pear tree-orlicz space is defined as
![{\ displaystyle L_ {Q} (X, \ mu): = \ left \ {f \ colon X \ to \ mathbb {K}: f \, {\ rm {is \ measurable}} \ ,, \ | f \ | _ {Q} <\ infty \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eccff8380429325f028593cc8d45e34cc80a3808)
(or in short as ), i.e. as the space of all measurable functions that have a finite Orlicz norm.
![{\ displaystyle L_ {Q}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad9f06487e6c1ffd46f8356df1a14131ea06ad37)
Luxembourg norm
An equivalent norm called the Luxemburg norm is obtained by
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.
The following norm results for a random variable :
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
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.
It applies to the Luxembourg-standard: .
![{\ displaystyle \ | \ emptyset \ | _ {\ phi} = \ inf \ left \ {\ emptyset \ right \} = \ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/696c253f07466490f611184981ab94b066f0060a)
properties
- The following applies to inclusion:
![p \ in [1, \ infty)](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c107de2d3bb99b7eb3f4ca1b7f68e0c57a0be3b)
If one takes for , one obtains the L p -spaces .
![{\ displaystyle \ phi _ {p} (x): = x ^ {p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb04a75b26ec7cad5d827633a3306c04174b58bb)
Individual evidence
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↑ On the generalization of the concept of mutually conjugated potencies Studia Mathematica 3, pp. 1–67, 1931.