Birnbaum–Orlicz space - Alchetron, The Free Social Encyclopedia

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

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

Φ ( x ) x → ∞ , a s x → ∞ , Φ ( x ) x → 0 , a s x → 0.

Let L Φ † be the set of measurable functions f : X → R such that the integral

∫ X Φ ( | f | ) d μ

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 L Φ † is the Birnbaum–Orlicz space, denoted L Φ .

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

Ψ ( x ) = ∫ 0 x ( Φ ′ ) − 1 ( t ) d t .

Note that Young's inequality holds:

a b ≤ Φ ( a ) + Ψ ( b ) .

The norm is then given by

∥ f ∥ Φ = sup { ∥ f g ∥ 1 ∣ ∫ Ψ ∘ | g | d μ ≤ 1 } .

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

An equivalent norm (Rao & Ren 1991, §3.3) is defined on L_Φ by

∥ f ∥ Φ ′ = inf { k ∈ ( 0 , ∞ ) ∣ ∫ X Φ ( | f | / k ) d μ ≤ 1 } ,

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

Example

Here is an example where L Φ † is not a vector space and is strictly smaller than L Φ . 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 L Φ but is only in the set L Φ † if |a|<1.

Properties

Orlicz spaces generalize L^p spaces (for 1 < p < ∞ ) in the sense that if φ ( t ) = | t | p , then ∥ u ∥ L φ ( X ) = ∥ u ∥ L p ( X ) , so L φ ( X ) = L p ( X ) .

The Orlicz space L φ ( X ) is a Banach space — a complete normed vector space.

Relations to Sobolev spaces

Certain Sobolev spaces are embedded in Orlicz spaces: for X ⊆ R n open and bounded with Lipschitz boundary ∂ X ,

W 0 1 , p ( X ) ⊆ L φ ( X )

for

φ ( t ) := exp ⁡ ( | t | p / ( p − 1 ) ) − 1.

This is the analytical content of the Trudinger inequality: For X ⊆ R n open and bounded with Lipschitz boundary ∂ X , consider the space W 0 k , p ( X ) , k p = n . There exist constants C 1 , C 2 > 0 such that

∫ X exp ⁡ ( ( | u ( x ) | C 1 ∥ D k u ∥ L p ( X ) ) p / ( p − 1 ) ) d x ≤ C 2 | X | .

Orlicz Norm of a Random Variable

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

∥ X ∥ Ψ ≜ inf { k ∈ ( 0 , ∞ ) ∣ E [ Ψ ( | X | / k ) ] ≤ 1 } .

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

When Ψ ( x ) = x p , 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 Ψ q ( x ) = exp ⁡ ( x q ) − 1 (for q ≥ 1 ). A random variable with finite Ψ 2 norm is said to be "sub-Gaussian" and a random variable with finite Ψ 1 norm is said to be "sub-exponential". Indeed, the boundedness of the Ψ p norm characterizes the limiting behavior of the probability density function:

∥ X ∥ Ψ p = c → lim x → ∞ f X ( x ) exp ⁡ ( | x / c | p ) = 0 ,

so that the tail of this probability density function asymptotically resembles, and is bounded above by exp ⁡ ( − | x / c | p ) .

The Ψ 1 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 M X ( t ) = ( 1 − 2 t ) − K / 2 , so that the inverse of the Ψ 1 norm is related to the functional inverse of the moment-generating function:

∥ X ∥ Ψ 1 − 1 = M X − 1 ( 2 ) = ( 1 − 4 − 1 / K ) / 2.

References

Birnbaum–Orlicz space Wikipedia

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