Birnbaum–Orlicz space

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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[edit]

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 (Rao & Ren 1991, §3.3) is defined on LΦ by

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

Example[edit]

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[edit]

  • Orlicz spaces generalize Lp spaces (for ) in the sense that if , then , so .
  • The Orlicz space is a Banach space — a complete normed vector space.

Relations to Sobolev spaces[edit]

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[edit]

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[edit]

  • Birnbaum, Z. W.; Orlicz, W. (1931), "Über die Verallgemeinerung des Begriffes der zueinander Konjugierten Potenzen", Studia Mathematica, 3: 1–67  PDF.
  • Bund, Iracema (1975), "Birnbaum–Orlicz spaces of functions on groups", Pacific Mathematics Journal, 58 (2): 351–359 .
  • Hewitt, Edwin; Stromberg, Karl, Real and abstract analysis, Springer-Verlag .
  • Krasnosel'skii, M.A.; Rutickii, Ya.B. (1961), Convex Functions and Orlicz Spaces, Groningen: P.Noordhoff Ltd 
  • Rao, M.M.; Ren, Z.D. (1991), Theory of Orlicz Spaces, Pure and Applied Mathematics, Marcel Dekker, ISBN 0-8247-8478-2 .
  • Zygmund, Antoni, "Chapter IV: Classes of functions and Fourier series", Trigonometric series, Volume 1 (3rd ed.), Cambridge University Press .
  • Ledoux, Michel; Talagrand, Michel, Probability in Banach Spaces, Springer-Verlag .