Definition
Definition 1. The mean oscillation of a locally integrable function u (i.e. a function belonging to ) over a hypercube Q in ℝn is defined as the following integral:
where
- |Q| is the volume of Q, i.e. its Lebesgue measure
- uQ is the average value of u on the cube Q, i.e.
-
- .
Definition 2. A BMO function is any function u belonging to whose mean oscillation has a finite supremum over the set of all cubes Q contained in ℝn.
Note. The use of cubes Q in ℝn as the integration domains on which the mean oscillation is calculated, is not mandatory: Wiegerinck (2001) uses balls instead and, as remarked by Stein (1993, p. 140), in doing so a perfectly equivalent of definition of functions of bounded mean oscillation arises.
Read more about this topic: Bounded Mean Oscillation
Famous quotes containing the word definition:
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)