Besov space

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, the Besov space (named after Oleg Vladimirovich Besov) is a complete quasinormed space which is a Banach space when 1 ≤ p, q ≤ ∞. These spaces, as well as the similarly defined Triebel–Lizorkin spaces, serve to generalize more elementary function spaces such as Sobolev spaces and are effective at measuring regularity properties of functions.

Definition[edit]

Several equivalent definitions exist. One of them is given below.

Let

and define the modulus of continuity by

Let n be a non-negative integer and define: s = n + α with 0 < α ≤ 1. The Besov space contains all functions f such that

Norm[edit]

The Besov space is equipped with the norm

The Besov spaces coincide with the more classical Sobolev spaces .

If and is not an integer, then , where denotes the Sobolev–Slobodeckij space.

References[edit]

  • Triebel, H. "Theory of Function Spaces II".
  • Besov, O. V. "On a certain family of functional spaces. Embedding and extension theorems", Dokl. Akad. Nauk SSSR 126 (1959), 1163–1165.
  • DeVore, R. and Lorentz, G. "Constructive Approximation", 1993.
  • DeVore, R., Kyriazis, G. and Wang, P. "Multiscale characterizations of Besov spaces on bounded domains", Journal of Approximation Theory 93, 273-292 (1998).