Besov space

In mathematics, the Besov space (named after Oleg Vladimirovich Besov) formula_1 is a complete quasinormed space which is a Banach space when . 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.
Several equivalent definitions exist. One of them is given below.
Let
and define the modulus of continuity by
Let be a non-negative integer and define: with . The Besov space formula_1 contains all functions such that
Norm.
The Besov space formula_1 is equipped with the norm
The Besov spaces formula_8 coincide with the more classical Sobolev spaces formula_9.
If formula_10 and formula_11 is not an integer, then formula_12, where formula_13 denotes the Sobolev–Slobodeckij space.