In mathematics, the Besov space (named after Oleg Vladimirovich Besov) is a complete quasinormed space which is a Banach space when . These spaces, as well as the similarly defined TriebelâÂÂLizorkin spaces, serve to generalize more elementary function spaces such as Sobolev spaces and are effective at measuring regularity properties of functions.
Several equivalent definitions exist. One of them is given below. This definition is quite limited because it does not extend to the range .
Let
and define the modulus of continuity by
Let be a non-negative integer and define: with . The Besov space contains all functions such that
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 SobolevâÂÂSlobodeckij space.