In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Besov space (named after Oleg Vladimirovich Besov) is a complete quasinormed space which is a Banach space

Banach space

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

 when It, as well as the similarly defined Triebel–Lizorkin space

Triebel–Lizorkin space

In the mathematical discipline known as functional analysis, a Triebel–Lizorkin space is a generalization of many standard function spaces such as Lp spaces and Sobolev spaces. It is named after Hans Triebel and Petr Ivanovich Lizorkin....

, serve to generalize more elementary function spaces and are effective at measuring (in a sense) smoothness properties of functions.

Let



and the modulus of continuity

Modulus of continuity

In mathematical analysis, a modulus of continuity is a function\omega:[0,\infty]\to[0,\infty]used to measure quantitatively the uniform continuity of functions. So, a function f:I\to\R admits \omega as a modulus of continuity if and only if|f-f|\leq\omega,for all x and y in the domain of f...

is defined by



Let with , the Besov space
contains all functions such that

and

The Besov space is equipped with the norm


If , the Besov spaces coincide with the more classical Sobolev spaces .