# Besov space

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In mathematics, the Besov space (named after Oleg Vladimirovich Besov) B p , q s ( R ) 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

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

Let

Δ h f ( x ) = f ( x h ) f ( x )

and define the modulus of continuity by

ω p 2 ( f , t ) = sup | h | t Δ h 2 f p

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

f W n , p ( R ) , 0 | ω p 2 ( f ( n ) , t ) t α | q d t t < .

## Norm

The Besov space B p , q s ( R ) is equipped with the norm

f B p , q s ( R ) = ( f W n , p ( R ) q + 0 | ω p 2 ( f ( n ) , t ) t α | q d t t ) 1 q

The Besov spaces B 2 , 2 s ( R ) coincide with the more classical Sobolev spaces H s ( R ) .

If p = q and s is not an integer, then B p , p s ( R ) = W ¯ s , p ( R ) , where W ¯ s , p ( R ) denotes the Sobolev–Slobodeckij space.

## References

Besov space Wikipedia

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