Approximation by quasi-projection operators in Besov spaces

In this paper, we investigate approximation of quasi-projection operators in Besov spaces Bp,qμ, μ>0μ>0, 1≤p,q≤∞1≤p,q≤∞. Suppose II is a countable index set. Let (ϕi)i∈I(ϕi)i∈I be a family of functions in Lp(Rs)Lp(Rs), and let (ϕ̃i)i∈I be a family of functions in Lp̃(Rs), where 1/p+1/p̃=1. Let QQ be the quasi-projection operator given by Qf=∑i∈I〈f,ϕ̃i〉ϕi,f∈Lp(Rs). For h>0h>0, by σhσh we denote the scaling operator given by σhf(x):=f(x/h)σhf(x):=f(x/h), x∈Rsx∈Rs. Let Qh:=σhQσ1/hQh:=σhQσ1/h. Under some mild conditions on the functions ϕiϕi and ϕ̃i (i∈Ii∈I), we establish the following result: If 0<μ<ν0h>0, where CC is a constant independent of hh and ff. Density of quasi-projection operators in Besov spaces is also discussed.