Approximation by quasi-projection operators in Besov spaces

In this paper, we investigate approximation of quasi-projection operators in Besov spaces B(p,q)(mu), mu > 0, 1 <= p, q <= infinity. Suppose I is a countable index set. Let (Phi(i))(i is an element of 1), be a family of functions in Lp(R(s)), and let ((Phi) over bar (i))(i is an element of 1) be a family of functions in L (p) over tilde (R(s)), where 1/p + 1/(p) over tilde = 1. Let Q be the quasi-projection operator given by Qf = Sigma(i is an element of I)< f, (Phi) over bar (i)>Phi(i), f is an element of L(p)(R(s)). For h > 0, by sigma(h) we denote the scaling operator given by sigma(h) f(x) := f(x) := f(x/h), x is an element of R(s). Let Q(h) := sigma(h) Q sigma(1)/h. Under some mild conditions on the functions phi(i) and (phi) over tilde (i) (i is an element of 1), we establish the following result: If 0 < mu < nu < k, and if Qg = g for all polynomials 017 degree at most k - 1, then the estimate vertical bar f - Q(h)f vertical bar B(p,q)(mu) <= Ch(nu-mu)vertical bar f vertical bar B(p,q)(nu) for all f is an element of B(p,q)(nu)(R(s)) is valid for all h > 0, where C is a constant independent of h and f. Density of quasi-projection operators in Besov spaces is also discussed. (C) 2009 Elsevier Inc. All rights reserved.