Strang Fix Equivalence

Suppose we are interested in just the approximation order L of a synthesis function without caring much about the details of E. In this case, the explicit computation of (10) is not necessary. Instead, Strang and Fix [37] have proposed a series of conditions that are equivalent to (11). The interest of these equivalent conditions is that they can be readily tested. They are valid for all synthesis functions with sufficient decay—sinc is one of the very rare cases where these conditions are not satisfied. We mention three equivalent 1D Strang-Fix conditions as follows:

(1) Lth-order zeros in the Fourier domain:

(2) Reproduction of all monomials of degree n < N = L — 1 :

(3) Discrete moments:

where ^n depends on n only.

Under mild hypothesis, any of these conditions is equivalent to e(h) < Const x hL||/

0 0

Post a comment