Approximation properties of the expansions ∑k ℤdLf(M-j.)(-k)φ(Mjx + k), where L is a linear differential operator and M is a matrix dilation, are studied. The sampling expansions are a special case of such differential expansions. Error estimations in Lp-norm, 2 ;le& p ;le, are given in terms of the Fourier transform of f. The approximation order depends on the smoothness of f, the order of L, the order of Strang-Fix condition for φ and M. A wide class of φ including both band-limited and compactly supported functions is considered, but a special condition of compatibility φ with L is required. Such differential expansions may be useful for engineers.
Scopus subject areas
- Applied Mathematics