Multivariate sampling-type approximation

Approximation properties of the expansions [Formula: see text], where [Formula: see text] is a linear differential operator and [Formula: see text] is a matrix dilation, are studied. The sampling expansions are a special case of such differential expansions. Error estimations in [Formula: see text]-norm, [Formula: see text], are given in terms of the Fourier transform of [Formula: see text]. The approximation order depends on the smoothness of [Formula: see text], the order of [Formula: see text], the order of Strang–Fix condition for [Formula: see text] and [Formula: see text]. A wide class of [Formula: see text] including both band-limited and compactly supported functions is considered, but a special condition of compatibility [Formula: see text] with [Formula: see text] is required. Such differential expansions may be useful for engineers.

MULTI-CHANNEL SAMPLING ON SHIFT-INVARIANT SPACES WITH FRAME GENERATORS

Let φ be a continuous function in L2(ℝ) such that the sequence {φ(t - n)}n∈ℤ is a frame sequence in L2(ℝ) and assume that the shift-invariant space V(φ) generated by φ has a multi-banded spectrum σ(V). The main aim in this paper is to derive a multi-channel sampling theory for the shift-invariant space V(φ). By using a type of Fourier duality between the spaces V(φ) and L2[0, 2π] we find necessary and sufficient conditions allowing us to obtain stable multi-channel sampling expansions in V(φ).