Abstract
In this paper we characterize all subspaces of analytic functions in finitely generated shift-invariant spaces with compactly supported generators and provide explicit descriptions of their elements. We illustrate the differences between our characterizations and Strang-Fix or zero conditions on several examples. Consequently, we depict the analytic functions generated by scalar or vector subdivision with masks of bounded and unbounded support. In particular, we prove that exponential polynomials are indeed the only analytic limits of level dependent scalar subdivision schemes with finitely supported masks.
Highlights
Shift-invariant spaces ⎧ ⎫ ⎨n ⎬ S =⎩c( j, k)φ j (· − k) : c( j, ·) ∈ (Z)⎭ j=1 k∈ZCommunicated by Marcin Bownik.Sections 1 and 5 are written by M
We address the question of the existence of subspaces of S comprised of analytic functions other than polynomials or exponential polynomials
We show that the full variety of analytic functions in shift-invariant spaces is very rich, even for a shift-invariant space generated by one function φ = φ1
Summary
Polynomial subspaces generated by vector subdivision schemes and approximation properties of the related multi-wavelets were thoroughly studied in the stationary case in [6,15,33,38,40,42,43] and references therein. 3, we consider the shift-invariant space Sφ generated by a single compactly supported not necessarily refinable distribution φ and show that the analytic subspace H of Sφ has a simpler structure. Φ is a limit of the level dependent subdivision scheme with masks a j , j ∈ N In this case, Theorem 4.1 states that the sequences in (2) contain only finitely many non-zero elements. Theorem 3 Every analytic limit of a level dependent scalar subdivision scheme with finitely supported masks is an exponential polynomial
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