Hermite-Like Interpolating Refinable Function Vector and Its Application in Signal Recovering

Abstract

Interpolating refinable function vectors with compact support are of interest in applications such as sampling theory, numerical algorithm, and signal processing. Han et al. (J. Comput. Appl. Math. 227:254–270, 2009), constructed a class of compactly supported refinable function vectors with (d,r)-interpolating property. A continuous d-refinable function vector ϕ=(ϕ1,…,ϕr)T is (d,r)-interpolating if $$\phi_{\ell}\biggl(\frac{m}{r}+k\biggr)=\delta_{k}\delta_{\ell-1-m},\quad \forall k\in\mathbb{Z},\ m=0,1,\ldots,r-1,\ \ell=1,\ldots,r.$$ In this paper, based on the (d,r)-interpolating refinable function vector ϕ∈(C1(ℝ))r, we shall construct r functions ϕr+1,…,ϕ2r such that the new d-refinable function vector ϕ♮=(ϕT,ϕr+1,…,ϕ2r)T belongs to (C1(ℝ))2r and has the Hermite-like interpolating property: Open image in new window Then any function f∈C1(ℝ) can be interpolated and approximated by Open image in new window That is, \(\widetilde{f}^{(\kappa)}(k+\frac{m}{r})=f^{(\kappa)}(k+\frac{m}{r})\), ∀κ∈{0,1}, ∀k∈ℤ, and m=0,1,…,r−1. When ϕ has symmetry, it is proved that so does ϕ♮ by appropriately selecting some parameters. Moreover, we address the approximation order of ϕ♮. A class of Hermite-like interpolating refinable function vectors with symmetry are constructed from ϕ such that they have higher approximation order than it. Several examples of Hermite-like interpolating refinable function vectors are given to illustrate our results. The truncated error estimate of the interpolating series above is given in Sect. 3. A numerical example of recovering signal is given in Sect. 5 to check the efficiency of the interpolating formula above.