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Reconstruction in time-warped weighted shift-invariant spaces with application to spline subspaces. (English) Zbl 1068.41020
A time-warped weighted shift-invariant space $$V^p_{m,\gamma } (\phi )$$ is given by $V^p_{m,\gamma} (\phi ) := \{f_\gamma : f_\gamma (\cdot ) = f(\gamma (\cdot )) , \; f \in V^p (\phi ) \},$ where $$V^p (\phi ) := \{ f = \sum_{k \in \mathbb Z^d}c(k) \phi (\cdot - k) : c \in l^p_m\}\subset L^p_m (\mathbb R^d)$$ is a weighted shift-invariant space with $$\|c \|_{l_{m}^p} := \| mc\|_{l^p}$$ and $$\| f\|_{L_m^p} :=\| mf \|_{L^p}.$$ The continuous and invertible function $$\gamma : \mathbb R^n\to \mathbb R^n$$ is called warping function, and the weight $$m$$ is a nonnegative continuous function on $$\mathbb R^n$$.
In this paper, the reproducing kernel structure in shift-invariant spaces $$V^2 (\phi)$$ and $$V^2_{1,\gamma}(\phi )$$ is discussed. Further, the reconstructon formula in $$V^p_{m,\gamma}$$ is derived. As a special example, a reconstruction formula in time-warped spline subspaces is obtained.
##### MSC:
 41A15 Spline approximation 42C15 General harmonic expansions, frames 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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