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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.
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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