×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI EuDML