Using a natural deconvolution for analysis of perturbed integer sampling in shift-invariant spaces
Let \(V \subset L^2(\mathbb R)\) be a shift-invariant space spanned by a Riesz basis of integer-translates of a single function. The authors discuss the existence of a numerically stable reconstruction formula that recovers any \(f\in V\) from the samples \(f(k + \varepsilon_k)\) \((k \in \mathbb Z)\) with perturbed integer sampling points \(k + \varepsilon_k\), where \(\sup_k |\varepsilon_k| \leq \varepsilon\) for some jitter error bound \(\varepsilon \in (0,\, \frac{1}{2})\). It is shown that under mild assumptions, \(V\) can also be generated by integer-translates of a function \(\varphi\), where its Fourier transform reads \[ {\hat \varphi}(v) = \int_{v-\pi}^{v+\pi} g(x) \, dx \] for some \(g\) with \(\int_{-\infty}^{\infty} g(x)\, dx = 1\). The authors adapt the analysis techniques from their previous paper [Int. J. Wavelets Multiresolut. Inf. Process. 3, No. 3, 301--319 (2005; Zbl 1075.42013)] into new sampling theorems with \(g\) as a design parameter. The method is demonstrated on different shift-invariant spaces for which \(g\) has compact support and bounded variation (including all spaces generated by Meyer wavelet scaling functions). Further, jitter error bounds \(\varepsilon\) are computed for particular choices of \(\varphi\).
scientific article; zbMATH DE number 5804460
scientific article