Treffer: Iterative Reconstruction of Multivariate Band-Limited Functions from Irregular Sampling Values: Iterative reconstruction of multivariate band-limited functions from irregular sampling values

Summary: This paper describes a real analysis approach to the problem of complete reconstruction of a band-limited, multivariate function \(f\) from irregularly spaced sampling values \((f(x_ i))_{i\in I}\). The required sampling density of the set \(X= (x_ i)_{i\in I}\) depends only on the spectrum \(\Omega\) of \(f\). The proposed reconstruction methods are iterative and stable and converge for a given function \(f\) with respect to any weighted \(L^ p\)-norm, \(1\leq p\leq \infty\), for which \(f\) belongs to the corresponding Banach space \(L^ p_ v(\mathbb{R}^ m)\). It is also shown that any band-limited function \(f\) can be represented as a series of translates \(L_{y_ j} g\) (with complex coefficients) for a given integrable, band-limited function \(g\) if the Fourier transform satisfies \(\widehat g(t)\neq 0\) over \(\Omega\) and the family \(Y=(y_ j)_{j\in J}\) is sufficiently dense. Moreover, the behavior of the coefficients (such as weighted \(p\)-summability) corresponds precisely to the global behavior of \(f\) (i.e., membership in the corresponding weighted \(L^ p\)-space). The proofs are based on a careful analysis of convolution relations, spline approximation operators, and discretization operators (approximation of functions by discrete measures). In contrast to Hilbert space methods, the techniques used here yield pointwise estimates. Special cases of the algorithms presented provide a theoretical basis for methods suggested recently in the engineering literature. Numerical experiments have demonstrated the efficiency of these methods convincingly.