Optimality of certain iterative and non-iterative data extrapolation procedures
When the Fourier transform of a compactly supported function is given on a finite interval then the function is uniquely determined. If only samples at a finite number of points are available this is no longer the case. Gerchberg, Papoulis and Cadzow [(GPC); cf. \textit{R. W. Gerchberg}, Super-resolution through energy reduction, Opt-Acta 21, 709-720 (1974); \textit{A. Papoulis}, A new algorithm in spectral analysis and band-limited extrapolation, IEEE Trans. Circuits Systems CAS-22, 735-742 (1975) and \textit{J. A. Cadzow}, IEEE Trans. Acoust. Speech Signal Process. ASSP-27, 4-12 (1979; Zbl 0413.94007)] suggested a method which iteratively determines a function of minimum \(L_ 2\)-norm consistent with both the data and support constraints. The authors modify this procedure by using only a finite version of the Fourier transform. The restriction of the limit function to the given data coincides with the restriction of the function having minimum \(L_ 2\)-norm on the rest of the real line. If the interval, on which the Fourier transform in the procedure is computed, tends to the real axis then the limit converges to the limit in the GPC procedure. Finally if only samples of the searched-for function have to be computed a linear system of equations has to be solved leading to a minimization in a weighted \(L_ 2\)-norm, the weight depends on the extrapolation kernel.
scientific article; zbMATH DE number 3951963