A note on 'The Backus-Gilbert inversion method and the processing of sampled data'

摘要

We show a sequence of interpolation formulas for the Backus-Gilbert (BG), published in 1967, method with /spl delta/-function kernels and penalty functions J(t, t')=(t-t')/sup 2k/ for integers X<0. We show that the interpolation in the limit sense of X/spl rarr//spl infin/ is the Haar representation The interpolation formulas are generalizations of the one obtained by Caccin et al.(see ibid., vol.40, no.11, p.2823, 1992). We investigate the possibility of the BG method with /spl delta/-function kernels so that it is exactly the same as the Shannon sampling formula. We also examined the possibility of the exact reconstruction by the BG method for bandlimited signals.