Functional estimation problem error bounds

It should be clear that the extrapolation methods suggested in this section do not have rigorous error bounds like those developed in Section III. However, the extrapolation methods do furnish estimates of the spectral density itself, rather than only averages of the spectral density. Furthermore, these estimates satisfy all known conditions on the spectral density discussed in Section II. They are (a) positive functions, (b) with correct moments, insofar as they are known, and (c) satisfy any known asymptotic behavior at the ends of the frequency intervals. In a number of test cases with known positive continuous functions with known asymptotic behavior, estimates generally were correct to within a few per cent, even when only a few (say 10) moments were given. A typical spectral density obtained in this way for a lattice vibration problem is plotted in Figure 4.34 The results are similar to those obtained numerically for the same problem, by solving a random sample of secular equations for the lattice vibrations.36... 

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