Moment‐theory approximations constructed from finite numbers of spectral power moments are described for continuous, nonnegative spectral densities and associated Stieltjes integrals. Derivatives of the mean (Stieltjes) values of the nth‐order Tchebycheff bounds on nondecreasing distributions provide the appropriate approximations to the associated spectral densities. The nth‐order Tchebycheff density so defined is shown to be real, nonnegative, and continuous on the real axis, to have 2n−4 continuous derivatives there, and to support 2n‐2 positive‐integer power moments. Related approximations to the associated Stieltjes integral are obtained from corresponding principal‐value quadratures. The Tchebycheff densities are convergent in the limit of large numbers of spectral moments for determined moment problems, but they are not solutions of reduced moment problems of appropriate finite order. An illustrative application in the case of normal‐mode lattice vibrations in a diatomic chain indicates that the Tchebycheff densities are suitably convergent, and provide faithful images of the forbidden band gap and Van Hove singularities present.

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In the present section, and in subsequent ones, we make extensive use of well‐known results from the classical theory of moments, described in detail by J. A. Shohat and J. D. Tamarkin, Ref. 2, referred to as (S & T) from now on.
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26.
For notational simplicity we suppress explicit indication of the dependence of Ãτ(n)(z),n−1(z), and n(z) on ε, its presence being indicated by the upper tilde.
27.
The radius of the circle bounds on RΦ(z) provided by Eq. (20) is rn = (2|y|i = 0n−1|Ni−1i(z)|2)−1, where |y| is the distance from the real axis of the complex point z (S & T, p. 48). Consequently, as we have indicated above, when y→0,rn diverges for any finite n. However, since the series i = 0|Ni−1i(z)|2 diverges for a determined moment problem (S & T, p. 49), it may be possible to obtain bounds on RΦ(z) on the real axis from a suitable limiting process in which y→0 as n→∞. We are unaware of published applications of such a limiting procedure in constructing approximations to a Stieltjes integral on the real axis from a given sequence of power moments.
28.
In contrast to the roots of Eq. (13a), which are all real and positive, (n−1) of the roots of Eq. (25a) must be real and positive, while the remaining root need only be real, even though the density of Eq. (3) vanishes on the negative real axis. The weight [Eq. (25b)] associated with a negative root is found to be generally small, however, and it vanishes in the limit n→∞. The behavior of the roots of Eq. (25a) as ε is varied over the real axis is discussed in detail in Sec. 5.
29.
Although gt(n)(ε) supports 2n−2 power moments, their numerical values do not necessarily agree with those of Eq. (4), particularly for small n. Numerical studies (Refs. 18 and 19 and Sec. 6) indicate, however, that the kth moment of Eq. (4) is accurately reproduced by gt(n)(ε) for n≫k.
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In computational applications the Tchebycheff derivatives of Eq. (37) are evaluated at each of the steps of a Tchebycheff distribution of given order. Consequently, the fixed point ε need be varied only over one of the intervals εi−1(n−1)<ε<εi(n−1) to obtain gt(n)(ε) over its entire domain. Matrix diagonalization procedures similar to those described by
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