Power moment problem

Here, we shall develop a recurrence algorithm for solving a general tridiagonal inhomogeneous system of n linear equations. This will be illustrated for two important classes of problems, such as the power moment problem and spectral analysis, as frequently encountered in physics and chemistry, as well as in linear algebra. 

In the power moment problem [69], there is only one nonzero element of the inhomogeneous column vector D = D, from Eq. (284), that is, D a 5 i, where is the Kronecker 5-symbol. Thus,... 

It is well known that the determination of abscissas and weights of the Gaussian quadrature from power moments is an exponentially ill-conditioned problem due to the presence of rounding errors. 

The modified moments are useful whenever it is possible to guess an auxiliary distribution n E) closely simulating the actual one. However, in the construction of the N matrix, the principal ingredients are just the modified moments, whose determination from the power moment is still a notoriously ill-conditioned problem. 

By the moment method we mean the technique of directly using power moments to determine the Green s function and to reconstruct the spectral density. From a mathematical point of view, this problem goes back to the last century (see Chapter III), but the applications in several branches of physics have more recent origin. 

The general problem of reconstructing a function from its power moments (i.e. the Moment Problem),... 

Research of these factors and creation of a design procedure of water resistance is the momentous problem in decrease in metal consumption of apparatuses and power inputs on gas clearing. 

Nevertheless, when this article is continued after two decades, it should not remain restricted to stereochemical problems. During that time, other very powerful methods will have been developed which are mostly easier and often also more reliable. Particularly, the recent development in X-ray analysis gives a completely reliable proof of configuration valid even for the isolated molecule in the case of conformation it may appear necessary to prove that it is unchanged in solution (see for instance Reference 3). While X-ray is the most reliable method, NMR spectroscopy is the fastest. It still uses some empirical rules and comparison with model compounds, but in a modem version (2-D NMR, NOE) it is also completely trustworthy. Therefore, many recent dipole-moment studies investigated compounds whose steric arrangement was already known, and attention was focused on the electron distribution on the individual bonds, or in conjugated systems. The difference in the point of view may be explained as follows. 

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