Stieltjes

All difficulties of this nature could be avoided in subsequent discussions by replacing integrals of the form J 4(()Px(()< ( hy their corresponding Lebesque-Stieltjes integrals... 

In most cases of interest, this n + m order derivative can be written as an ordinary n + mth order derivative and some Dirac delta functions. Situations do exist in which this is not true, but they do not seem to have any physical significanpe and we shall ignore them. In any event, all difficulties of this nature could be avoided by replacing integrals involving probability density functions by their corresponding Lebesque-Stieltjes integrals. 

The Stieltjes integral appearing in Eq. (3-239) can be rewritten (at least formally) in the form... 

These conditions automatically exclude from consideration exactly those time functions X(t) that we have been concerned with in this chapter namely, functions that have nontrivial (not simply a unit jump at the origin) probability distribution functions, and, therefore, do not go to zero as t - oo as required by either Eq. (3-301) or Eq. (3-302). These integrability conditions can be waived in favor of weaker ones that do not require the functions involved to go to zero as t - + oo by making use of a representation in the form of a Stieltjes-like integral66... 

Moments of this conditional distribution can be written as standard Riemann Integrals of the pdf fx(z (N)) or as Stieltjes integrals of the cdf Fx(zf(N)) For example, the conditional expectation is written ... 

Comparing equation (46.15) with equation (46.7) we see the relation between II [x) and 6 x). It may be seen from these equations that is not a function l ut a Stieltjcs measure, and thnt the use of the Dirac delta function could be avoided entirely by a systematic use of Stieltjes integration. 

But there is some complication arising from the double Stieltjes integral. 

V. Averbukh, L.S. Cederbaum, Ab initio calculation of interatomic decay rates by a combination of the Fano ansatz, Green s-function methods, and the Stieltjes imaging technique, J. Chem. Phys. 123 (2005) 204107. 

P.W. Langhoff, Stieltjes-Tchebycheff moment-theory approach to molecular photoionization studies, in T. Rescigno, V. McKoy, B. Schneider (Eds.), Electron-Molecule and Photon-Molecule Collisions, Plenum, New York, 1979 A.U. Hazi, Stieltjes-moment-theory technique for calculating resonance widths, in T. Rescigno, V. McKoy, B. Schneider (Eds.), Electron-Molecule and Photon-Molecule Collisions, Plenum, New York, 1979. 

V. Carravetta, H. Agren, Stieltjes imaging method for molecular Auger transition rates Application to the Auger spectrum of water, Phys. Rev. A 35 (1987) 1022. 

R.K. Nesbet, Stieltjes imaging method for computation of oscillator-strength distributions for complex atoms, Phys. Rev. A 14 (1976) 1065. 

K. Gokhberg, V. Vysotskiy, L.S. Cederbaum, L. Storchi, F. Tarantelli, V. Averbukh, Molecular photoionization cross sections by Stieltjes-Chebyshev moment theory applied to Lanczos pseudospectra, J. Chem. Phys. 130 (2009) 064104. 

For enumeration of many of the Stieltjes integral properties, cf. [605] (p.105). In the following, we present some useful convolution relationships ... 

The more desirable approach is to determine f(Q) from an assumed 0(P,T,Q) and the experimental adsorption isotherm. Sips (16) showed that Equation 1 could be treated by a Stieltjes transform, so that in principle an explicit function could be written for f(Q), provided the experimental isotherm function, 0, could be expressed in analytical form. Subsequently, Honig and coworkers (10, 11, 12) investigated this approach further. The difficulty is that only for certain types of assumed functions 0 and 0 is the approach practical. As a consequence the procedure has been first to restrict the choice of 0 to the Langmuir equation, and second to assume certain simple functions for 0 such as the Freundlich and Temkin isotherm equations. The system is thus forced into an arbitrary mold and again it is not certain how much reliance should be placed on the site energy distributions obtained. 

The passage from the Stieltjes to the Cauchy integral via the Dirac-Lebesgue measure for discretization of inner products 183... 

The space structure via the scalar product as the generalized Stieltjes integral... 

These zeros uk of QK(u) coincide with the eigenvalues of both the evolution matrix U and the corresponding Hessenberg matrix H from Eqs. (131) and (130), respectively. The zeros of Qk(u) are called eigenzeros. The structure of CM is determined by its scalar product for analytic functions of complex variable z or u. For any two regular functions/(m) and g(u) from CM, the scalar product in CM is defined by the generalized Stieltjes integral ... 

The solution (299) is the continued fraction of the order n. The system (293) is encountered in statistical mechanics [69] when applying the method of power moments to obtain a sequence of approximations to the partition function Q(/3) defined as the Stieltjes integral /0°° exp (-pE)dcp(E), where fi is a parameter proportional to the reciprocal temperature of the investigated system, d[Pg.218]

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