P. W. Langhoff1 has introduced Stieltjes and Tchebycheff imaging techniques for smoothing a discrete L2 approximation to the continuous photoionization cross section. These techniques are discussed at length in Langhoff’s accompanying presentation.

A modification of the langhoff imaging technique

Selected bibliography on atomic collisions: data collections, bibliographies, review articles, books, and papers of particular tutorial value

Biography - Ernest Roy Davidson

Part I: Convergence Theory: The continued fraction as a product of linear fractional transformations Convergence theorems Convergence of continued fractions whose partial denominators are equal to unity Introduction to the theory of positive definite continued fractions Some general convergence theorems Stieltjes type continued fractions Extensions of the parabola theorem The value region problem Part II: Function Theory: J-fraction expansions for rational functions Theory of equations J-fraction expansions for power series Matrix theory of continued fractions Continued fractions and definite integrals The moment problem for a finite interval Bounded analytic functions Hausdorff summability The moment problem for an infinite interval The continued fraction of Gauss Stieltjes summability The Pade table Bibliography Index.

Analytic Theory of Continued Fractions

Moment-theory methods for the construction of photoabsorption and dispersion profiles from associated dipole spectral moments are described and applied to simple atoms and ions. A previously devised (Stieltjes) moment approach, which provides convergent histogram approximations to absorption and dispersion profiles, is refined and extended to the use of arbitrarily large numbers of spectral moments, and an improved (Tchebycheff) moment approach is introduced which gives profiles that are continuous in the photoionization region and exhibit the 8-function-like behavior associated with discrete transitions at the appropriate frequencies. Recurrence relations for the polynomials orthogonal and quasiorthogonal with respect to the distributions are employed in solving the necessary moment problems involving large numbers of spectral moments. The methods are applied in illustrative calculations of absorption and dispersion profiles in oneand two-electron atoms and ions. In the case of one-electron atomic systems the necessary polynomial recurrence coefficients are obtained in closed form from the known spectral moments, allowing the construction of distributions which reproduce the known profiles with high accuracy, employing as many as 100 spectral moments. Variational calculations using large basis sets of square-integrable functions, including the special functions required to satisfy sum rules, provide accurate spectral moments for atomic helium and the negative hydrogen ion. A simple moment-extension procedure is devised to interpolate the associated recurrence coefficients to infinite order employing their known asymptotic values. The associated Stieltjes and Tchebycheff approximations to the absorption and dispersion profiles obtained in these cases are in excellent agreement with available measurements and previous accurate calculations employing discrete and continuum ;;ave functions.

A modification of the langhoff imaging technique

Citations

Selected bibliography on atomic collisions: data collections, bibliographies, review articles, books, and papers of particular tutorial value

Biography - Ernest Roy Davidson

References

Analytic Theory of Continued Fractions

Moment-theory investigations of photoabsorption and dispersion profiles in atoms and ions