This paper surveys the closely related topics included in the title. Emphasis is given to the parallelism between the approach using (Perron-Caratheodory) continued fractions to solve the trigonometric moment problem, and the alternate development that proceeds from the sequence of moments {ti n }™«>> to the linear functional ft, to the Szego polynomials and their reciprocal and associated polynomials, and to the quadrature formula for fi and the solution of the moment problem. CONTENTS

A solut ion is given to the problem of r epresentin g the complex proper ties of ionospheri c cha racteristics on a worldwide scale, including their diurnal variation, by num eri cal analysis of ionospher ic data as t h ey a re obtained fr om a n etwork of so undin g statio ns, wit hout prior han d operations. The problem is co mpli cated by t wo bas ic diffi culties : (1) t he da ta ar e affected by noise (ran do m fiuctuation) an d (2) t he stations ar e irreg ularl y posit ion ed in t h e two s pace dim ens ions. The seco nd diffi culty is over co me by a gener al met hod for constructing fun ctions orthogonal relative to the dist ri bution of t he sta t ions . S pccial filterin g p ro cesses a re employed for t hc optimum separation of noise fr om real p hys ical var iat ions . The en d produ ct of t he a na lys is is a table of numerical coeffi cie nts de finin g a fun ction r (A, 0, I) of t hr ec variables, latit ud e (i--) , longitude (0) and ti me (I) , w hich can be used to compute t he io nospheri c characte ristic at a ny des ir ed location or instant of t ime. The method a pplies to a ny io nospheri c characte ristic; however, a s a means of illus tration we use in t h e p resent p a p er only t he characterist ic, monthly med ian of t he F2-laye r cri t ical fr eq uency (fo F,).December 1957 96 stations.

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Abstract.This paper is concerned with double sequences of complex numbers C = [cn}^x and with formal Laurent series Lq(C) = 2f -c_mzm and /."(C) = 2o°cmz~m generated by them. We investigate the following related problems: (1) Does there exist a holomorphic function having L0(C) and LX(C) as asymptotic expansions at z -0 and z = oo, respectively? (2) Does there exist a real-valued bounded, monotonically increasing function (f) with infinitely many points of increase on [0, oo) such that, for every integer n, c" = / 5°(-0" dty(t)l The latter problem is called the strong Stieltjes moment problem. We also consider a modified moment problem in which the function ¡f/(t) has at most a finite number of points of increase. Our approach is made through the study of a special class of continued fractions (called positive 7"-fractions) which correspond to L0(C) at z = 0 and LX(C) at z = oo. Necessary and sufficient conditions are given for the existence of these corresponding continued fractions. It is further shown that the even and odd parts of these continued fractions always converge to holomorphic functions which have Lq(C) and LX(C) as asymptotic expansions. Moreover, these holomorphic functions are shown to be represented by Stieltjes integral transforms whose distributions ^°\t) and ^'"'(O solve the strong Stieltjes moment problem. Necessary and sufficient conditions are given for the existence of a solution to the strong Stieltjes moment problem. This moment problem is shown to have a unique solution if and only if the related continued fraction is convergent. Finally it is shown that the modified moment problem has a unique solution if and only if there exists a terminating positive T-fraction that corresponds to both Lq(C) and ¿"(C).References are given to other moment problems and to investigations in which negative, as well as positive, moments have been used.