Upon a convergence result in the theory of the Padé table
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- by P. Wynn PDF
- Trans. Amer. Math. Soc. 165 (1972), 239-249 Request permission
Abstract:
The main theorem of this paper is the following: Let ${M_\nu },{b_\nu }(\nu = 1,2, \ldots ,n)$ be two sets of finite positive real numbers, with ${b_1} > {b_2} > \cdots > {b_n}$, and let $\sigma (\varsigma )$ be a bounded nondecreasing function for $a \leqq \varsigma \leqq b$ where $0 \leqq a \leqq b < {b_n}$; denote the Padé quotients derived from the series expansion of the function \[ f(z) = \sum \limits _{\nu = 1}^n {\frac {{{M_\nu }}}{{(1 + {b_\nu }z)}} + \int _a^b {\frac {{d\sigma (\varsigma )}}{{1 + z\varsigma }}} } \] in ascending powers of z by $\{ {R_{i,j}}(z)\} ;$ let $\mathfrak {D}$ be the open disc $|z| < {b^{ - 1}}$ cut along the real segment $( - {b^{ - 1}}, - b_1^{ - 1}];$ define a progressive sequence of Padé quotients to be one in which the successor ${R_{i'',j''}}(z)$ to ${R_{i’,j’}}(z)$ is such that either $i'' > i’$ and $j'' \geqq j’$ or $i'' \geqq i’$ and $j'' > j’;$ then any infinite progressive sequence of quotients $\{ {R_{i,j}}(z)\}$ for which $i \geqq n$ and $j \geqq n$ converges uniformly for $z \in \mathfrak {D}$ to $f(z)$. The proof proceeds in a number of stages; we first consider those progressive sequences bounded by the main diagonal sequence ${R_{r,r}}(z)(r = n,n + 1, \ldots )$ and the row sequence ${R_{n,n + r}}(z)(r = 0,1, \ldots )$. It follows from a result of Markoff that all diagonal sequences of the form ${R_{n + r,n + n’ + r}}(z)$, where $n’$ is a finite nonnegative integer and $r = 0,1, \ldots ,$ converge uniformly for $z \in \mathfrak {D}$ to $f(z)$. From a theorem of de Montessus de Ballore the row sequence ${R_{n,n + r}}(z)(r = 0,1, \ldots )$ converges uniformly for $z \in \mathfrak {D}$ to $f(z)$. From a result of the author the backward diagonal sequences ${R_{n + r,2m - n - r}}(z)(r = 0,1, \ldots ,m - n)$ and ${R_{n + r,2m - n - r + 1}}(z)(r = 0,1, \ldots ,m - n + 1)$, where m is a finite positive integer, are, when z is real and positive, respectively monotonically decreasing and monotonically increasing. Hence the result of the theorem is true for the restricted progressive sequences in question when z is real and positive. Using the result of de Montessus de Ballore, and extending a result of Nevanlinna to the theory of the Padé table in question, it is shown that there exists a finite positive integer $r’$ such that all quotients ${R_{n + r,n + r'' + r}}(r = 0,1, \ldots ;r'' = r’,r’ + 1, \ldots )$ are uniformly bounded for $z \in \mathfrak {D}’$, where $\mathfrak {D}’$ is that part of $\mathfrak {D}$ from which points lying in the neighborhood of the negative real axis have been excluded. Thus, using the Stieltjes-Vitali theorem, all progressive sequences of Padé quotients taken from the latter double array converge uniformly for $z \in \mathfrak {D}’$ to $f(z)$. That the diagonal sequences of the complementary set ${R_{n + r,n + r'' + r}}(z)(r = 0,1, \ldots ;r'' = 0,1, \ldots ,r’ - 1)$ each converge uniformly for $z \in \mathfrak {D}’$ to $f(z)$ follows from Markoff’s result. Hence the result of the theorem is true for the restricted progressive sequences when $z \in \mathfrak {D}’;$ that this result also holds for values of $z \in \mathfrak {D}$ lying in the neighborhood of the negative real axis (and not, therefore, belonging to $\mathfrak {D}’)$ is proved by the use of a theorem of Tschebyscheff. The Padé quotients lying below the principal diagonal can be associated with a function $\hat f(z)$ having many of the properties of $f(z)$, and the proof outlined above may be extended to the progressive sequences bounded by the principal diagonal and the column sequence ${R_{n + r,n}}(z)(r = 0,1, \ldots )$. The two partial results are then combined.References
- Oskar Perron, Die Lehre von den Kettenbrüchen. Dritte, verbesserte und erweiterte Aufl. Bd. II. Analytisch-funktionentheoretische Kettenbrüche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1957 (German). MR 0085349
- H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Co., Inc., New York, N. Y., 1948. MR 0025596 H. Padé, Sur la représentation approchée d’une fonction par des fractions rationelles, Ann. Sci. École Norm. Sup. 9 (1892), (supplement), 1-93. C. G. J. Jacobi, Über die Darstellung einer Reihe gegebner Werthe durch einer gebrochnen rationale Funktion, J. Reine Angew. Math. 30 (1845), 127-156. G. Frobenius, Uber Relationen zwischen den Näherungsbruchen von Potenzreihen, J. Reine Angew. Math. 90 (1881), 1-17.
- T. J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Math. (6) 4 (1995), no. 1, Ji–Jiv, J1–J35 (French). Reprint of the 1894 original; With an introduction by Jean Cassinet. MR 1344720, DOI 10.5802/afst.789
- Edward Burr Van Vleck, On an extension of the 1894 memoir of Stieltjes, Trans. Amer. Math. Soc. 4 (1903), no. 3, 297–332. MR 1500644, DOI 10.1090/S0002-9947-1903-1500644-3
- Hubert S. Wall, On the Padé approximants associated with the continued fraction and series of Stieltjes, Trans. Amer. Math. Soc. 31 (1929), no. 1, 91–116. MR 1501470, DOI 10.1090/S0002-9947-1929-1501470-X
- Peter Wynn, Upon the Padé table derived from a Stieltjes series, SIAM J. Numer. Anal. 5 (1968), 805–834. MR 239734, DOI 10.1137/0705060
- R. de Montessus, Sur les fractions continues algébriques, Bull. Soc. Math. France 30 (1902), 28–36 (French). MR 1504403 J. Hadamard, Essai sur l’étude des fonctions données par leur developpement de Taylor, J. Math. 8 (1892), 101-186. P. Tschebyscheff, Sur le developpement des fonctions á une seule variable, Bull. Acad. Imp. Sci. St. Petersburg 1 (1860).
- André Markoff, Deux démonstrations de la convergence de certaines fractions continues, Acta Math. 19 (1895), no. 1, 93–104 (French). MR 1554864, DOI 10.1007/BF02402872 H. Hamburger, Ueber eine Erweiterung des Stieltjes’schen Momentenproblems, Math. Ann. 81 (1920), 235-319; ibid. 82 (1921), 120-164, 168-187. R. Nevanlinna, Asymptotische Entwickelungen beschränkter Funktionen und das Stieltjes’sche Momentenproblem, Ann. Acad. Sci. Fenn. Ser. A I 18 (5) (1922).
- Peter Henrici and Pia Pfluger, Truncation error estimates for Stieltjes fractions, Numer. Math. 9 (1966), 120–138. MR 212991, DOI 10.1007/BF02166031
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 165 (1972), 239-249
- MSC: Primary 30A82
- DOI: https://doi.org/10.1090/S0002-9947-1972-0293106-9
- MathSciNet review: 0293106