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Upon a convergence result in the theory of the Padé table


Author: P. Wynn
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
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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

$\displaystyle 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 $ \vert z\vert < {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 [Enhancements On Off] (What's this?)

  • [1] O. Perron, Die Lehre von den Kettenbrüchen, Dritte, Verbesserte und erweiterte Aufl. Bd. II, Analystich-funktionentheoretische Kettenbrüche, Teubner Verlagsgesellschaft, Stuttgart, 1957. MR 19, 25. MR 0085349 (19:25c)
  • [2] H. S. Wall, Analytic theory of continued fractions, Van Nostrand, Princeton, N. J., 1948. MR 10, 32. MR 0025596 (10:32d)
  • [3] H. Padé, Sur la représentation approchée d'une fonction par des fractions rationelles, Ann. Sci. École Norm. Sup. 9 (1892), (supplement), 1-93.
  • [4] C. G. J. Jacobi, Über die Darstellung einer Reihe gegebner Werthe durch einer gebrochnen rationale Funktion, J. Reine Angew. Math. 30 (1845), 127-156.
  • [5] G. Frobenius, Uber Relationen zwischen den Näherungsbruchen von Potenzreihen, J. Reine Angew. Math. 90 (1881), 1-17.
  • [6] T. J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Univ. Toulouse 8 (1894), 1-22; ibid. 9 (1895), 1-47. MR 1344720 (97i:01054a)
  • [7] E. B. Van Vleck, On an extension of the 1894 memoir of Stieltjes, Trans. Amer. Math. Soc. 4 (1903), 297-332. MR 1500644
  • [8] H. S. Wall, On the Padé approximants associated with the continued fraction and series of Stieltjes, Trans. Amer. Math. Soc. 31 (1929), 91-116. MR 1501470
  • [9] P. Wynn, Upon the Padé table derived from a Stieltjes series, SIAM J. Numer. Anal. 5 (1968), 805-834. MR 39 #1091. MR 0239734 (39:1091)
  • [10] R. de Montessus de Ballore, Sur les fractions continues algébriques, Bull. Soc. Math. France 30 (1902), 28-36. MR 1504403
  • [11] J. Hadamard, Essai sur l'étude des fonctions données par leur developpement de Taylor, J. Math. 8 (1892), 101-186.
  • [12] P. Tschebyscheff, Sur le developpement des fonctions á une seule variable, Bull. Acad. Imp. Sci. St. Petersburg 1 (1860).
  • [13] A. Markoff, Deux demonstrations de la convergence de certaines fractions continues, Acta Math. 19 (1895), 93-104. MR 1554864
  • [14] H. Hamburger, Ueber eine Erweiterung des Stieltjes'schen Momentenproblems, Math. Ann. 81 (1920), 235-319; ibid. 82 (1921), 120-164, 168-187.
  • [15] R. Nevanlinna, Asymptotische Entwickelungen beschränkter Funktionen und das Stieltjes'sche Momentenproblem, Ann. Acad. Sci. Fenn. Ser. A I 18 (5) (1922).
  • [16] P. Henrici and P. Pfluger, Truncation error estimates for Stieltjes fractions, Numer. Math. 9 (1966), 120-138. MR 35 #3856. MR 0212991 (35:3856)

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DOI: https://doi.org/10.1090/S0002-9947-1972-0293106-9
Article copyright: © Copyright 1972 American Mathematical Society

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