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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0293106-9
PII: S 0002-9947(1972)0293106-9
Article copyright: © Copyright 1972 American Mathematical Society