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 be two sets of finite positive real numbers, with , and let be a bounded nondecreasing function for where ; denote the Padé quotients derived from the series expansion of the function

in ascending powers of z by let be the open disc cut along the real segment define a progressive sequence of Padé quotients to be one in which the successor to is such that either and or and then any infinite progressive sequence of quotients for which and converges uniformly for to .

The proof proceeds in a number of stages; we first consider those progressive sequences bounded by the main diagonal sequence and the row sequence . It follows from a result of Markoff that all diagonal sequences of the form , where is a finite nonnegative integer and converge uniformly for to . From a theorem of de Montessus de Ballore the row sequence converges uniformly for to . From a result of the author the backward diagonal sequences and , 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 such that all quotients are uniformly bounded for , where is that part of 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 to . That the diagonal sequences of the complementary set each converge uniformly for to follows from Markoff's result. Hence the result of the theorem is true for the restricted progressive sequences when that this result also holds for values of lying in the neighborhood of the negative real axis (and not, therefore, belonging to is proved by the use of a theorem of Tschebyscheff. The Padé quotients lying below the principal diagonal can be associated with a function having many of the properties of , and the proof outlined above may be extended to the progressive sequences bounded by the principal diagonal and the column sequence . The two partial results are then combined.

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