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 | References | Similar Articles | Additional Information
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




![$ ( - {b^{ - 1}}, - b_1^{ - 1}];$](images/img10.gif)











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