Upon a convergence result in the theory of the Padé table

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.