Convergence versus correspondence for sequences of rational functions
Let f be meromorphic in the plane and analytic at 0. Then its diagonal sequence {[ n/n]} ∞ n = 1 of Padé approximants need not converge pointwise. We ask whether by reducing the order of contact (or correspondence) of [ n/n] with f at 0, namely 2 n + 1, we can ensure locally uniform convergence. In...
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| Published in | Indagationes mathematicae Vol. 12; no. 2; pp. 213 - 219 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
18.06.2001
|
| Online Access | Get full text |
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| Summary: | Let
f be meromorphic in the plane and analytic at 0. Then its diagonal sequence {[
n/n]}
∞
n = 1
of Padé approximants need not converge pointwise. We ask whether by reducing the order of contact (or correspondence) of [
n/n] with
f at 0, namely 2
n + 1, we can ensure locally uniform convergence. In particular, we show that there exist rational functions
R
n
of type (
n, n),
n ≥ 1, and a sequence of positive integers {
l
n
}
∞
n = 1
with limit ∞, depending on
f, such that
R
n
has contact of order
n +
l
n
+ 1 with
f at 0, and which converge locally uniformly to
f. Moreover, for any given sequence {
l
n
}
∞
n = 1
, there exists an entire
f for which order of contact higher than
n +
l
n
is incompatible with convergence. |
|---|---|
| ISSN: | 0019-3577 1872-6100 |
| DOI: | 10.1016/S0019-3577(01)80026-4 |