Math 2.0

Approximations and expansions Rational and Padé approximation Conjecture

Baker–Gammel–Wills conjecture

Builds on Padé approximantPadé tableMeromorphic functionPoleUniform convergence

Statement

Let D={z∈ℂ:∣z∣<1}D = \set{z \in \C : \abs{z} < 1} be the unit disc. Let ff be a meromorphic function on DD that is analytic at 00, and let PP be its set of poles in DD. Let [n/n]f[n/n]_f be the diagonal Padé approximants of ff at 00. Then there is an infinite sequence n1<n2<⋯n_1 < n_2 < \cdots such that [nj/nj]f[n_j/n_j]_f converges to ff uniformly on every compact subset of D∖PD \setminus P.

Remarks

The conjecture is false in this generality. In 2003 D. S. Lubinsky published a counterexample built from the Rogers–Ramanujan continued fraction. Convergence of the full diagonal fails in general, because the approximants can have spurious poles that wander through the disc; this is why the conjecture only asks for a subsequence. A weaker form holds: for functions meromorphic in the whole plane, the diagonal sequence converges in logarithmic capacity (the Nuttall–Pommerenke theorem). For columns of the Padé table with a fixed number of poles, Montessus de Ballore's theorem gives uniform convergence.

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To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Meromorphic function, concept, no evidence levelMeromorphic functionconceptPadé approximant, concept, no evidence levelPadé approximantconceptPadé table, concept, no evidence levelPadé tableconceptPole, concept, no evidence levelPoleconceptUniform convergence, concept, no evidence levelUniform convergenceconceptBaker–Gammel–Wills conjecture, conjecture, no evidence levelBaker–Gammel–Wills co…conjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Meromorphic function concept
  2. Padé approximant concept
  3. Padé table concept
  4. Pole concept
  5. Uniform convergence concept
  6. Baker–Gammel–Wills conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Pad%C3%A9_approximant
About this page
TagYPTNBRWRY7
URIm2://YPTNBRWRY7
Kindconjecture
WritingW0 stub
MSC41A21, 30E10
StatusThis copy of the site has no build data. Run m2 status YPTNBRWRY7.
AboutPadé approximant
UsesPadé approximant, Padé table, Meromorphic function, Pole, Uniform convergence

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