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Multipoint Padé approximants used for piecewise rational interpolation and for interpolation to functions of Stieltjes' type
Umeå University, Faculty of Science and Technology, Department of mathematics.
1978 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

A multipoint Padë approximant, R, to a function of Stieltjes1 type is determined.The function R has numerator of degree n-l and denominator of degree n.The 2n interpolation points must belong to the region where f is analytic,and if one non-real point is amongst the interpolation points its complex-conjugated point must too.The problem is to characterize R and to find some convergence results as n tends to infinity. A certain kind of continued fraction expansion of f is used.From a characterization theorem it is shown that in each step of that expansion a new function, g, is produced; a function of the same type as f. The function g is then used,in the second step of the expansion,to show that yet a new function of the same type as f is produced. After a finite number of steps the expansion is truncated,and the last created function is replaced by the zero function.It is then shown,that in each step upwards in the expansion a rational function is created; a function of the same type as f.From this it is clear that the multipoint Padê approximant R is of the same type as f.From this it is obvious that the zeros of R interlace the poles, which belong to the region where f is not analytical.Both the zeros and the poles are simple. Since both f and R are functions of Stieltjes ' type the theory of Hardy spaces can be applied (p less than one ) to show some error formulas.When all the interpolation points coincide ( ordinary Padé approximation) the expected error formula is attained. From the error formula above it is easy to show uniform convergence in compact sets of the region where f is analytical,at least wien the interpolation points belong to a compact set of that region.Convergence is also shown for the case where the interpolation points approach the interval where f is not analytical,as long as the speed qî approach is not too great.

##### Place, publisher, year, edition, pages
Umeå: Umeå universitet , 1978. , p. 14
##### Series
University of Umeå, Department of Mathematics ; 1978:8
##### Keyword [en]
Continued fractions, Hardy space, Padé approximant, series of Stieltjes
Mathematics
##### Identifiers
OAI: oai:DiVA.org:umu-78987DiVA, id: diva2:638352
##### Public defence
1979-01-19, Samhällsvetarhuset, hörsal C, Umeå universitet, Umeå, 10:15
##### Projects
digitalisering@umu Available from: 2013-07-30 Created: 2013-07-30 Last updated: 2013-07-30Bibliographically approved
##### List of papers
1. Piecewise rational interpolation
Open this publication in new window or tab >>Piecewise rational interpolation
##### Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1975. p. 13
##### Series
University of Umeå, Department of Mathematics, ISSN 0345-3928 ; 1975:9
Mathematics
##### Identifiers
urn:nbn:se:umu:diva-78986 (URN)
##### Projects
digitalisering@umu Available from: 2013-07-30 Created: 2013-07-30 Last updated: 2013-07-30Bibliographically approved
2. Rational interpolation to functions of Stieltjes' type
Open this publication in new window or tab >>Rational interpolation to functions of Stieltjes' type
##### Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1978. p. 86
##### Series
University of Umeå, Department of Mathematics, ISSN 0345-3928 ; 1978:6
Mathematics
##### Identifiers
urn:nbn:se:umu:diva-78984 (URN)
##### Projects
digitalisering@umu Available from: 2013-07-30 Created: 2013-07-30 Last updated: 2013-08-02Bibliographically approved

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Cite
Citation style
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Other style
More styles
Language
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Other locale
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• text
• asciidoc
• rtf