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Multipoint Padé approximants used for piecewise rational interpolation and for interpolation to functions of Stieltjes' typePrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 1978 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### 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
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:umu:diva-78987OAI: oai:DiVA.org:umu-78987DiVA, id: diva2:638352
##### Public defence

1979-01-19, Samhällsvetarhuset, hörsal C, Umeå universitet, Umeå, 10:15
#####

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##### Projects

digitalisering@umu
Available from: 2013-07-30 Created: 2013-07-30 Last updated: 2013-07-30Bibliographically approved
##### List of papers

A multipoint Padë approximant, R, to a function of Stieltjes^{1} 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.

1. Piecewise rational interpolation$(function(){PrimeFaces.cw("OverlayPanel","overlay638349",{id:"formSmash:j_idt480:0:j_idt484",widgetVar:"overlay638349",target:"formSmash:j_idt480:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Rational interpolation to functions of Stieltjes' type$(function(){PrimeFaces.cw("OverlayPanel","overlay638347",{id:"formSmash:j_idt480:1:j_idt484",widgetVar:"overlay638347",target:"formSmash:j_idt480:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

urn-nbn$(function(){PrimeFaces.cw("Tooltip","widget_formSmash_j_idt1141",{id:"formSmash:j_idt1141",widgetVar:"widget_formSmash_j_idt1141",showEffect:"fade",hideEffect:"fade",showDelay:500,hideDelay:300,target:"formSmash:altmetricDiv"});});

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