Computing

, Volume 27, Issue 4, pp 319–337 | Cite as

On positive function series

  • K. S. Kölbig
  • F. Schwarz
Article

Abstract

Function series of the form
$$f(x) = \sum\limits_{n = 0}^N {c_n f_n (x)} $$
are considered under the constraintf(x)≥0 in a given intervala≤x≤b. The cone in teh spaceRN+1 of the coefficientscn which is determined by the positivity constraint is approximated numerically by a polyhedral cone. A numerical estimate for the error involved is given and it is shown how it may be reduced. A special series of Jacobi polynomials is discussed and new estimates for the range of parameters for which this series is non-negative are obtained.

Key words

Linear inequalities inequalities polyhedral cones positivity of functions Jacobi polynomials approximation 

AMS Classification

15 A 39 33 A 65 65 D 99 

Über positive Funktionenreihen

Zusammenfassung

Funktionenreihen der Form
$$f(x) = \sum\limits_{n = 0}^N {c_n f_n (x)} $$
werden im Intervalla≤x≤b unter der Nebenbedingungf(x)≥0 betrachtet. Der durch diese Bedingung bestimmte Kegel der Koeffizientencn imRN+1 wird numerisch durch einen polyedrischen Kegel angenähert. Numerische Werte für die entstehenden Fehler und Wege zu ihrer Verbesserung werden angegeben. Die Diskussion einer speziellen Reihe mit Jakobipolynomen führt zu neuen Aussagen über Koeffizientenbereiche für welche die Reihe nicht-negativ ist.

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Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • K. S. Kölbig
    • 1
  • F. Schwarz
    • 2
  1. 1.c/o CERNCH-1211GenèveSwitzerland
  2. 2.FB PhysikUniversität KaiserslauternKaiserslauternFederal Republic of Germany

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