headerpos: 12198
 
 
 

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
Publisher
Journal Information
» Editorial Board
» Editorial Policy
» Archival Policy
» Article Publication Charges
» Copyright and Licensing Policy
Guidelines for Authors
» For Authors
» Instructions to Authors
» LaTex style files
Guidelines for Reviewers
» For Reviewers
» Review Form
Open Access
List of Issues
» 2019
» 2018
» 2017
» 2016
» 2015
» 2014
» 2013
» 2012
» 2011
» 2010
» 2009
» 2008
Vol. 57, Issue 4
Vol. 57, Issue 3
Vol. 57, Issue 2
Vol. 57, Issue 1
» Back Issues Phys. Math.
» Back Issues Chemistry
» Back issues (full texts)
  in Google. Phys. Math.
» Back issues (full texts)
  in Google. Chemistry
» Back issues (full texts)
  in Google Engineering
» Back issues (full texts)
  in Google Ecology
» Back issues in ETERA Füüsika, Matemaatika jt
Subscription Information
» Prices
Internet Links
Support & Contact
Publisher
» Staff
» Other journals

Comparison of speeds of convergence in Riesz-type families of summability methods; pp. 70–80

(Full article in PDF format) doi: 10.3176/proc.2008.2.02


Authors

Anna Šeletski, Anne Tali

Abstract

We deal with Riesz-type families (see Proc. Estonian Acad. Sci. Phys. Math., 2002, 51, 18–34 and Acta Sci. Math. (Szeged), 2004, 70, 639–657) of summability methods Aα for converging functions and sequences. The methods Aα in a Riesz-type family depend on a continuous parameter α, and are connected through certain generalized integral Nörlund methods. By extending and applying the results of Stadtmüller and Tali (Anal. Math., 2003, 29, 227–242), we compare speeds of convergence in a Riesz-type family. As expected, the speed of convergence cannot increase if we switch from one summability method to a stronger one. Comparative estimations for speeds are found. In particular, the families of integral Riesz methods, generalized integral Nörlund methods, and Abel- and Borel-type summability methods are considered. Numerical examples are given.

Keywords

Riesz-type family of summability methods, Riesz methods, speed of convergence, generalized integral Nörlund methods, Abel-type methods, Borel-type methods.

References

1. Borwein , D. On a scale of Abel-type summability methods. Proc. Cambridge Phil. Soc. , 1957 , 53 , 318–322.

2. Borwein , D. On Borel-type methods of summability. Mathematica , 1958 , 5 , 128–133.

3. Borwein , D. and Shawyer , B. L. R. On Borel-methods. Tôhoku Math. J. , 1966 , 18 , 283–298.
doi:10.2748/tmj/1178243418

4. Kangro , G. On the summability factors of the Bohr–Hardy for a given rapidity. I. Eesti NSV Tead. Akad. Toim. Füüs. Mat. , 1969 , 18 , 137–146 (in Russian).

5. Kangro , G. Summability factors for the series l-bounded by the methods of Riesz and Cesàro. Tartu Ülik. Toimetised , 1971 , 277 , 136–154 (in Russian).

6. Pavlova , V. and Tali , A. On the convexity theorem of M. Riesz. Proc. Estonian Acad. Sci. Phys. Math. , 2002 , 51 , 18–34.

7. Stadtmüller , U. and Tali , A. Comparison of certain summability methods by speeds of convergence. Anal. Math. , 2003 , 29 , 227–242.
doi:10.1023/A:1025419305735

8. Stadtmüller , U. and Tali , A. Strong summability in certain families of summability methods. Acta Sci. Math. (Szeged) , 2004 , 70 , 639–657.

9. Hardy , G. H. Divergent Series. Oxford Press , 1949.

10. Kuttner , B. On “translated quasi-Cesàro” summability. Proc. Cambridge Phil. Soc. , 1966 , 62 , 705–712.

11. Tali , A. Zero-convex families of summability methods. Tartu Ülik. Toimetised , 1981 , 504 , 48–57 (in Russian).

12. Meronen , O. and Tammeraid , I. Generalized Nörlund method and convergence acceleration. Math. Model. Anal. , 2007 , 12 , 195–204.

 
Back

Current Issue: Vol. 68, Issue 3, 2019




Publishing schedule:
No. 1: 20 March
No. 2: 20 June
No. 3: 20 September
No. 4: 20 December