Skip to main content Accessibility help
×
Home

On Absolute Summability by Riesz and Generalized Cesàro Means. II

  • H.-H. Körle (a1)
  • Please note a correction has been issued for this article.

Extract

1. We will use the terminology of part I [9], including the general assumptions of [9, § 1]. In that paper we had proved that |R, λ, κ| = |C, λ, κ| in case that κ is an integer. Now, we turn to non-integral orders κ.

As to ordinary summation, the following inclusion relations (in the customary sense; see [9, end of § 1]) for non-integral κ have been established so far. (Since we are comparing Riesz methods of the same type λ and order κ only, (R, λ, κ) is written (R), etc., for the moment.) (R) ⊆ (C) is a result by Borwein and Russell [2]. (C) ⊆ (R) was proved by Jurkat [3] in the case 0 < κ < 1, and, after Borwein [1], it holds in the case 1 < κ < 2 if

(1)

(2)

(i.e. decreases in the wide sense).

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On Absolute Summability by Riesz and Generalized Cesàro Means. II
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On Absolute Summability by Riesz and Generalized Cesàro Means. II
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On Absolute Summability by Riesz and Generalized Cesàro Means. II
      Available formats
      ×

Copyright

References

Hide All
1. Borwein, D., On generalized Cesaro summability, Indian J. Math. 9 (1967), 5564.
2. Borwein, D. and Russell, D. C., On Riesz and generalised Cesàro summability of arbitrary positive order, Math. Z. 99 (1967), 171177.
3. Jurkat, W., Über Rieszsche Mittel mit unstetigem Parameter, Math. Z. 55 (1951), 812.
4. H.-H., Körle, Über unstetige absolute Riesz-Summierung. I, Math. Ann. 176 (1968), 4552.
5. H.-H., Körle, Über unstetige absolute Riesz-Summierung. II, Math. Ann. 177 (1968), 230234.
6. H.-H., Körle, Qn the equivalence of continuous and the discontinuous absolute Riesz means. I, Indian J. Math. 11 (1969), 1116.
7. H.-H., Körle, On the equivalence of continuous and the discontinuous absolute Riesz means. II, Indian J. Math. 11 (1969), 8390.
8. H.-H., Körle, A Tauberian theorem for the discontinuous absolute Riesz means, Indian J. Math. 12 (1970), 1320.
9. H.-H., Körle, On absolute summability by Riesz and generalized Cesàro means. I, Can. J. Math. 22 (1970), 202208.
10. Norlund, N. E., Vorlesungen Über Differenzenrechnung (Springer, Berlin, 1924).
11. Peyerimhoff, A., On discontinuous Riesz means, Indian J. Math. 6 (1964), 6991.
12. Wilansky, A. and Zeller, K., Abschnittsbeschrdnkte Matrixtransformationen; starke Limitierbarkeit, Math. Z. 64 (1956), 258269.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

On Absolute Summability by Riesz and Generalized Cesàro Means. II

  • H.-H. Körle (a1)
  • Please note a correction has been issued for this article.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed

A correction has been issued for this article: