Access the full text.
Sign up today, get DeepDyve free for 14 days.
By P. ERD()S (Budapest), corresponding member of the Academy To the memory of O. SzAsz 1 ~ aj~. If ~... a;. Let .,.,~a~,, be an infinite series. Put al-- 2 ~'`+1 ~ J,-o ]i'=f converges, it is defined as the Euler sum of .~, "~at,. It is easy to see that if I; = 0 Co 03 at,: converges, so does ~, a~',,, i. e. Euler summability is regular. Euler summa- bility was first investigated systematically by KNOPP. 1 MEYER-KONIG 2 proved the following high-indices theorem for Euler summability: Assume that a~,, = 0 except if (1) k=ni where //.i+l ~ c > 1. nj Then if 2a7,, is Euler summable, it is convergent. MEwR-KONI6 further con- l;=O jectured that the theorem remains true if (1)is replaced by the much weaker 1,2 condition ni+l--n J > c nf where c > 0 is any constant. It is not hard to see that MF~yF.R-KONIG'S conjecture if true is certainly best possible. I succeeded in proving the following somewhat weaker theorem: '~ Let co at,: be Euler summable, further at,, = 0 except if k:-n< where n.i+~--ni > Cn) ''~ k=0 co where C is a sufficiently large constant. Then .~at,:
Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals
Published: Jul 16, 2005
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.