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- Publisher
- de Gruyter
- Copyright
- © by İbrahim Çanak
- ISSN
- 0420-1213
- eISSN
- 2391-4661
- DOI
- 10.1515/dema-2008-0208
- Publisher site
- See Article on Publisher Site
Abstract
DEMONSTRATIO MATHEMATICAVol. XLINo 22008Ibrahim ÇanakSTRUCTURE OF TAYLOR COEFFICIENTSB Y EQUIVALENCE OF TAUBERIAN CONDITIONSAbstract. From the equivalent statement of a sequence (u n ) whose general controlmodulo of the oscillatory behavior of integer order m is (C, 1) slowly oscillating, we obtainsome conclusions regarding the structure of the general control modulo of the oscillatorybehavior of integer order k, k < m, of (un) and investigate subsequential convergence ofsome sequences related to (u n ).1. IntroductionLet the sequence of the backward differences of a sequence u — (un)be denoted by Au = (Aun) where Aun = un — u n _i for η > 1, andAu0 = uq for η — 0. Stanojevic [11] showed that the Hardy-LittlewoodTauberian condition [8](1.1)Vn(\Au\,p)1n= - y2k*>\Auk\i>= 0(1),η —> oo,ρ > 1,which is needed for recovering convergence of (u n ) out of Abel limitability of (u n ) is equivalent toη(1-2)=\ogvnk= 1for some O-Regularly varying sequence (v n ). Later, Stanojevic [12]obtained structural information of Taylor coefficients of power seriesfrom the equivalent form (1.2) of (1.1). After the concept of the general control modulo of the oscillatory behavior of integer order m of asequence is introduced by Stanojevic [14], Çanak and Totur [5] proved2000 Mathematics
Journal
Demonstratio Mathematica – de Gruyter
Published: Apr 1, 2008