Düzenli olarak üretilen diziler için Tauber tipi teoremler

Abstract

Klasik Tauber teorisinin temel amac.lar³ndan biri, Abel'in gerek ko»sulu yada genelle»stirilmesi olarak bilinen limitin varl³~g³ndan ve sal³n³m davran³»slar³n³ kon- trol eden ko»sullar ile ³raksakl³~g³ kontrol edilebilen dizilerin yak³nsakl³~g³na ula»s³lma s³d³r. Bu ko»sullara Tauber ko»sullar³ ve bu tip teoremlere Tauber teoremleri denir. Stanojevi¶c taraf³ndan tan³mlanan tamsay³ mertebeli kontrol modÄulolar ile klasik Tauber teoremleri incelenecektir. Tauber, bir dizinin klasik kontrol modÄulosu s³f³r dizisi ise Abel'in gerek ko»sulundan dizinin yak³nsakl³~g³na ula»s³ld³~g³na gÄostermi»stir. Littlewood, Tauber'in ko»sulundan daha zay³f olan dizinin klasik kontrol modÄulosu nun s³n³rl³l³~g³ ile de~gi»stirebilece~gini gÄostermi»stir. Sonra Schmidt yava»s sal³n³ml³ dizileri tan³tm³»s ve genelle»stirilmi»s Littlewood teoremi olarak bilinen daha genel teoremi ispatlam³»st³r. Dik, Karamata'n³n temel toreminden yararlanarak ve Stano- jevi¶c taraf³ndan tan³t³lan dÄuzenli olarak Äuretilen dizi kavram³n³ kullanarak Abel limitleme metodu ic.in yeni Tauber tipi teoremler elde etmi»s ve klasik Tauber tipi teoremleri genelle»stirmi»stir. Dik ve C. anak dÄuzenli olarak Äuretilen dizi kavram³ yard³m³yla Abel limitleme metodu ic.in klasik ve klasik olmayan yeni Tauber tipi teoremler ispatlam³»st³r. C. anak ve Dik verilen bir dizinin Äuretec.lerinin yada geri farklar³n³n yava»s sal³n³ml³ olmas³ durumunda dizinin hangi ko»sullar alt³nda yak³nsak yada altdizisel yak³nsak oldu~gu ara»st³rm³»st³r. One of the main objectives of the classical Tauberian theory is to recover con- vergence of sequences, whose divergence is manageable, out of the existence of certain limits which is known as Abel's necessary condition or its generalizations, and certain additional conditions that control the oscillatory behavior. These conditions are called Tauberian conditions and this type of theorems are called Tauberian theorems. In terms of the control modulo of oscillatory behavior of integer order, introduced by Stanojevi¶c, we now summarize the classical results. Tauber proved that if the classical control modulo of a sequence is a null sequence, then one obtains convergence of sequence out of its Abel's necessary condition. Littlewood showed that Tauber's condition can be replaced by the boundedness of the classical control modulo of a sequence. Later Schmidt introduced the slowly oscillating sequences and proved the more general theorem, which is known as the generalized Littlewood theorem. Using Karamata's Hauptsatz and employing the concept of regularly generated sequences introduced by Stanojevi¶c, Dik ob- tained new Tauberian theorems and generalized classical Tauberian theorems for Abel limitable method. Dik ve C. anak proved classical and nonclassical Taube- rian theorems for Abel limitable method by the concept of regularly generated sequences. C. anak ve Dik investigated under which conditions sequences converges or converges subsquentially, provided that its generator sequence or sequence of its backword di®erences is slowly oscillating.