Özet:
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.