Contraction properties of sequence transformations (Q1112532)

The author introduces the concept of contraction of a sequence transformation: Let T be a sequence transformation; we say that T is a contraction iff \(\forall (S_ n)\), sequence converging to S, \(\exists K_ 1,K_ 2\), \(-1<K_ 1\leq K_ 2<1\), \(\exists N\) such that \(\forall n\geq N\), \(K_ 1\leq (T_ n-S)/(S_ N-S)\leq K_ 2.\) The motivations are the difficulties (and even the impossibility) of finding acceleration methods for some sets of sequences. Some general results are given and Aitken's \(\Delta^ 2\) process is examined in more details. Also, the theta-procedure is studied and some conclusions and perspectives are evocated.