On absolute almost convergence

Advances in Almost Convergence

In this paper, we first give the concept of properly distributed sequence, and prove that it is almost convergent with F-limit expressed as a formal integral. Basing on these, we review the work of Feng and Li, which is shown to be a special case of our generalized theory. Then we generalize Banach limit to Banach limit functional, which is the minimum requirement to characterize strong almost ...

Absolute Convergence in Ordered Fields

We explore the distinction between convergence and absolute convergence of series in both Archimedean and non-Archimedean ordered fields and find that the relationship between them is closely connected to sequential (Cauchy) completeness.

Almost Everywhere Convergence of Poisson Integrals on Generalized Half-planes

On Distribution and Almost Convergence of Bounded Sequences

In this paper, we give the concepts of properly distributed and simply distributed sequences, and prove that they are almost convergent. Basing on these, we review the work of Feng and Li [Feng, B. Q. and Li, J. L., Some estimations of Banach limits, J. Math. Anal. Appl. 323(2006) No. 1, 481-496. MR2262220 46B45 (46A45).], which is shown to be a special case of our generalized theory. 1. prelim...

On Almost Everywhere Strong Convergence of Multidimensional Continued Fraction Algorithms

We describe a strategy which allows one to produce computer assisted proofs of almost everywhere strong convergence of Jacobi-Perron type algorithms in arbitrary dimension. Numerical work is carried out in dimension three to illustrate our method. To the best of our knowledge this is the rst result on almost everywhere strong convergence in dimension greater than two.