Pringsheim Convergence and the Dirichlet Function

Thomas Beatty,Bradley Hansen

doi:10.4236/apm.2016.66031

Abstract

Double sequences have some unexpected properties which derive from the possibility of commuting limit operations. For example, may be defined so that the iterated limits and exist and are equal for all x, and yet the Pringsheim limit does not exist. The sequence is a classic example used to show that the iterated limit of a double sequence of continuous functions may exist, but result in an everywhere discontinuous limit. We explore whether the limit of this sequence in the Pringsheim sense equals the iterated result and derive an interesting property of cosines as a byproduct.

Highlights

The problem of convergence of a doubly indexed sequence presents some interesting phenomena related to the order of taking iterated limits as well as subsequences where one index is a function of the other
Convergence of a double sequence in the sense of Pringsheim is a strong enough condition to allow us to characterize the behavior of the iterated limits as well as the limits of ordinary sequences induced by collapsing the two indices into one according to a am = 2−m3 )
We can extend the notion of Pringsheim convergence of numerical sequences to pointwise convergence in the Pringsheim sense for functions

Summary

Introduction

The problem of convergence of a doubly indexed sequence presents some interesting phenomena related to the order of taking iterated limits as well as subsequences where one index is a function of the other. We can extend the notion of Pringsheim convergence of numerical sequences to pointwise convergence in the Pringsheim sense for functions. (2016) Pringsheim Convergence and the Dirichlet Function. Hansen tions of the form fmn ( x) = (cos πm!x)2n in this context. One iterated limit of this sequence, namely limm→∞ limn→∞ fmn ( x) , is a well-known example of the construction of the Dirichlet “salt-and-pepper” function δ ( x). Which the ordinary sequence (cos πm!x)2m does not converge to zero

Background

Main Theorem

Dirichlet Function

Postscript