Constructing Singular Functions via Farey Fractions

Author: Girgensohn R.  

Publisher: Academic Press

ISSN: 0022-247X

Source: Journal of Mathematical Analysis and Applications, Vol.203, Iss.1, 1996-10, pp. : 127-141

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

Abstract

To illustrate some points about continued fractions, H. Minkowski in 1904 introduced the so-called ?-function. This function and some generalizations of it are known to be singular, i.e., strictly monotone with derivative 0 almost everywhere. They can be characterized by systems of functional equations, such as fbiggl({xover x+1}biggr)=tf(x),qquad fbiggl({1over x+1}biggr)=1-(1-t)f(x)qquadhbox{for all $xin[0,1]$},eqno{hbox{(F)}} where f :[0, 1]-< R is the unknown, and rbiggl({xover x+1}biggr)=tr(x),qquad rbiggl({1over 2-x}biggr)=t+(1-t)r(x)qquadhbox{for all $xin[0,1]$.}eqno{hbox{(R)}} where r : [0, 1]-< R is the unknown. In both cases, t∈ (0, 1) is a given parameter. In the present note we give a general construction of singular functions, based on the Farey fractions and including, as a special case, the Minkowski function and its generalizations. In contrast to earlier proofs, the methods presented here do not make explicit use of the theory of continued fractions.