ACTA issues

Prevalence of chaotic differences for unpredictable functions

J. W. Neuberger

Acta Sci. Math. (Szeged) 61:1-4(1995), 181-196

Abstract. Denote the difference of order $n$ $$ \sum ^n_{k=0} C(n,k)(-1)^{n-k}f(u+k\delta ) $$ by $\Delta_f(n;u,\delta )$ provided $[u,u+n\delta ] \subset D(f)$ where $C(n,k)$ is the binomial coefficient ${n\choose k}$. Call a function $f$ with domain [0,1] unpredictable if $f$ is continuous and is identically zero on $[0,1/2]$ but not identically zero on any longer subinterval of $[0,1]$. We deal with the question: For $f$ unpredictable, how large are differences $\Delta_f(n;0,\delta )$ where $n\delta\in (1/2,1)$? We show that higher order differences of such a function exhibit a much more extreme asymptotic character than was previously known.

AMS Subject Classification (1991): 26E10

Received December 28, 1994. (Registered under 5679/2009.)