Prevalence of ``nowhere analyticity'' (Q2919697)

Let \(C^\infty[0,1]\) denote the linear space of functions of the class \(C^\infty\) on \([0,1]\). Then \(C^\infty[0,1]\) is a Fréchet space with the metric NEWLINE\[NEWLINEd(f,g)= \sum^\infty_{k=0} {2^{-k}p_k(f-g)\over 1+ p_k(f-g)},NEWLINE\]NEWLINE where NEWLINE\[NEWLINEp_k(f)= \sup_{0\leq j\leq k}\,\Biggl[\sup_{x\in [0,1]}|f^{(j)}(x)|\Biggr].NEWLINE\]NEWLINE For a \(C^\infty\) function \(f\) on an open interval containing \(x_0\) and its Taylor series at \(x_0\) given by NEWLINE\[NEWLINET(f,x_0)(x)= \sum^\infty_{n=0} {f^{(n)}(x_0)\over n!} (x- x_0)^n,NEWLINE\]NEWLINE we say that \(f\) is analytic at \(x_0\) if \(T(f,x_0)\) converges to \(f\) on an open neighborhood of \(x_0\). If \(f\) is not analytic at \(x_0\), then \(f\) is said to have a singularity at \(x_0\). A function with a singularity at each point of an interval is called nowhere analytic on the interval. This paper examines the set of nowhere analytic functions with regard to their prevalence, defined in various senses.NEWLINENEWLINE For example, a Borel set \(B\) in a complete metric space \(E\) is said to be shy if there exists a Borel probability measure \(\mu\) on \(E\) with compact support such that \(\mu(B+x)= 0\) for all \(x\in E\). More generally, a set is called shy if it is contained in a shy Borel set. The complement of a shy set is called prevalent. The authors prove that the set of nowhere analytic functions is prevalent in \(C^\infty[0,1]\).NEWLINENEWLINE Additionally, let \(s>0\) and \(\Omega\subset\mathbb{R}\) be open, then, for \(f\in C^\infty(\Omega)\), we say that \(f\) is Gevrey differentiable of order \(s\) at \(x_0\in\Omega\) if there exist a compact neighborhood \(I\) of \(x_0\) and constants \(C,h>0\) such that NEWLINE\[NEWLINE\sup_{x\in I} |f^{(n)}(x)|\leq Ch^n(n!)^s,\text{ for all }n\in \mathbb N_0.NEWLINE\]NEWLINE The case \(s=1\) corresponds to analyticity. A function \(f\in C^\infty[0,1]\) is nowhere Gevrey differentiable on \([0,1]\) if \(f\) is not Gevrey differentiable of order \(s\) at \(x_0\) for any \(x_0\in [0,1]\) and \(s\geq 1\). The authors prove that the set of nowhere differentiable functions is a prevalent subset of \(C^\infty[0,1]\). It is also proved that the set of nowhere Gevrey differentiable functions is a residual subset of \(C^\infty[0,1]\), that is, it contains a countable intersection of dense open sets in \(C^\infty[0,1]\).NEWLINENEWLINEOther generalizations are also discussed.