On circular parameters (Q1097442)

Let \(a_ n\), \(b_ n\) be the Fourier coefficients of f, and let \(\rho_ n=(a^ 2_ n+b^ 2_ n)^{1/2}\). It is known that (*) \(\sum^{n}_{k=1}\rho_ n=o(\log n)\) is necessary and sufficient for f, of class V, \(1<p<2\), to be continuous. For functions of bounded \(\Phi\)-variation (class \(V_{\Phi})\), the same is true if \(\lim_{x\to 0}x^ 2/\Phi (x)=0.\) The author raises the question of what happens if \(\liminf x^ 2/\Phi (x)>0\) but \(\limsup x^ 2\Phi (x)=0.\) Say that f is almost increasing on (0,\(\delta)\) if, for some \(c>1\), \(f(x)<cf(u)\) for \(u>x\) on (0,\(\delta)\). The author shows that if \(u^{-1}\Phi (u)\) is almost increasing on some (0,\(\delta)\) there is a function f in \(V_{\Phi}\cap C\) for which (*) is not satisfied. However, if the extra condition on \(u^{-1}\Phi (u)\) is dropped, there is an \(f\in V_{\Phi}\) that satisfies (*), but for which \(V_{\Phi}\cap C\) consists only of constants (so that (*) is trivially satisfied).