Functions of \((m,\Phi)\)-bounded variation and the convergence of Fourier series (Q1842484)

There is given the definition of an \((m,\Phi)\)-variation of a given function \(f\) in an interval \([a, b]\) for a sequence \(\Phi= (\varphi_ n)\) of strictly increasing and convex functions \(\varphi_ n: [0, \infty)\to [0, \infty)\) such that \(\varphi_{n+ 1}(x)\leq \varphi_ n(x)\) and \(\sum^ \infty_{n= 1} \varphi_ n(x)= \infty\) for all \(x> 0\), by the formula \[ V_{m, \Phi}(f)= \sup \sum^ \infty_{n= 1} \varphi_ n(|\Delta^ m f(I_ n)|), \] where \((I_ n)\) is an arbitrary sequence of disjoint subintervals of \([a, b]\) and \(\Delta^ m\) is the symbol of \(m\)th difference of \(f\) is \(I_ n\). There are shown properties of this \((m, \Phi)\)-variation and an application to convergence everywhere of Fourier series.