On some convergence properties of Haar-Fourier series in the classes \(\phi\) (L) (Q789659)

The author considers the Haar-Fourier Series. Let \(\Phi\) be the set of functions \(\phi\) :\(R\to R\) with the properties \(0=\phi(0)=\lim_{t\to 0}\phi(t)<\phi(t)=\phi(-t)\leq \phi(\tau)<\lim_{t\to \infty}\phi(t)=\infty\) where \(0<t<\tau<\infty\). The class \(\phi\) (L) consists of all Lebesgue-measurable functions f:[0,1]\(\to R\) satisfying \(\| f\|_{\phi}=\int^{1}_{0}\phi(f(x))dx<\infty\). A sequence of measurable functions \(f_ n\), \(n\geq 1\), is said to be \(\psi\)- convergent to the measurable function f, if \(f-f_ n\) belongs to \(\psi\) (L) for large n and \(\lim_{n\to \infty}\| f-f_ n\|_{\psi}=0.\) Let \(h(\psi)\) denote the class \(H(\psi)=\{f\in L^ 1,\| s_ nf- f\|_{\psi}\to 0\}\quad as\quad n\to \infty.\) Here \(s_ nf(x)=\sum^{n}_{j=1}a_ j(f)X_ j(x),a_ n(f)=\int^{1}_{0}f(x)X_ n(x)dx\) is the partial sum of the Haar- Fourier series of a function \(f\in L^ 1\). We now state Theorem 1. Let \(\phi\in \Phi\) and \(\phi(L)\subset L^ 1\). Then the inclusion \(\phi\) (L)\(\subset H(\phi)\) is valid if and only if \(\phi\) satisfies the condition \(\phi(2t)=0(\phi(t))\), and \(\phi(t)\asymp \phi_ c(t)\), \(t\to \infty\) where \(\phi_ c(t)\) is the convex minorant with reference to \(\phi\) (t). These conditions are equivalent to the following one: The class \(\phi\) (L) can be identified with a separable Orlicz space. Theorem 2 gives a somewhat more general result than Theorem 1.