Some properties of generalized measures on the dyadic group (Q1088102)

If \(m_{\mu}\) is a pseudo measure, that is, \({\hat \mu}\)(k)\(=O(1)\) as \(k\to \infty\), then for each \(\epsilon >0\) \(m_{\mu}(I_ n(x))=o (n(\log n)^{1+\epsilon}/\sqrt{2^ n})\) as \(n\to \infty\) quasi- uniformly, where \(I_ n(x)=I^ p_ n\) is the dyadic interval of length \(1/2^ n\) containing x. Put \(m_ f(I)=\int_{I}f(x)dx\) for an integrable function f. Then \(\sum^{2^ n-1}_{p=0}| \Delta m_ f(I^ p_ n)| =o (1)\) as \(n\to \infty\) where \(\Delta m_ f(I^ p_ n)=m_ f(I^{2p}_{n+1})-m_ f(I_{n+1}^{2p+1}).\) This result is stronger than the Riemann-Lebesgue theorem. Some fundamental theorems are stated.