Almost everywhere convergence of mock Fourier series for the middle-fourth Cantor measure

Abstract

In 1998, Jorgensen and Pedersen constructed the first example of a singular continuous spectral measure. Precisely, they proved that the self-similar measure generated by the iterated function system $\{\frac{x-1}{4},\frac{x+1}{4}\}$ with equal weights, denoted by ${\mu }_{1/4}$, is a spectral measure with a spectrum ${\Lambda }_4$, called the canonical spectrum,\[ Λ_4 := \set{ \sum_{j=0}^{m-1}\varepsilon_j4^j: m\ge 1,\ \varepsilon_j\in\{0,1\} }. \] For $f\in L^1({\mu }_{1/4})$, let $S_{n}f$ be the $n$-th partial sum of its Mock Fourier series with respect to ${\Lambda }_4$. We prove that the associated maximal operator $S^{\ast }f={\sup }_n|S_{n}f|$ is of weak type $(1,1)$. Consequently, $S_{n}f\to f$ ${\mu }_{1/4}$-almost everywhere on $\supp(μ_{1/4})$. This solves a long-standing open problem of Strichartz \cite[p.~341]{Str06}.

15 pages