One and two dimensional Cantor-Lebesgue type theorems

Abstract:

Let $\varphi (n)$ be any function which grows more slowly than exponentially in $n,$ i.e., $\limsup _{n\rightarrow \infty }\varphi (n)^{1/n}\leq 1.$ There is a double trigonometric series whose coefficients grow like $\varphi (n),$ and which is everywhere convergent in the square, restricted rectangular, and one-way iterative senses. Given any preassigned rate, there is a one dimensional trigonometric series whose coefficients grow at that rate, but which has an everywhere convergent partial sum subsequence. There is a one dimensional trigonometric series whose coefficients grow like $\varphi (n),$ and which has the everywhere convergent partial sum subsequence $S_{2^j}.$ For any $p>1,$ there is a one dimensional trigonometric series whose coefficients grow like $\varphi (n^{\frac {p-1}p}),$ and which has the everywhere convergent partial sum subsequence $S_{[j^p]}.$ All these examples exhibit, in a sense, the worst possible behavior. If $m_j$ is increasing and has arbitrarily large gaps, there is a one dimensional trigonometric series with unbounded coefficients which has the everywhere convergent partial sum subsequence $S_{m_j}.$