One and two dimensional Cantor-Lebesgue type theorems

Abstract

Let φ ( n ) \varphi (n) be any function which grows more slowly than exponentially in n , n, i.e., lim sup n → ∞ φ ( n ) 1 / n ≤ 1. \limsup _{n\rightarrow \infty }\varphi (n)^{1/n}\leq 1. There is a double trigonometric series whose coefficients grow like φ ( n ) , \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 φ ( n ) , \varphi (n), and which has the everywhere convergent partial sum subsequence S 2 j . S_{2^j}. For any p > 1 , p>1, there is a one dimensional trigonometric series whose coefficients grow like φ ( n p − 1 p ) , \varphi (n^{\frac {p-1}p}), and which has the everywhere convergent partial sum subsequence S [ j p ] . S_{[j^p]}. All these examples exhibit, in a sense, the worst possible behavior. If m j 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 . S_{m_j}.