One and Two Dimensional Cantor-lebesgue Type Theorems

Let φ(n) be any function which grows more slowly than exponentially in n, i.e., limsup n→∞ φ(n)1/n ≤ 1. There is a double trigonometric series whose coefficients grow like φ(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), and which has the everywhere convergent partial sum subsequence S2j . For any p > 1, there is a one dimensional trigonometric series whose coefficients grow like φ(n p−1 p ), and which has the everywhere convergent partial sum subsequence S[jp]. All these examples exhibit, in a sense, the worst possible behavior. If mj 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 Smj .