CONVERGENCE AND DIVERGENCE OF DECREASING REARRANGED FOURIER-SERIES

In a number of useful applications, e.g., data compression, the approp riate partial sums of the Fourier series are formed by taking into con sideration the size of the coefficients rather than the size of the fr equencies involved. The purpose of this paper is to show the limitatio ns of that method of summation. We use several results from the number theory to construct counterexamples to L-p-convergence for p < 2. We also show how to obtain positive results if we combine the two points of view, i.e., cutting on frequencies and the size of coefficients at the same time. This can be considered as a kind of uncertainty princip le for Fourier sums.