1996/04/01 by Mustafa Akcoglu, M. A. Akcoglu, Alexandra Bellow +6 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Approximation Theory and Sequence Spaces #Mathematical Approximation and Integration
paper · doi:10.1017/s0143385700008798
openalex publication_date 1996/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/21
Abstract In this paper we establish conditions on a sequence of operators which imply divergence. In fact, we give conditions which imply that we can find a set B of measure as close to zero as we like, but such that the operators applied to the characteristic function of this set have a lim sup equal to 1 and a lim inf equal to 0 a.e. (strong sweeping out). The results include the fact that ergodic averages along lacunary sequences, certain convolution powers, and the Riemann sums considered by Rudin are all strong sweeping out. One of the criteria for strong sweeping out involves a condition on the Fourier transform of the sequence of measures, which is often easily checked. The second criterion for strong sweeping out involves showing that a sequence of numbers satisfies a property similar to the conclusion of Kronecker's lemma on sequences linearly independent over the rationals.