Mathematicians map the limits of a 90-year-old signal processing tool
Researchers have identified precisely when a classical mathematical technique for analyzing complex signals breaks down and when it works reliably. The findings could refine how engineers filter data in telecommunications, audio processing, and other signal-heavy industries where accuracy at scale determines system performance.
Originaltitel: Convergence Almost Everywhere of Partial Sums and Féjer Means of Vilenkin-Fourier Series
<p>We characterize subsequences {S<sub>nk</sub>} of partial sums with respect to (bounded or unbounded) Vilenkin systems of f ∈ L<sup>1</sup>(G<sub>m</sub>) for which almost everywhere convergence holds. Moreover, we construct an explicit f ∈ L<sup>p</sup>(G<sub>m</sub>), 1 ≤ p < ∞ whose partial sums (satisfying the same conditions which guarantee almost everywhere convergence) diverges on any set of measure zero. We also prove a similar divergence result for Vilenkin-Féjer means.</p>