Low-discrepancy sequences for piecewise smooth functions on the two-dimensional torus
arXiv:1503.06655
Originality Synthesis-oriented
AI Analysis
Provides a theoretical advance in numerical integration for a specific class of piecewise smooth functions on the torus, but the result is incremental and domain-specific.
The authors construct explicit low-discrepancy sequences for integrating smooth periodic functions over convex domains with positive curvature in R^2, using simultaneous Diophantine approximation and a generalized Erdős–Turán inequality.
We produce explicit low-discrepancy infinite sequences which can be used to approximate the integral of a smooth periodic function restricted to a convex domain with positive curvature in R^2. The proof depends on simultaneous diophantine approximation and a general version of the Erdos-Turan inequality.