Positive quadrature and mobile sampling of multivariate trigonometric polynomials

Abstract

The geometric properties of quadrature points in one and multiple dimensions are a classical topic in numerical analysis. Recently, generalized quadrature methods, where discrete points and weights are replaced by integration along curves, have attracted growing interest. This paper focuses on positive generalized quadratures that are exact for multivariate trigonometric polynomials. We derive upper bounds on the covering radius of such quadratures using sign-localized test functions, a technique that also extends naturally to algebraic polynomials on intervals, spherical polynomials, and hyperbolic cross trigonometric polynomials. For lower bounds on the length of quadrature curves, we compare two recent approaches in the context of multivariate trigonometric polynomials. The second part of the paper studies a well-known, geometrically simple curve that underlies rank-1 Korobov lattice rules for periodic functions. We prove that these curves are quasi-optimal with respect to multiple optimality criteria and demonstrate its adaptability to the 2-sphere.