Beyond Tchakaloff Quadrature: Positive Functionals, Frames and Widths
Martin Schäfer, Tino Ullrich
TL;DR
This work generalizes exact discretization beyond traditional nonnegative quadrature by proving a complex-valued extension of $Tchakaloff$ discretization for finite-dimensional subspaces and by characterizing when positive discretizations are possible for linear functionals. It introduces strict $S$-positivity as a key criterion, connects discretization to Kolmogorov widths and singular numbers, and applies the framework to $L_p$-Marcinkiewicz-Zygmund equalities as well as frame discretization, yielding both theoretical bounds and constructive procedures. The results yield explicit bounds on Tchakaloff widths $\kappa_n^+$, establish conditions for exact weighted frame subsampling, and provide a $D$-optimal design scheme to choose discretization points, with potential algorithmic impact for quadrature, norm discretization, and frame analysis. Collectively, the paper unifies discretization theory across quadrature, norm approximation, and frames, and offers practical routes to construct discretizations via convex geometry and determinant optimization.
Abstract
Tchakaloff's theorem from 1957 asserts the existence of exact quadrature rules with non-negative weights for any polynomial space of finite degree on $\mathbb{R}^d$ if the underlying measure is positive, compactly supported, and absolutely continuous with respect to the Lebesgue measure. This classical result coined the term Tchakaloff quadrature for quadrature that is exact and only uses non-negative weights. It has been a long-standing endeavor, under which conditions such rules exist. A final answer was given in 2012 by Bisgaard with the insight that, in fact, every finite-dimensional space of integrable functions on a positive measure space admits them. In this article we recall this result and provide a major extension to the question of positive discretizability of $\mathbb{C}$-linear functionals on finite-dimensional spaces. We introduce the notion of strict $S$-positivity for such functionals, where $S$ are subsets of the functional's domain, and show the equivalence of positive discretizability to being strictly $S$-positive for a suitable choice of $S$. We further investigate consequences for other discretization problems. One fundamental implication is the guaranteed existence of $L_p$-Marcinkiewicz-Zygmund equalities in finite-dimensional spaces of $p$-integrable functions in case that $p$ is an even integer, another the exact discretizability of any frame in $\mathbb{K}^n$, where $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$, if a rescaling of the frame elements is allowed. In addition, we provide bounds for Tchakaloff quadrature widths $κ_n^+$ and, addressing the question of constructibility of discretization points, establish a connection to $D$-optimal design.