[Paper Review] On Koksma-Hlawka inequality

Luca Brandolini, Leonardo Colzani+2 more|arXiv (Cornell University)|Jun 23, 2011

Functional Equations Stability Results15 references3 citations

TL;DR

This paper extends the classical Koksma-Hlawka inequality to piecewise smooth functions $ f\chi_\Omega $, where $ f $ is smooth and $ \Omega \subset [0,1]^d $ is a Borel set. It introduces a discrepancy measure $ \mathcal{D}(\Omega, x_j) $ based on intersections with axis-aligned intervals, and proves that the integration error is bounded by $ \mathcal{D}(\Omega, x_j) \mathcal{V}(f) $, with $ \mathcal{V}(f) $ being the total variation of $ f $. The result applies to $ L^p $-$ L^q $ norms and extends to compact manifolds like spheres.

ABSTRACT

The classical Koksma Hlawka inequality does not apply to functions with simple discontinuities. Here we state a Koksma Hlawka type inequality which applies to piecewise smooth functions $fχ_Ω$, with $f$ smooth and $Ω$ a Borel subset of $[0,1]^{d}$.

Motivation & Objective

• To address the limitation of the classical Koksma-Hlawka inequality, which fails for functions with simple discontinuities such as characteristic functions of non-rectangular sets.
• To develop a Koksma-Hlawka-type inequality applicable to piecewise smooth functions $ f\chi_\Omega $, where $ f $ is smooth and $ \Omega \subset [0,1]^d $ is a Borel set.
• To define a geometric discrepancy $ \mathcal{D}(\Omega, x_j) $ based on the distribution of points within intersections of $ \Omega $ with axis-aligned intervals, cubes, or balls.
• To extend the inequality to compact Riemannian manifolds, particularly spheres, using harmonic analysis and spherical harmonics.

Proposed method

• Define a new discrepancy $ \mathcal{D}(\Omega, x_j) = 2^d \sup_I \left| N^{-1} \sum_{j=1}^N \chi_{\Omega \cap I}(x_j) - |\Omega \cap I| \right| $, measuring the deviation of point distribution in $ \Omega \cap I $ from uniformity.
• Introduce an $ L^p $-$ L^q $ norm version of the inequality with $ 1/p + 1/q = 1 $, using $ \mathcal{D}_q(\Omega, \mu) $ and $ \mathcal{V}_p(f) $, where $ \mathcal{V}_p(f) $ is the $ L^p $-total variation of $ f $.
• Prove the inequality on the torus $ \mathbb{T}^d $ using Fourier analysis and properties of spherical harmonics, particularly for functions on spheres.
• Extend results to compact manifolds by using the spectral theory of the Laplacian and $ L^2 $-norms of measures restricted to spherical caps.
• Use the function $ \varphi(n) $, related to spherical Bessel functions, to control the growth of Fourier coefficients and derive $ L^2 $-bounds on discrepancy.
• Establish bounds involving $ \gamma > 5/2 $ and $ \gamma > (d+3)/2 $ for spheres of dimension $ d $, ensuring convergence of the series in the error estimate.

Experimental results

Research questions

• RQ1 Can the classical Koksma-Hlawka inequality be extended to functions with discontinuities, such as $ f\chi_\Omega $, where $ f $ is smooth and $ \Omega $ is a Borel set?
• RQ2 What is an appropriate geometric discrepancy measure $ \mathcal{D}(\Omega, x_j) $ that captures the distribution of points within $ \Omega \cap I $ for axis-aligned intervals $ I $?
• RQ3 How can the inequality be generalized to $ L^p $-$ L^q $ norms with $ 1/p + 1/q = 1 $, and what are the resulting bounds on the integration error?
• RQ4 Can the inequality be extended to compact Riemannian manifolds such as spheres, and what role do spherical harmonics and spectral theory play in this extension?
• RQ5 What is the quantitative dependence of the error bound on the number of points $ N $, the smoothness of $ f $, and the geometry of $ \Omega $?

Key findings

• The paper establishes a Koksma-Hlawka-type inequality for $ f\chi_\Omega $, with error bounded by $ \mathcal{D}(\Omega, x_j) \mathcal{V}(f) $, where $ \mathcal{D}(\Omega, x_j) $ is the discrepancy over intersections of $ \Omega $ with axis-aligned intervals.
• For $ L^p $-$ L^q $ norms, the error is bounded by $ \mathcal{D}_q(\Omega, \mu) \mathcal{V}_p(f) $, with $ \mathcal{D}_q $ defined via $ L^q $-norms of point distribution in $ \Omega \cap I $, and $ \mathcal{V}_p(f) $ the $ L^p $-total variation of $ f $.
• On the sphere $ \mathcal{S} $, the inequality holds for almost every $ \vartheta \in (0, \pi) $ with $ \gamma > 5/2 $, yielding a bound involving $ \left\{ \int_{\mathcal{S}} |\mu(B(x,\vartheta) \cap \Omega)|^2 dx \right\}^{1/2} $.
• For almost every $ \vartheta \in (0, \pi/2) $, a stronger bound is obtained using both $ B(x,\vartheta) $ and $ B(x,2\vartheta) $, with the error controlled by a sum of two $ L^2 $-discrepancy terms.
• In the application to the simplex $ \Sigma $, the bound for $ \Omega = \Sigma $ is $ c\varepsilon^{-d} N^{-1} \log^d(N) $, which improves upon the general convex case bound of $ c\varepsilon^{-d} N^{-2/(d+1)} \log^\gamma(N) $.
• The results extend to higher-dimensional spheres with $ \gamma > (d+3)/2 $, indicating that the spectral decay of spherical harmonics governs the convergence rate.