Remarks on numerical integration, discrepancy, and diaphony

TL;DR

This paper unifies the formulation of numerical integration problems across approximation and discrepancy theories and demonstrates how approximation techniques can establish lower bounds for new types of discrepancy measures.

Contribution

It introduces a unified framework for numerical integration and applies approximation theory methods to analyze smooth discrepancy.

Findings

Unified formulation of numerical integration problems.

Application of approximation theory to lower bounds in discrepancy.

Insights into smooth discrepancy measures.

Abstract

The goal of this paper is twofold. First, we present a unified way of formulating numerical integration problems from both approximation theory and discrepancy theory. Second, we show how techniques, developed in approximation theory, work in proving lower bounds for recently developed new type of discrepancy -- the smooth discrepancy.