We study the convergence of a family of numerical integration methods where the numerical integral is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, the convergence has already been established using the theory of strong operator convergence. In this article, we consider unbounded functions and domains which pose several difficulties compared to the bounded case. A natural choice of method for this study is the theory of strong resolvent convergence which has previously been mostly applied to study the convergence of approximations of differential operators. The existing theory already includes convergence theorems that can be used as proofs as such for a limited class of functions and extended for wider class of functions in terms of function growth or discontinuity. The extended results apply to all self-adjoint operators, not just multiplication operators. We also show how Jensen's operator inequality can be used to analyse the convergence of an improper numerical integral of a function bounded by an operator convex function.

翻译：我们研究了一类数值积分方法的收敛性，其中数值积分被构造为乘法算子的有限矩阵逼近。对于有界函数，已利用强算子收敛理论建立了收敛性。在本文中，我们考虑无界函数和定义域，这相比于有界情形带来了若干困难。本研究自然选择的方法论是强预解收敛理论，该理论此前主要应用于研究微分算子逼近的收敛性。现有理论已包含可对有限类函数直接用作证明的收敛定理，并可通过函数增长性或不连续性扩展到更广泛的函数类。扩展后的结果适用于所有自伴算子，而不仅仅是乘法算子。我们还展示了如何使用Jensen算子不等式来分析由算子凸函数控制的有界函数的不当数值积分的收敛性。