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:----
关于数值积分作为有限矩阵逼近乘法算子的收敛性
翻译后的摘要:
我们研究一类数值积分方法的收敛性,其中数值积分被构成为乘法算子的有限矩阵逼近。对于有界函数,已经使用强算子收敛理论来证明了其收敛性。在本文中,我们考虑未被界定的函数和定义域,相对于有界情况,这种情况存在一些困难。在这种情况下,自然的方法选择是使用强解析收敛理论。这个理论以前主要应用于微分算子逼近的收敛性研究上。现有理论已经包含了可以用于有限函数类证明和扩展收敛定理。拓展结果在所有自伴算子上都适用,而不仅仅是乘法算子上。我们还展示了如何使用Jensen算子不等式来分析函数被一个算子凸函数所界限的不合适的数值积分收敛。