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We study the convergence of a family of numerical integration methods wherethe numerical integral is formulated as a finite matrix approximation to amultiplication operator. For bounded functions, the convergence has alreadybeen established using the theory of strong operator convergence. In thisarticle, we consider unbounded functions and domains which pose severaldifficulties compared to the bounded case. A natural choice of method for thisstudy is the theory of strong resolvent convergence which has previously beenmostly applied to study the convergence of approximations of differentialoperators. The existing theory already includes convergence theorems that canbe used as proofs as such for a limited class of functions and extended forwider class of functions in terms of function growth or discontinuity. Theextended results apply to all self-adjoint operators, not just multiplicationoperators. We also show how Jensen's operator inequality can be used to analysethe convergence of an improper numerical integral of a function bounded by anoperator convex function.

On the Convergence of Numerical Integration as a Finite Matrix Approximation to Multiplication Operator