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Weak Convergence and Banach Space-Valued Functions: Improving the Stability Theory of Feynman’s Operational Calculi

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Abstract

In this paper we investigate the relation between weak convergence of a sequence \(\left\{ \mu_{n}\right\} \) of probability measures on a Polish space S converging weakly to the probability measure μ and continuous, norm-bounded functions into a Banach space X. We show that, given a norm-bounded continuous function f:S→X, it follows that \(\lim_{n\to\infty}\int_{S}f\, d\mu_{n}=\int_{S}f\, d\mu\)—the limit one has for bounded and continuous real (or complex)—valued functions on S. This result is then applied to the stability theory of Feynman’s operational calculus where it is shown that the theory can be significantly improved over previous results.

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  1. Department of Mathematics, Creighton University, Omaha, NE, 68178, USA

    Lance Nielsen

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  1. Lance Nielsen
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Correspondence to Lance Nielsen.

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Nielsen, L. Weak Convergence and Banach Space-Valued Functions: Improving the Stability Theory of Feynman’s Operational Calculi. Math Phys Anal Geom 14, 279–294 (2011). https://doi.org/10.1007/s11040-011-9097-z

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  • DOI: https://doi.org/10.1007/s11040-011-9097-z

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