Weak Convergence and Banach Space-Valued Functions

Improving the Stability Theory of Feynman's Operational Calculi

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3 Citations (Scopus)

Abstract

In this paper we investigate the relation between weak convergence of a sequence {μn} 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 limn→∞Sf dμn = ∫S f dμ-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.

Original languageEnglish
Pages (from-to)279-294
Number of pages16
JournalMathematical Physics Analysis and Geometry
Volume14
Issue number4
DOIs
StatePublished - Dec 2011

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Operational Calculus
Stability Theory
Weak Convergence
Banach space
Probability Measure
Norm
Polish Space
Continuous Function

All Science Journal Classification (ASJC) codes

  • Mathematical Physics
  • Geometry and Topology

Cite this

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AB - In this paper we investigate the relation between weak convergence of a sequence {μn} 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 limn→∞ ∫Sf dμn = ∫S f dμ-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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