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 language | English |
|---|---|
| Pages (from-to) | 279-294 |
| Number of pages | 16 |
| Journal | Mathematical Physics Analysis and Geometry |
| Volume | 14 |
| Issue number | 4 |
| DOIs | |
| State | Published - Dec 2011 |
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All Science Journal Classification (ASJC) codes
- Mathematical Physics
- Geometry and Topology
Cite this
Weak Convergence and Banach Space-Valued Functions : Improving the Stability Theory of Feynman's Operational Calculi. / Nielsen, Lance.
In: Mathematical Physics Analysis and Geometry, Vol. 14, No. 4, 12.2011, p. 279-294.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Weak Convergence and Banach Space-Valued Functions
T2 - Improving the Stability Theory of Feynman's Operational Calculi
AU - Nielsen, Lance
PY - 2011/12
Y1 - 2011/12
N2 - 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.
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.
UR - http://www.scopus.com/inward/record.url?scp=82955187685&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=82955187685&partnerID=8YFLogxK
U2 - 10.1007/s11040-011-9097-z
DO - 10.1007/s11040-011-9097-z
M3 - Article
AN - SCOPUS:82955187685
VL - 14
SP - 279
EP - 294
JO - Mathematical Physics Analysis and Geometry
JF - Mathematical Physics Analysis and Geometry
SN - 1385-0172
IS - 4
ER -