Integrals devised for special purposes

McShane, E. J.

doi:10.1090/S0002-9904-1963-10964-7

Integrals devised for special purposes
HTML articles powered by AMS MathViewer

by E. J. McShane PDF

Bull. Amer. Math. Soc. 69 (1963), 597-627

References

S. G. Brush, Functional integrals and statistical physics, Rev. Mod. Phys. 33 (1961), 79–92. MR 0134253, DOI 10.1103/revmodphys.33.79
R. H. Cameron, Some examples of Fourier-Wiener transforms of analytic functionals, Duke Math. J. 12 (1945), 485–488. MR 12707
R. H. Cameron, A “Simpson’s rule” for the numerical evaluation of Wiener’s integrals in function space, Duke Math. J. 18 (1951), 111–130. MR 40589, DOI 10.1215/S0012-7094-51-01810-8
R. H. Cameron, The translation pathology of Wiener space, Duke Math. J. 21 (1954), 623–627. MR 65033, DOI 10.1215/S0012-7094-54-02165-1
R. H. Cameron, The generalized heat flow equation and a corresponding Poisson formula, Ann. of Math. (2) 59 (1954), 434–462. MR 61261, DOI 10.2307/1969711
R. H. Cameron, A family of integrals serving to connect the Wiener and Feynman integrals, J. Math. and Phys. 39 (1960/61), 126–140. MR 127776, DOI 10.1002/sapm1960391126
R. H. Cameron, The Ilstow and Feynman integrals, J. Analyse Math. 10 (1962/63), 287–361. MR 150845, DOI 10.1007/BF02790311
R. H. Cameron and R. E. Fagen, Nonlinear transformations of Volterra type in Wiener space, Trans. Amer. Math. Soc. 75 (1953), 552–575. MR 59476, DOI 10.1090/S0002-9947-1953-0059476-X
R. H. Cameron and C. Hatfield Jr., On the summability of certain orthogonal developments of nonlinear functionals, Bull. Amer. Math. Soc. 55 (1949), 130–145. MR 28534, DOI 10.1090/S0002-9904-1949-09186-3
R. H. Cameron and C. Hatfield, On the summability of certain series for unbounded nonlinear functionals, Proc. Amer. Math. Soc. 4 (1953), 375–387. MR 54848, DOI 10.1090/S0002-9939-1953-0054848-7
R. H. Cameron and W. T. Martin, Transformations of Wiener integrals under translations, Ann. of Math. (2) 45 (1944), 386–396. MR 10346, DOI 10.2307/1969276
R. H. Cameron and W. T. Martin, Transformations of Wiener integrals under a general class of linear transformations, Trans. Amer. Math. Soc. 58 (1945), 184–219. MR 13240, DOI 10.1090/S0002-9947-1945-0013240-1
R. H. Cameron and W. T. Martin, Fourier-Wiener transforms of analytic functionals, Duke Math. J. 12 (1945), 489–507. MR 12708
R. H. Cameron and W. T. Martin, Fourier-Wiener transforms of functionals belonging to $L_2$ over the space $C$, Duke Math. J. 14 (1947), 99–107. MR 20231, DOI 10.1215/S0012-7094-47-01409-9
R. H. Cameron and W. T. Martin, The transformation of Wiener integrals by nonlinear transformations, Trans. Amer. Math. Soc. 66 (1949), 253–283. MR 31196, DOI 10.1090/S0002-9947-1949-0031196-6
Robert Cogburn, Conditional probability operators, Ann. Math. Statist. 33 (1962), 634–658. MR 142133, DOI 10.1214/aoms/1177704587
S. H. Coleman, Separability of stochastic processes, Proc. Amer. Math. Soc. 14 (1963), 172–176. MR 143248, DOI 10.1090/S0002-9939-1963-0143248-3
Jacques Dixmier, Représentations intégrables du groupe de De Sitter, Bull. Soc. Math. France 89 (1961), 9–41. MR 140614, DOI 10.24033/bsmf.1558
H. A. Dye, The Radon-Nikodým theorem for finite rings of operators, Trans. Amer. Math. Soc. 72 (1952), 243–280. MR 45954, DOI 10.1090/S0002-9947-1952-0045954-5
E. B. Dynkin, Markoff processes and analysis problems connected with them, Russian Math. Surveys 15 (1960), no. 2, 1–21. MR 0119254, DOI 10.1070/RM1960v015n02ABEH004215
R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Modern Physics 20 (1948), 367–387. MR 0026940, DOI 10.1103/revmodphys.20.367
K. O. Friedrichs, Mathematical aspects of the quantum theory of fields, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1953. MR 0056468
K. O. Friedrichs and H. N. Shapiro, Integration over Hilbert space and outer extensions, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 336–338. MR 108711, DOI 10.1073/pnas.43.4.336
K. O. Friedrichs and H. N. Shapiro, Integration of functionals, Courant Institute of Mathematical Sciences, New York University, New York, 1976. A seminar; Reprinting. MR 0486397
I. M. Gel′fand and A. M. Jaglom, Integration in functional spaces and its applications in quantum physics, J. Mathematical Phys. 1 (1960), 48–69. MR 112604, DOI 10.1063/1.1703636
Leonard Gross, Integration and nonlinear transformations in Hilbert space, Trans. Amer. Math. Soc. 94 (1960), 404–440. MR 112025, DOI 10.1090/S0002-9947-1960-0112025-3
Kiyosi Itô, Stochastic integral, Proc. Imp. Acad. Tokyo 20 (1944), 519–524. MR 14633
Kiyosi Itô, On a stochastic integral equation, Proc. Japan Acad. 22 (1946), no. nos. 1-4, 32–35. MR 36958, DOI 10.2183/pjab1945.22.32
Kiyosi Itô, Stochastic differential equations in a differentiable manifold, Nagoya Math. J. 1 (1950), 35–47. MR 38596, DOI 10.1017/S0027763000022819
Kiyosi Itô, Brownian motions in a Lie group, Proc. Japan Acad. 26 (1950), no. 8, 4–10. MR 46603
Kiyosi Itô, Multiple Wiener integral, J. Math. Soc. Japan 3 (1951), 157–169. MR 44064, DOI 10.2969/jmsj/00310157
Kiyosi Itô, On a formula concerning stochastic differentials, Nagoya Math. J. 3 (1951), 55–65. MR 44063, DOI 10.1017/S0027763000012216
Kiyosi Ito, On stochastic differential equations, Mem. Amer. Math. Soc. 4 (1951), 51. MR 40618
Kiyosi Itô, Complex multiple Wiener integral, Jpn. J. Math. 22 (1952), 63–86 (1953). MR 63609, DOI 10.4099/jjm1924.22.0_{6}3
Kiyosi Itô, Stationary random distributions, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 28 (1954), 209–223. MR 65060, DOI 10.1215/kjm/1250777359
Kiyosi Itô, Stochastic differential equations in a differentiable manifold. II, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 28 (1953), 81–85. MR 60180, DOI 10.1215/kjm/1250777513
Kiyosi Itô, Spectral type of the shift transformation of differential processes with stationary increments, Trans. Amer. Math. Soc. 81 (1956), 253–263. MR 77017, DOI 10.1090/S0002-9947-1956-0077017-0
Kiyosi Itô, Wiener integral and Feynmann integral, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, Univ. California Press, Berkeley, Calif., 1961, pp. 227–238. MR 0132616
M. Kac, On distributions of certain Wiener functionals, Trans. Amer. Math. Soc. 65 (1949), 1–13. MR 27960, DOI 10.1090/S0002-9947-1949-0027960-X
M. Kac, On some connections between probability theory and differential and integral equations, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley-Los Angeles, Calif., 1951, pp. 189–215. MR 0045333
Shizuo Kakutani, Determination of the spectrum of the flow of Brownian motion, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 319–323. MR 35924, DOI 10.1073/pnas.36.5.319
A. N. Kolmogorov, Foundations of the Theory of Probability, Chelsea Publishing Co., New York, N. Y., 1950. MR 0032961
E. J. McShane, Remark concerning integration, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 46–49. MR 27831, DOI 10.1073/pnas.35.1.46
E. J. McShane, Stochastic integrals and non-linear processes, J. Math. Mech. 11 (1962), 235–283. MR 0148105
E. J. McShane, Weak topologies for stochastic processes, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1148–1151. MR 139193, DOI 10.1073/pnas.48.7.1148
E. J. McShane, A weak topology for stochastic processes, Bol. Soc. Mat. São Paulo 14 (1962), 83–101 (1962). MR 157400
E. J. McShane, Families of measures and representations of algebras of operators, Trans. Amer. Math. Soc. 102 (1962), 328–345. MR 137002, DOI 10.1090/S0002-9947-1962-0137002-X
Elliott W. Montroll, Markoff chains, Wiener integrals, and quantum theory, Comm. Pure Appl. Math. 5 (1952), 415–453. MR 52721, DOI 10.1002/cpa.3160050403
Shu-Teh Chen Moy, Characterizations of conditional expectation as a transformation on function spaces, Pacific J. Math. 4 (1954), 47–63. MR 60750, DOI 10.2140/pjm.1954.4.47
Margaret Owchar, Wiener integrals of multiple variations, Proc. Amer. Math. Soc. 3 (1952), 459–470. MR 47257, DOI 10.1090/S0002-9939-1952-0047257-7
M. Rosenblatt, On a class of Markov processes, Trans. Amer. Math. Soc. 71 (1951), 120–135. MR 43406, DOI 10.1090/S0002-9947-1951-0043406-9
Gian-Carlo Rota, Une généralisation de l’espérance mathématique conditionnelle qui se présente dans la théorie statistique de la turbulence, C. R. Acad. Sci. Paris 251 (1960), 624–626 (French). MR 116718
Gian-Carlo Rota, On the representation of aeraging operators, Rend. Sem. Mat. Univ. Padova 30 (1960), 52–64. MR 112041
I. E. Segal, Decompositions of operator algebras. I, Mem. Amer. Math. Soc. 9 (1951), 67. MR 44749
I. E. Segal, Equivalences of measure spaces, Amer. J. Math. 73 (1951), 275–313. MR 41191, DOI 10.2307/2372178
I. E. Segal, A non-commutative extension of abstract integration, Ann. of Math. (2) 57 (1953), 401–457. MR 54864, DOI 10.2307/1969729
I. E. Segal, Abstract probability spaces and a theorem of Kolmogoroff, Amer. J. Math. 76 (1954), 721–732. MR 63602, DOI 10.2307/2372714
I. E. Segal, Tensor algebras over Hilbert spaces. I, Trans. Amer. Math. Soc. 81 (1956), 106–134. MR 76317, DOI 10.1090/S0002-9947-1956-0076317-8
I. E. Segal, Tensor algebras over Hilbert spaces. II, Ann. of Math. (2) 63 (1956), 160–175. MR 77908, DOI 10.2307/1969994
I. E. Segal, Distributions in Hilbert space and canonical systems of operators, Trans. Amer. Math. Soc. 88 (1958), 12–41. MR 102759, DOI 10.1090/S0002-9947-1958-0102759-X
Thomas I. Seidman, Linear transformations of a functional integral. I, Comm. Pure Appl. Math. 12 (1959), 611–622. MR 110009, DOI 10.1002/cpa.3160120404
M. H. Stone, Notes on integration. IV, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 50–58. MR 27830, DOI 10.1073/pnas.35.1.50
Norbert Wiener, Generalized harmonic analysis, Acta Math. 55 (1930), no. 1, 117–258. MR 1555316, DOI 10.1007/BF02546511
Norbert Wiener, The Homogeneous Chaos, Amer. J. Math. 60 (1938), no. 4, 897–936. MR 1507356, DOI 10.2307/2371268

Additional Information

Journal: Bull. Amer. Math. Soc. 69 (1963), 597-627
DOI: https://doi.org/10.1090/S0002-9904-1963-10964-7
MathSciNet review: 0162905