Some fundamental formulas for an integration on infinite dimensional Hilbert spaces. (English) Zbl 07639059
Summary: In this paper, we define an integral on (infinite-dimensional) Hilbert spaces via the ideas and the concepts introduced by Hayek (Integ Trans Spec Funct 13:373-378, 2002; Appl Math Comput 218:773-776, 2011; Integ Trans Spec Funct 24:1-8, 2013), Negrin (J Funct Anal 141:37-44, 1996), Segal (Tran Amer Math Soc 81:106-134, 1956; Ann Math 63:160-175, 1956), Yeh (J Math 15:37-46, 1971; Stochastic Processes and the Wiener Integral. Marcel Dekker Inc, New York, 1973). We then obtain the existence of the integral for polynomial functions. Finally we establish the composition formula, the change of scale formula and the translation theorem. The bulk of the present paper is concerned with the integration of functionals on Hilbert spaces. The peculiarity of this paper is that all relationships and formulas can be obtained by use of the concept of the eigenvalues of operators on Hilbert spaces.
MSC:
| 47-XX | Operator theory |
References:
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