Abstract
We are concerned with the question whether and how a random function f(t,ω) can be uniquely determined by its image of SFT (i.e. stochastic Fourier transformation). The question was first posed by the author in the study of various problems of stochastic analysis (Ogawa 1979, 1985) and has been studied again recently in the papers Ogawa (1986), Ogawa and Uemura (2013, 2014) where some affirmative answers to the question as well as the schemes for inversion of SFT are given. In these papers the problem has been studied in the framework of Homogeneous Chaos which we feel clumsy partly because there all statements and schemes for the inversion are expressed in terms of the infinite sequence of representing kernels of the given Wiener functional and partly because for the execution of the inversion scheme developed there we need complete data of the underlying Brownian motion. The aim of the present note is to show an elementary approach to the problem, that does not rely on such heavy frameworks like Homogeneous Chaos, and give a direct formula for the inversion of the SFT.
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Ogawa, S. A Direct Inversion Formula for SFT. Sankhya A 77, 30–45 (2015). https://doi.org/10.1007/s13171-014-0056-1
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DOI: https://doi.org/10.1007/s13171-014-0056-1