Sankhya A

pp 1–13 | Cite as

Direct Inversion Formulas for the Natural SFT



The stochastic Fourier transform, or SFT for short, is an application that transforms a square integrable random function f(t, ω) to a random function defined by the following series; \({\mathcal T}_{\epsilon , \varphi }f(t,\o ):= {\sum }_{n} \epsilon _{n} \hat {f}_{n}(\o )\varphi _{n}(t)\) where {𝜖 n } is an 2-sequence such that 𝜖 n ≠  0, ∀n and \(\hat {f}_{n}\) is the SFC (short for “stochastic Fourier coefficient”) defined by \(\hat {f}_{n}(\o )={{\int }_{0}^{1}} f(t,\o )\overline {\varphi _{n}(t)}dW_{t}\), a stochastic integral with respect to Brownian motion W t . We have been concerned with the question of invertibility of the SFT and shown affirmative answers with concrete schemes for the inversion. In the present note we aim to study the case of a special SFT called “natural SFT” and show some of its basic properties. This is a follow-up of the preceding article (Ogawa,S.,“A direct inversion formula for SFT”, Sankhya-A 77-1 (2015)).

Keywords and phrases

Brownian motion Stochastic integrals Fourier series 

AMS (2000) subject classification.

Primary: 60H05, 60H07 Secondary: 42A61 


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Copyright information

© Indian Statistical Institute 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesRitsumeikan UniversityShigaJapan

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