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Abstract
Given a random function and an orthonormal basis
in the
, we are concerned with the basic properties of its stochastic Fourier coefficient (SFC) which is defined by the stochastic integral with respect to the Brownian motion,
. More precisely we are concerned in this note with the problem of reconstructing the function
from its SFCs. We will show as our main result the affirmative answer when the random function belongs to a certain restricted but wide enough subclass, which is the class of causal Wiener functionals.
Acknowledgements
The author wish to express his cordial gratitude to Professors Gerard Kerkyacharian and Dominique Picard with whom he could have many valuable discussions while he was invited to LPMA of University Paris-6 and 7 in 2011. The author would also like to thank Prof. Hideaki Uemura of Aichi University of Education who kindly read the article and gave the author useful criticisms.