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Journal Article
Direct Inversion Formulas for the Natural SFT
2018
Overview
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;
T
∈
,
φ
f
(
t
,
w
)
:
=
∑
n
∈
n
f
n
(
ω
)
φ
n
(
t
)
Where
{
∈
n
}
is an
l
2
-sequence such that
∈
n
≠
0
,
∀
n
and
f
^
n
is the SFC (short for “stochastic Fourier coefficient”) defined by
f
^
n
(
ω
)
=
∫
0
1
f
(
t
,
ω
)
φ
n
(
t
)
¯
d
W
t
, a stochastic 0 integral with respect to Brownian motion Wt. 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)).
Publisher
Springer Science + Business Media,Springer India
