IJMMS 29:10 (2002) 591–608
PII. S0161171202006361
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
RELATIONSHIPS OF CONVOLUTION PRODUCTS, GENERALIZED
TRANSFORMS, AND THE FIRST VARIATION
ON FUNCTION SPACES
SEUNG JUN CHANG and JAE GIL CHOI
Received 9 December 2000
We use a generalized Brownian motion process to define the generalized Fourier-Feynman
transform, the convolution product, and the first variation. We then examine the various
relationships that exist among the first variation, the generalized Fourier-Feynman transform, and the convolution product for functionals on function space that belong to a
Banach algebra S(Lab [0, T ]). These results subsume similar known results obtained by
Park, Skoug, and Storvick (1998) for the standard Wiener process.
2000 Mathematics Subject Classification: 28C20.
1. Introduction. The concept of L1 analytic Fourier-Feynman transform (FFT) was
introduced by Brue in [1]. In [3], Cameron and Storvick introduced an L2 analytic FFT.
In [13], Johnson and Skoug developed an Lp analytic FFT for 1 ≤ p ≤ 2, which extended
the results in [1, 3] and gave various relationships between the L1 and L2 theories. In
[10], Huffman et al. defined a convolution product for functionals on Wiener space and
obtained, in [11, 12], various results involving the FFT and the convolution product.
Both the FFT and the convolution product are defined in terms of a Feynman integral. In this paper, we extend the ideas of [10, 11, 12] from the Wiener process to
more general stochastic process. We note that the Wiener process is free of drift and is
stationary in time. However, the stochastic process considered in this paper is subject
to drift and nonstationary in time.
In Section 2, we consider the function space induced by a generalized Brownian
motion process and define several concepts. In Section 3, we examine all relationships involving exactly two of three concepts of transform, convolution product and
first variation of functionals in S(Lab [0, T ]). In Section 4, we examine all relationships
involving all three of theses concepts where each concept is used exactly once.
2. Definitions and preliminaries. Let D = [0, T ] and let (Ω, Ꮾ, P ) be a probability
measure space. A real-valued stochastic process Y on (Ω, Ꮾ, P ) and D is called a generalized Brownian motion process if Y (0, ω) = 0 a.e.; and for 0 ≤ t0 < t1 < · · · < tn ≤ T ,
the n-dimensional random vector (Y (t1 , ω), . . . , Y (tn , ω)) is normally distributed with
the density function
−1/2
n
= (2π )n
b tj − b tj−1
K t, η
j=1
n
1
· exp −
2 j=1
2 ηj − a tj − ηj−1 − a tj−1
,
b tj − b tj−1
(2.1)
592
S. J. CHANG AND J. G. CHOI
= (η1 , . . . , ηn ), η0 = 0, t = (t1 , . . . , tn ), t0 = 0, and a(t) is a continuous realwhere η
valued function of bounded variation with a(0) = 0, and b(t) is a strictly increasing,
continuous real-valued function with b(0) = 0.
As explained in [19, pages 18–20], Y induces a probability measure µ on the measurable space (RD , ᏮD ) where RD is the space of all real-valued functions x(t), t ∈ D, and
ᏮD is the smallest σ -algebra of subsets of RD with respect to which all the coordinate
evaluation maps, et (x) = x(t), defined on RD are measurable. The triple (RD , ᏮD , µ)
is a probability measure space.
We note that, the generalized Brownian motion process Y determined by a(·) and
b(·) is a Gaussian process with mean function a(t) and covariance function r (s, t) =
min{b(s), b(t)}. By [19, Theorem 14.2, page 187], the probability measure µ induced
by Y , taking a separable version, is supported by Cab [0, T ] (which is equivalent to
the Banach space of continuous functions x on [0, T ] with x(0) = 0 under the sup
norm). Hence (Cab [0, T ], Ꮾ(Cab [0, T ]), µ) is the function space induced by Y where
Ꮾ(Cab [0, T ]) is the Borel σ -algebra of Cab [0, T ]. Note that we can also express x in
the form
x(t) = w b(t) + a(t),
0 ≤ t ≤ T,
(2.2)
where w(·) is the standard Brownian motion process [6, 7].
Let Lab [0, T ] be the Hilbert space of functions on [0, T ] which are Lebesgue measurable and square integrable with respect to the Lebesgue Stieltjes measures on [0, T ]
induced by a(·) and b(·), that is,
T
Lab [0, T ] = v :
0
T
v 2 (s)db(s) < ∞ and
0
v 2 (s)d|a|(s) < ∞ ,
(2.3)
where |a| denotes the total variation [6, 7].
A subset B of Cab [0, T ] is said to be scale-invariant measurable [8, 14] provided that
ρB is Ꮾ(Cab [0, T ])-measurable for all ρ > 0, and a scale-invariant measurable set N
is said to be scale-invariant null set provided that µ(ρN) = 0 for all ρ > 0. A property
that holds except on a scale-invariant null set, is said to hold on scale-invariant almost
everywhere (s-a.e.). If two functionals F and G defined on Cab [0, T ] are equal s-a.e.,
we write F ≈ G.
We denote the function space integral of a Ꮾ(Cab [0, T ])-measurable functional F by
E[F ] =
F (x)dµ(x)
(2.4)
Cab [0,T ]
whenever the integral exists.
We are now ready to state the definitions of the generalized analytic Feynman integral.
Definition 2.1. Let C denote the complex numbers. Let C+ = {λ ∈ C : Re λ > 0}
and C̃+ = {λ ∈ C : λ ≠ 0 and Re λ ≥ 0}. Let F : Cab [0, T ] → C be such that, for each
λ > 0, the function space integral
F λ−1/2 x dµ(x)
(2.5)
J(λ) =
Cab [0,T ]
593
RELATIONSHIPS OF CONVOLUTION PRODUCTS . . .
exists for all λ > 0. If there exists a function J ∗ (λ) analytic in C+ such that J ∗ (λ) = J(λ)
for all λ > 0, then J ∗ (λ) is defined to be the analytic function space integral of F over
Cab [0, T ] with parameter λ, and for λ ∈ C+ we write
E anλ [F ] = J ∗ (λ).
(2.6)
Let q ≠ 0 be a real number and let F be a functional such that E anλ [F ] exists for
all λ ∈ C+ . If the following limit exists, we call it the generalized analytic Feynman
integral of F with parameter q and we write
E anf q [F ] = lim E anλ [F ],
(2.7)
λ→−iq
where λ approaches −iq through C+ .
Next we state the definition of the generalized FFT (GFFT).
Definition 2.2. For λ > 0 and y ∈ Cab [0, T ], let
anλ F (y + x) .
Tλ (F )(y) = Ex
(2.8)
In the standard Fourier theory, the integrals involved are often interpreted in the mean;
a similar concept is useful in the FFT theory [13, page 104]. Let p ∈ (1, 2] and let p
and p be related by 1/p + 1/p = 1. Let {Hn } and H be scale-invariant measurable
functionals such that for each ρ > 0,
p = 0.
lim E Hn (ρy) − H(ρy)
(2.9)
n→∞
Then we write
H ≈ l. i. m. Hn
(2.10)
n→∞
and we call H the scale-invariant limit in the mean of order p . A similar definition
is understood when n is replaced by the continuously varying parameter λ. Let real
(p)
q ≠ 0 be given. For 1 < p ≤ 2, we define the Lp analytic GFFT, Tq (F ) of F , by the
formula (λ ∈ C+ )
(p)
Tq (F )(y) = l. i. m. Tλ (F )(y)
λ→−iq
(2.11)
(1)
if it exists. We define the L1 analytic GFFT, Tq (F ) of F , by the formula (λ ∈ C+ )
Tq(1) (F )(y) = lim Tλ (F )(y)
λ→−iq
(2.12)
if it exists.
(p)
(p)
We note that for 1 ≤ p ≤ 2, Tq (F ) is defined only s-a.e. We also note that if Tq (F )
exists and if F ≈ G, then
(p)
Tq (G)
exists and
(p)
Tq (G)
≈
(p)
Tq (F ).
594
S. J. CHANG AND J. G. CHOI
Definition 2.3. Let F and G be functionals on Cab [0, T ]. For λ ∈ C̃+ we define the
convolution product (if it exists) by

y −x
y +x
anλ


√
√
G
dµ(x),
F

 Cab [0,T ]
2
2
(F ∗ G)λ (y) =

anfq
y −x
y +x


 C [0,T
√
G √
dµ(x),
]F
ab
2
2
λ ∈ C+
(2.13)
λ = −iq.
Remark 2.4. When λ = −iq, we denote (F ∗ G)λ by (F ∗ G)q .
We finish this section by giving the definition of the first variation δF of the functional F [2, 5].
Definition 2.5. Let F be a Ꮾ(Cab [0, T ])-measurable functional on Cab [0, T ] and
let w ∈ Cab [0, T ]. Then
∂
δF (x | w) =
(2.14)
F (x + hw)
∂h
h=0
(if it exists) is called the first variation of F .
The following analytic Feynman integration formula is used throughout:
1/2
i
i 2 E anf q exp i
v, x = exp −
(v, a)
v ,b +i
2q
q
(2.15)
for all real q = 0, where v ∈ Lab [0, T ], v, x denotes the Paley-Wiener-Zygmund
T
stochastic integral 0 v(s)dx(s), and (v 2 , b) denotes the Lebesgue Stieltjes integral
T 2
0 v (s)db(s).
3. Relationships involving exactly two of three concepts of transform, convolution, and first variation. Let M(Lab [0, T ]) be the space of C-valued, countably additive finite Borel measures on Lab [0, T ]. The Banach algebra S(Lab [0, T ]) consists of
functionals F on Cab [0, T ] expressible in the form
exp i
u, x df (u),
(3.1)
F (x) =
Lab [0,T ]
for s-a.e. x ∈ Cab [0, T ], where f is an element of M(Lab [0, T ]). Further works on
S(Lab [0, T ]) show that it contains many functionals of interest in Feynman integration
theory [3, 4, 6, 7, 13, 14, 15, 16, 17].
Also, let
A ≡ y ∈ Cab [0, T ] : y is absolutely continuous on [0, T ] with y ∈ Lab [0, T ] . (3.2)
Remark 3.1. Throughout, we choose the variance function b(·) which is strictly
increasing such that the function p defined by p(t) = 1/b (t), p(0) = p(T ) = 0 is of
bounded variation on [0, T ]. For any u ∈ Lab [0, T ], let
ub =
1/2
T
2
u (s)db(s)
0
=
u2 , b ,
(3.3)
595
RELATIONSHIPS OF CONVOLUTION PRODUCTS . . .
T
then · b is a norm on Lab [0, T ] since
u, v ∈ Lab [0, T ],
0
u2 (s)db(s) < ∞. Hence we have for any
(uv, b) ≤ ub vb .
(3.4)
This will insure that the first variation, δF (· | w) of F in S(Lab [0, T ]), that arises will
exist for all w ∈ A (see [9]).
Let G in S(Lab [0, T ]) be given by
G(x) =
exp i
v, x dg(v)
(3.5)
Lab [0,T ]
for s-a.e. x ∈ Cab [0, T ] where g ∈ M(Lab [0, T ]).
In our first lemma, we obtain a formula for the first variation of functionals in
S(Lab [0, T ]).
Lemma 3.2. Let F ∈ S(Lab [0, T ]) be given by (3.1) with Lab [0,T ] ub |df (u)| < ∞.
Then for each w ∈ A and s-a.e. y ∈ Cab [0, T ],
δF (y | w) =
i
u, w exp i
u, y df (u).
(3.6)
Lab [0,T ]
Furthermore, as a function of y, δF (y | w) is an element of S(Lab [0, T ]).
Proof. By using the definition of the first variation, we see that
δF (y | w) =
=
∂
∂h
exp i
u, y + ih
u, w df (u) Lab [0,T ]
Lab [0,T ]
h=0
(3.7)
i
u, w exp u, y df (u)
for s-a.e. y ∈ Cab [0, T ]. Next, let φw (E) = E i
u, wdf (u) for each set E ∈ Ꮾ(Lab [0, T ]).
But
i
u, wdf (u) ≤ M w φ w ≤
ub df (u) < ∞,
(3.8)
b
Lab [0,T ]
Lab [0,T ]
where M = supt∈[0,T ] p(t). Hence δF (y | w) =
ment of S(Lab [0, T ]).
Lab [0,T ] exp{i
u, y}dφw (u)
is an ele-
In our next theorem, we obtain the transform of functional in S(Lab [0, T ]).
Theorem 3.3. Let F ∈ S(Lab [0, T ]) be given by (3.1) and let p ∈ [1, 2] be given.
(p)
Then the analytic generalized Fourier-Feynman transform Tq (F ) exists for all real
q ≠ 0 and is given by the formula
(p)
Tq (F )(y) =
Lab [0,T ]
for s-a.e. y ∈ Cab [0, T ].
exp i
u, y −
1/2
i 2 i
(u, a) df (u)
u ,b +i
2q
q
(3.9)
596
S. J. CHANG AND J. G. CHOI
Proof. By (2.8), the Fubini theorem, and (2.15), we have, for all λ > 0,
Tλ (F )(y) =
anλ
=
=
Cab [0,T ]
Lab [0,T ]
F (y + x)dµ(x)
anλ
Cab [0,T ]
exp i
u, y + i
u, x dµ(x)df (u)
(3.10)
1 2 i
u , b + √ (u, a) df (u)
exp i
u, y −
2λ
λ
Lab [0,T ]
for s-a.e. y ∈ Cab [0, T ]. But the last equation above is analytic throughout C+ and is
continuous on C̃+ , since f is a finite Borel measure. Thus (3.9) is established.
In the following theorem, we obtain the convolution product of functionals in
S(Lab [0, T ]).
Theorem 3.4. Let F ∈ S(Lab [0, T ]) be given by (3.1), and let G ∈ S(Lab [0, T ]) be
given by (3.5). Then their convolution product (F ∗ G)q exists for all real q ≠ 0 and is
given by the formula
i i
exp √ u + v, y −
(u − v)2 , b
2
4q
2
Lab [0,T ]
1/2
i
+i
(u − v, a) df (u)dg(v)
2q
(F ∗ G)q (y) =
(3.11)
for s-a.e. y ∈ Cab [0, T ].
Proof. By using (2.13), the Fubini theorem, and (2.15), we have that for all λ > 0,
(F ∗ G)λ (y)
anλ
=
y −x
y +x
√
G √
dµ(x)
2
2
Cab [0,T ]
anλ
i
i
=
exp √ u − v, x + √ u + v, y dµ(x)df (u)dg(v)
2
2
L2
ab [0,T ] Cab [0,T ]
i
i
1 (u − v)2 , b + √ (u − v, a) df (u)dg(v)
exp √ u + v, y −
=
4λ
2
2λ
L2
ab [0,T ]
(3.12)
F
for s-a.e. y ∈ Cab [0, T ]. But the last equation above is analytic throughout C+ , and is
continuous on C̃+ . Thus we have the desired result.
Next, we obtain the transform of the convolution product.
Theorem 3.5. Let F and G be given as in Theorem 3.4 and p ∈ [1, 2]. Then for all
(p)
real q ≠ 0, Tq ((F ∗ G)q ) exists and is given by the formula
y
y
(p) (p) (F ∗ G)q (y) = Tq F1 √ Tq G1 √
2
2
(p) Tq
for s-a.e. y ∈ Cab [0, T ], where F1 and G1 are given by (3.15) below.
(3.13)
RELATIONSHIPS OF CONVOLUTION PRODUCTS . . .
597
Proof. Let p ∈ [1, 2] and q ∈ R − {0}. Using (2.15), (3.9), and (3.11), we see that
(p) Tq
(F ∗ G)q (y)
anf q
=
(F ∗ G)q (y + z)dµ(z)
Cab [0,T ]
anf q
(y + z) − x
(y + z) + x
√
√
G
dµ(x)dµ(z)
2 [0,T ]
2
2
Cab
i i 1/2
=
exp −
(u − v, a)
(u − v)2 , b + i
4q
2q
L2
ab [0,T ]
anf q
i
i
·
exp √ u + v, z + √ u + v, y dµ(z)df (u)dg(v)
2
2
Cab [0,T ]
(3.14)
i i 1/2
i 2
2
=
exp −
(u − v, a) −
(u − v) , b + i
(u + v) , b
4q
2q
4q
L2
ab [0,T ]
i 1/2
i
+i
(u + v, a) + √ u + v, y df (u)dg(v)
2q
2
1/2
i
i
i 2 u ,b +i
exp √ u, y −
(u, a) df1 (u)
=
2q
q
2
Lab [0,T ]
1/2
i 2 i
i
·
exp √ v, y −
(v, a) dg1 (v)
v ,b +i
2q
q
2
Lab [0,T ]
y
y
(p) (p) = T q F1 √ T q G 1 √
2
2
=
F
for s-a.e. y ∈ Cab [0, T ], where
F1 (y) =
exp i
u, y df1 (u),
G1 (y) =
Lab [0,T ]
Lab [0,T ]
exp i
v, y dg1 (v),
(3.15)
1/2
i
i 1/2
exp − i
(u, a) + 2i
(u, a) df (u),
q
2q
E
1/2
i
(v, a) dg(v),
g1 (E) = exp − i
q
E
f1 (E) =
(3.16)
for every E ∈ Ꮾ(Lab [0, T ]), and so f1 ≤ f and g1 ≤ g.
In our next theorem, we have the convolution product of transforms of F and G in
S(Lab [0, T ]).
Theorem 3.6. Let F , G, and p be given as in Theorem 3.5. Then for all real q ≠ 0,
(p)
(p)
(Tq (F ) ∗ Tq (G))−q (y) exists and is given by the formula
(p)
(p)
Tq (F ) ∗ Tq (G)
(p)
−q
(y) = Tq
·
·
F2 √ G 2 √
(y)
2
2
for s-a.e. y ∈ Cab [0, T ] where F2 and G2 are given by (3.19) below.
(3.17)
598
S. J. CHANG AND J. G. CHOI
Proof. By using (3.9), (3.11), and the Fubini theorem, we have, for s-a.e. y ∈
Cab [0, T ],
(p)
(p)
Tq (F ) ∗ Tq (G)
anf−q
−q
(y)
y +x
y −x
(p)
(p)
Tq (F ) √
Tq (G) √
dµ(x)
2
2
Cab [0,T ]
1/2
i 2
i
i
u + v 2, b + i
exp √ u + v, y −
(u + v, a)
=
2q
q
2
L2
ab [0,T ]
anf−q
i
exp √ u − v, x dµ(x) df (u)dg(v)
·
2
Cab [0,T ]
1/2
i
i
i 2
=
u + v 2, b + i
exp √ u + v, y −
(u + v, a)
2q
q
2
L2
[0,T
]
ab
i i
(u − v, a) df (u)dg(v)
+
(u − v)2 , b + −
4q
2q
i
i 1/2
i
=
exp √ u + v, y −
(u + v, a)
(u + v)2 , b + i
4q
2q
2
L2
ab [0,T ]
i 1/2
i 1/2
(u + v, a) + i −
(u − v, a)
−i
2q
2q
1/2
i
+i
(u + v, a) df (u)dg(v)
q
i i
exp √ u + v, y −
=
(u + v)2 , b
2
4q
2
Lab [0,T ]
1/2
i
+i
(u + v, a) df2 (u)dg2 (v)
2q
·
·
(p)
= Tq
F2 √ G 2 √
(y),
2
2
=
(3.18)
where
F2 (y) =
G2 (y) =
Lab [0,T ]
Lab [0,T ]
exp i
u, y df2 (u),
exp i
v, y dg2 (v),
1/2 i 1/2
i 1/2
i
(u, a) df (u),
+ −
+
f2 (E) = exp i −
2q
2q
q
E
1/2 i 1/2
i 1/2
i
g2 (E) = exp i −
(v, a) dg(v),
− −
+
2q
2q
q
E
(3.19)
(3.20)
for every E ∈ Ꮾ(Lab [0, T ]), and so f2 ≤ f and g2 ≤ g.
In the next theorem, we obtain that the transform with respect to the first argument
of the variation equals the variation of the transform.
RELATIONSHIPS OF CONVOLUTION PRODUCTS . . .
599
Theorem 3.7. Let F be given as in Lemma 3.2, p ∈ [1, 2], q ∈ R − {0}, and w ∈ A
be given. Then
(p) Tq
(p)
δF (· | w) (y) = δTq (F )(y | w)
(3.21)
for s-a.e. y ∈ Cab [0, T ].
Also, both expressions in (3.21) are given by the expression
1/2
i 2 i
i
u, w exp i
u, y −
(u, a) df (u).
u ,b +i
2q
q
Lab [0,T ]
(3.22)
Proof. By using (3.6), the Fubini theorem, (2.15), and (3.9), we have that
(p) δF (· | w) (y)
anf q
δF (y + x | w)dµ(x)
=
Tq
Cab [0,T ]
=
anf q
Cab [0,T ]
=
Lab [0,T ]
i
u, w exp i
u, y + x df (u)dµ(x)
anf q
i
u, w exp i
u, y
Lab [0,T ]
exp i
u, x dµ(x) df (u)
Cab [0,T ]
(3.23)
1/2
i 2 i
=
i
u, w exp i
u, y −
(u, a) df (u)
u ,b +i
2q
q
Lab [0,T ]
1/2
i
i 2 ∂
(u, a) df (u)
u , b +i
exp i
u, y + hw−
=
2q
q
Lab [0,T ] ∂h
h=0
∂ (p)
Tq (F )(y + hw) =
∂h
h=0
(p)
= δTq (F )(y | w)
for s-a.e. y ∈ Cab [0, T ] as desired.
In the next theorem, we obtain the transform with respect to the second argument
of the variation.
Theorem 3.8. Let F , p, q, and w be given as in Theorem 3.7. Then, for s-a.e. y ∈
Cab [0, T ],
1/2 i
δF (y | ·) (w) = δF (y | w) + i
(u, a) exp i
u, y df (u). (3.24)
q
Lab [0,T ]
(p) Tq
Proof. Using (2.11) and (3.6), we obtain
(p) Tq
δF (y | ·) (w) =
=
anf q
Cab [0,T ]
anf q
Cab [0,T ]
δF (y | w + x)dµ(x)
Lab [0,T ]
i
u, w + x exp i
u, y df (u)dµ(x)
600
S. J. CHANG AND J. G. CHOI
anf q
exp i
u, y
=i
Lab [0,T ]
Cab [0,T ]
u, w + xdµ(x) df (u)
1/2
i
(u, a) exp i
u, y df (u)
q
Lab [0,T ]
1/2 i
= δF (y | w) + i
(u, a) exp i
u, y df (u)
q
Lab [0,T ]
=i
u, w +
(3.25)
for s-a.e. y ∈ Cab [0, T ]. In particular, if a ∈ A then (u, a) = u, a and so
(p) Tq
δF (y | ·) (w) = δF (y | w) +
1/2
i
δF (y | a).
q
(3.26)
In our next theorem, we obtain the first variation of convolution product of functionals F and G in S(Lab [0, T ]).
Theorem 3.9. Let F , p, q, and w be given as in Theorem 3.7 and let G be given
by (3.5) with Lab [0,T ] vb |dg(v)| < ∞. Then for s-a.e. y ∈ Cab [0, T ], we obtain the
formula
δ(F ∗ G)q (y | w)
i
i
i √ u + v, w exp √ u + v, y −
=
(u − v)2 , b
2
4q
2
2
Lab [0,T ]
1/2
i
+i
(u − v, a) df (u)dg(v)
2q
(3.27)
for s-a.e. y ∈ Cab [0, T ].
Proof. The proof of (3.27) can be obtained by using (3.6) and (3.11).
In our next theorem, we obtain the convolution product of the first variation with
respect to the first argument.
Theorem 3.10. Let F , G, p, q, and w be given as in Theorem 3.9. Then for s-a.e.
y ∈ Cab [0, T ], (δF (· | w) ∗ δG(· | w))q (y) exists and is given by the formula
δF (· | w) ∗ δG(· | w) q (y)
i
i =−
(u − v)2 , b
u, w
v, w exp √ u + v, y −
4q
2
L2
[0,T
]
ab
i 1/2
(u − v, a) df (u)dg(v).
+i
2q
Proof. By using (2.13), (3.6), the Fubini theorem, and (2.15), we have
δF (· | w) ∗ δG(· | w) q (y)
anf q
y +x y −x w dµ(x)
√ √
=
δF
w
δG
2
2
Cab [0,T ]
(3.28)
RELATIONSHIPS OF CONVOLUTION PRODUCTS . . .
=
anf q
Cab [0,T ]
=−
601
!
y +x
i
u, w exp i u, √
df (u)
2
Lab [0,T ]
!
y −x
i
v, w exp i v, √
dg(v)dµ(x)
·
2
Lab [0,T ]
anf q
u, w
v, w
L2
ab [0,T ]
Cab [0,T ]
i
i
· exp √ u − v, x + √ u + v, y dµ(x)df (u)dg(v)
2
2
i
i (u − v)2 , b
u, w
v, w exp √ u + v, y −
=−
2
4q
2
Lab [0,T ]
1/2
i
(u − v, a) df (u)dg(v)
+i
2q
(3.29)
for s-a.e. y ∈ Cab [0, T ].
In our next theorem, we obtain the convolution product of the first variation with
respect to the second argument.
Theorem 3.11. Let F , G, p, q, and w be given as in Theorem 3.9. Then
δF (y | ·) ∗ δG(y | ·) q (w)
i
(uv, b) exp i
u + v, y df (u)dg(v)
=
2q L2ab [0,T ]
1/2 w
i
a
+i
exp i
u, y df (u)
+ δF y u, √
√
q
2
2
Lab [0,T ]
1/2 w
a
i
√
√
· δG y v,
−
i
exp
i
v,
y
dg(v)
q
2
2
Lab [0,T ]
(3.30)
for s-a.e. y ∈ Cab [0, T ].
Proof. For each u, v ∈ Lab [0, T ], we have
anf q
Cab [0,T ]
u, x
v, xdµ(x) =
i
i
(u, a)(v, a) + (uv, b).
q
q
(3.31)
But, by using (2.13), (3.6), the Fubini theorem, and (3.31), we have
δF (y | ·) ∗ δG(y | ·) q (w)
w −x
w +x
√
√
δG
y
dµ(x)
δF y 2
2
Cab [0,T ]
anf q
1
u, w + x
v, w − x exp i
u + v, y dµ(x)df (u)dg(v)
=−
2 L2ab [0,T ] Cab [0,T ]
=
anf q
602
S. J. CHANG AND J. G. CHOI
=−
1
2
#
1/2
" 1/2
i
i
i
(u, a) + u, w v, w −
(v, a) − (uv, b)
q
q
q
L2
ab [0,T ]
· exp i
u + v, y df (u)dg(v)
1/2
i
i
√ u, w +
(u, a) exp i
u, y df (u)
q
2
Lab [0,T ]
1/2
i
i
√ v, w −
·
(v, a) exp i
v, y dg(v)
q
2
Lab [0,T ]
i
+
(uv, b) exp i
u + v, y df (u)dg(v)
2
2q Lab [0,T ]
1/2 w
a
i
= δF y √
u, √
+i
exp u, y df (u)
q
2
2
Lab [0,T ]
1/2 w
i
a
√
√
· δG y −i
exp v, y dg(v)
v,
q
2
2
Lab [0,T ]
i
+
(uv, b) exp i
u + v, y df (u)dg(v)
2
2q Lab [0,T ]
=
(3.32)
√
√
for s-a.e. y ∈ Cab [0, T ]. In particular, if a ∈ A then (u, a/ 2) = u, a/ 2 and (v,
√
√
a/ 2) = v, a/ 2 and so we have
δF (y | ·) ∗ δG(y | ·) q (w)
1/2
1/2 w
w
a
a
i
i
√
√
√
δF y,
δG y, √
= δF y δG y −i
+i
q
q
2
2
2
2
i
(uv, b) exp i
u + v, y df (u)dg(v).
+
2q L2ab [0,T ]
(3.33)
Also, by using (3.24), the alternative expression in (3.30) is given by
(p)
− Tq
·
·
(p)
δF y δG y (w)Tq
(−w)
√
√
2
2
i
(uv, b) exp i
u + v, y df (u)dg(v).
+
2
2q Lab [0,T ]
(3.34)
Thus we have the desired result.
4. Relationships involving three concepts. In this section, we look at all the relationships involving the transform, the convolution, and the first variation where each
operation is used exactly once.
In our next theorem, we obtain the formula for transform with respect to the first
argument of the variation of the convolution product which equals the variation of
the transform of the convolution product.
RELATIONSHIPS OF CONVOLUTION PRODUCTS . . .
603
Theorem 4.1. Let F , G, p, q, and w be given as in Theorem 3.9. Then for s-a.e.
(p)
y ∈ Cab [0, T ], Tq (δ(F ∗ G)q (· | w))(y) exists and is given by the formula
w
y
y
(p) (p)
√
δ(F ∗ G)q (· | w) (y) = Tq F1 √ Tq
δG1 · √
2
2
2
y
y
w
(p)
√ Tq(p) G1 √ ,
+ Tq
δF1 · √
2
2
2
(p) Tq
(4.1)
where F1 and G1 are given by (3.15).
Proof. By using (3.21) we have
(p) δ(F ∗ G)q (· | w) (y) = δTq (F ∗ G)q (y | w).
(p) Tq
(4.2)
Also, using (3.6) and (3.13), we obtain
(p) δTq
(F ∗ G)q (y | w)
y + hw
y + hw
∂
(p) (p) √
√
T q F1
Tq G1
=
∂h
2
2
h=0
1/2
i 2 ∂
i
i
=
exp √ u, y + hw −
(u, a) df1 (u)
u ,b +i
∂h
2q
q
2
Lab [0,T ]
1/2
i
i
i 2 ·
v ,b +i
exp √ v, y + hw −
(v, a) dg1 (v) 2q
q
2
Lab [0,T ]
h=0
1/2
i
i
i 2 u ,b +i
exp √ u, y −
(u, a) df1 (u)
2q
q
2
Lab [0,T ]
1/2
i
i
i 2 i
√ v, w exp √ v, y −
(v, a) dg1 (v)
·
v ,b +i
2q
q
2
2
Lab [0,T ]
1/2
i
i
i
i 2 √ u, w exp √ u, y −
+
(u, a) df1 (u)
u ,b +i
2q
q
2
2
Lab [0,T ]
1/2
i 2 i
i
exp √ v, y −
(v, a) dg1 (v)
·
v ,b +i
2q
q
2
Lab [0,T ]
y
y w
y w
y
(p) (p) (p) (p) + δTq F1 √ = Tq F1 √ δTq G1 √ √
√ Tq G1 √
2
2
2
2
2
2
y
w
y
(p)
(p)
√
δG1 · = T q F1 √ T q
√
2
2
2
w
y
y
(p)
√ Tq(p) G1 √
+ Tq
δF1 · √
2
2
2
(4.3)
=
for s-a.e. y ∈ Cab [0, T ].
In our next theorem, we obtain the transform with respect to the second argument
of the variation of the convolution product.
604
S. J. CHANG AND J. G. CHOI
Theorem 4.2. Let F , G, p, q, and w be given as in Theorem 4.1. Then for s-a.e.
y ∈ Cab [0, T ],
δ(F ∗ G)q (y | ·) (w)
#
"
1/2
i
i
(u + v, a)
u + v, w +
=√
q
2 L2ab [0,T ]
i
i i 1/2
· exp √ u + v, y −
(u − v)2 , b + i
(u − v, a) df (u)dg(v)
4q
2q
2
1/2
i
i
= δ(F ∗ G)q (y | w) + √
2 q
i i
·
(u + v, a) exp √ u + v, y −
(u − v)2 , b
2
4q
2
Lab [0,T ]
i 1/2
+i
(u − v, a) df (u)dg(v).
2q
(4.4)
(p) Tq
Proof. By using (3.11), (3.24), and (3.27), we obtain (4.4) above. In particular, if
a ∈ A, then (u + v, a) = u + v, a and hence we have
δ(F ∗ G)q (y | ·) (w) = δ(F ∗ G)q (y | w) +
(p) Tq
1/2
i
δ(F ∗ G)q (y | a).
q
(4.5)
Now we obtain formulas for the transforms of the convolution product with respect
to the first argument of the variations.
Theorem 4.3. Let F , G, p, q, and w be given as in Theorem 4.1. Then, for s-a.e.
y ∈ Cab [0, T ],
(p) Tq
δF (· | w) ∗ δG(· | w) q (y)
y
y
(p) (p) = Tq δF1 (· | w) √ Tq δG1 (· | w) √
2
2
y
y (p)
(p)
= δTq F1 √ w δTq G1 √ w ,
2
2
(4.6)
where F1 and G1 are given by (3.15) and
anf q
Cab [0,T ]
=
δF (· | w + x) ∗ δG(· | w + x) q (y)dµ(x)
y +x y −x (p)
· (w)dµ(x)
√ √
δG
·
(w)T
q
2
2
Cab [0,T ]
i
i i
(u − v)2 , b
(uv, b) exp √ u + v, y −
−
2
q Lab [0,T ]
4q
2
1/2
i
+i
(u − v, a) df (u)dg(v).
2q
anf q
(p)
Tq
δF
(4.7)
RELATIONSHIPS OF CONVOLUTION PRODUCTS . . .
605
Proof. By using (3.13) and (3.21), we obtain (4.6) above. To establish (4.7), we note
that, by the use of (2.15), (3.24), (3.28), and (3.31),
anf q
Cab [0,T ]
=−
δF (· | w + x) ∗ δG(· | w + x) q (y)dµ(x)
anf q
Cab [0,T ]
L2
ab [0,T ]
u, w + x
v, w + x
i
i (u − v)2 , b
· exp √ u + v, y −
4q
2
i 1/2
+i
(u − v, a) df (u)dg(v)dµ(x)
2q
#
1/2
"
1/2
i
i
i
(u, a) v, w +
(v, a) + (uv, b)
u, w +
=−
q
q
q
L2
ab [0,T ]
i i 1/2
i
(u − v, a) df (u)dg(v)
· exp √ u + v, y −
(u − v)2 , b + i
4q
2q
2
anf q !
1/2
i
y +x
df (u)
=
i u, w +
(u, a) exp i u, √
q
2
Cab [0,T ] Lab [0,T ]
!
1/2
i
y −x
dg(v)dµ(x)
·
i v, w +
(v, a) exp i v, √
q
2
Lab [0,T ]
i i
i
(u − v)2 , b
(uv, b) exp √ u + v, y −
−
q L2ab [0,T ]
4q
2
i 1/2
+i
(u − v, a) df (u)dg(v)
2q
anf q
y +x y −x (p)
(p)
· (w)dµ(x)
√ √
=
Tq
δF
δG
·
(w)T
q
2
2
Cab [0,T ]
i i
i
(uv, b) exp √ u + v, y −
(u − v)2 , b
−
q L2ab [0,T ]
4q
2
i 1/2
+i
(u − v, a) df (u)dg(v)
2q
(4.8)
for s-a.e. y ∈ Cab [0, T ].
In our next theorem, we obtain the transforms of the convolution product with
respect to the second argument of the variations.
Theorem 4.4. Let F , G, p, q, and w be given as in Theorem 4.1. Then for s-a.e.
y ∈ Cab [0, T ],
(p) Tq
δF (y | ·) ∗ δG(y | ·) q (w)
·
·
(p)
√
δG y (w)
= Tq
δF y √
2
2
606
S. J. CHANG AND J. G. CHOI
1/2
·
i
a
(p)
√
−i
Tq
δF y v, √
(w)
exp i
v, y dg(v)
q
2
2
Lab [0,T ]
1/2
a
i
·
(p)
√
√
(w)
exp i
u, y df (u)
Tq
δG y u,
+i
q
2
2
Lab [0,T ]
i
+
(u, a)(v, a) + (uv, b) exp i
u + v, y df (u)dg(v),
2q L2ab [0,T ]
anf q
Cab [0,T ]
δF (y + x | ·) ∗ δG(y + x | ·) q (w)dµ(x)
w
w
√
√
δF · δG
·
(y)
2
2
"
!
!
1/2 w
a
a
w
i
u, √
v, √
− u, √
v, √
+
q
2
2
2
2
L2
ab [0,T ]
#
1/2 a
i
a
1 i 1/2
(uv, b)
u, √
+
v, √
+
q
2 q
2
2
1/2
i i
· exp i
u + v, y −
(u + v, a) df (u)dg(v).
(u + v)2 , b + i
2q
q
(p)
= Tq
(4.9)
Proof. By using (2.11) and (3.30), we obtain (4.9).
Next, we obtain the variation of the convolution product of transforms.
Theorem 4.5. Let F , G, p, q, and w be given as in Theorem 4.1. Then
(p)
(p)
δ Tq (F ) ∗ Tq (G)
(p)
= δTq
=
−q
(y | w)
·
·
F2 √ G 2 √
(y)
2
2
i
√ u + v, w
2
i
i · exp √ u + v, y −
(u + v)2 , b
4q
2
1/2
i
−i 1/2
(u − v, a) + i
(u + v, a) df (u)dg(v)
+i
2q
q
L2
ab [0,T ]
(4.10)
for s-a.e. y ∈ Cab [0, T ] where F2 and G2 are given by (3.19).
Proof. By using (3.17) and the same calculation in the proof of Theorem 3.6, we
obtain (4.10).
Now, we obtain the formulas for convolution product of the variation of the transform. There are two cases; namely, we can take the convolution with respect to the
first argument or the second argument of the variation.
RELATIONSHIPS OF CONVOLUTION PRODUCTS . . .
607
Theorem 4.6. Let F , G, p, q, and w be given as in Theorem 4.1. Then, for s-a.e.
y ∈ Cab [0, T ],
· · w (y),
√
δF2 √ w
δG
2
−q
2
2
(4.11)
where F2 and G2 are given by (3.19); and if a ∈ A, then
(p)
(p)
δTq (F )(· | w) ∗ δTq (G)(· | w)
(p)
(y) = Tq
(p)
(p)
δTq (F )(y | ·) ∗ δTq (G)(y | ·)
−q
(w)
w
a
i 1/2 (p)
(p)
√
√
+
−
δT
(F
)
y
= δTq (F ) y q
q
2
2
1/2
w
a
i
(p)
(p)
√
δTq (G) y − −
· δTq (G) y √
q
2
2
i 2
i
(uv, b) exp i
u + v, y −
u + v 2, b
−
2q L2ab [0,T ]
2q
1/2
i
+i
(u + v, a) df (u)dg(v).
q
(4.12)
Proof. By using (3.22), (3.24), and the same calculation in the proof of Theorem
3.6, we obtain (4.11). Further, proceeding as in the proof of Theorem 3.8 and using
(3.33) and (3.22), we have (4.12).
Theorem 4.7. Let F , G, p, q, and w be given as in Theorem 4.1 and let a ∈ A. Then,
for s-a.e. y ∈ Cab [0, T ],
(p) (w)
δF (y | ·) ∗ Tq δG(y | ·)
(p) Tq
q
1/2
·
i
(p)
√
δF (y | a)Tq
δG y (−w)
q
2
1/2
·
i
i
(p)
√
δG(y | a)Tq
δF y (w) + δF (y | a)δG(y | a).
+
q
q
2
(4.13)
= δF (y | ·) ∗ δG(y | ·) q (w) −
Proof. By using (3.24) and a direct calculation, we obtain (4.13).
Acknowledgment. The first author was partially supported by Korea Science and
Engineering Foundation (KOSEF) in Korea in 2000.
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Seung Jun Chang: Department of Mathematics, Dankook University, Cheonan, 330714, Korea
E-mail address: sejchang@anseo.dankook.ac.kr
Jae Gil Choi: Department of Mathematics, Dankook University, Cheonan, 330-714,
Korea
E-mail address: jgchoi@anseo.dankook.ac.kr