Integral transforms, convolution products, and first variations

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PII. S0161171204305260 http://ijmms.hindawi.com © Hindawi Publishing Corp.

INTEGRAL TRANSFORMS, CONVOLUTION PRODUCTS,

AND FIRST VARIATIONS

BONG JIN KIM, BYOUNG SOO KIM, and DAVID SKOUG

Received 21 May 2003

We establish the various relationships that exist among the integral transformᏲα,βF, the convolution product(F∗G)α, and the ﬁrst variationδFfor a class of functionals deﬁned on K[0, T ], the space of complex-valued continuous functions on[0, T ]which vanish at zero.

2000 Mathematics Subject Classiﬁcation: 28C20.

1. Introduction and definitions. In a unifying paper [10], Lee deﬁned an integral transformᏲα,βof analytic functionals on an abstract Wiener space. For certain values of the parametersαandβand for certain classes of functionals, the Fourier-Wiener transform [2], the Fourier-Feynman transform [3], and the Gauss transform are special cases of his integral transformᏲα,β. In [5], Chang et al. established an interesting re-lationship between the integral transform and the convolution product for functionals on an abstract Wiener space. In this paper, we study the relationships that exist among the integral transform, the convolution product, and the ﬁrst variation [1,4,9,11].

LetC0[0, T ]denote one-parameter Wiener space, that is, the space of all real-valued

continuous functionsx(t)on[0, T ]withx(0)=0. Letᏹdenote the class of all Wiener measurable subsets ofC0[0, T ]and letmdenote Wiener measure. Then(C0[0, T ],ᏹ,

m)is a complete measure space and we denote the Wiener integral of a Wiener inte-grable functionalF by

C0[0,T ]

F (x)m(dx). (1.1)

Let K=K[0, T ] be the space of complex-valued continuous functions defined on [0, T ]which vanish att=0. Letαandβbe nonzero complex numbers. Next we state the definitions of the integral transformᏲα,βF, the convolution product(F∗G)α, and the first variationδF for functionals defined onK.

Definition1.1. LetF be a functional deﬁned onK. Then the integral transform Ᏺα,βF ofF is deﬁned by

Ᏺα,β(F )(y)≡Ᏺα,βF (y)≡

C0[0,T ]

F (αx+βy)m(dx), y∈K, (1.2)

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Definition 1.2. Let F and G be functionals deﬁned onK. Then the convolution

product(F∗G)αofF andGis deﬁned by

(F∗G)α(y)≡

C0[0,T ]

F

_y₊_αx

√ 2

G

_y₋_αx

√ 2

m(dx), y∈K, (1.3)

if it exists [5,7,13,14].

Definition_1.3. _Let_F _{be a functional deﬁned on}_K_{and let}_w∈_{K. Then the ﬁrst}

variationδFofF is deﬁned by

δF (y|w)≡ ∂

∂tF (y+tw)|t=0, y∈K, (1.4)

if it exists [1,4,11].

Let{θ1, θ2, . . .}be a complete orthonormal set of real-valued functions inL2[0, T ].

Furthermore, assume that eachθjis of bounded variation on[0, T ]. Also let Var(θj, [0, T ])denote the total variation ofθj on[0, T ]. Then for eachy∈Kandj∈ {1,2, . . .}, the Riemann-Stieltjes integralθj, y ≡

T

0 θj(t)dy(t)exists. Furthermore,

θj, y=

θj(T )y(T )−

T

0

y(t)dθj(t)

≤Cjy∞ (1.5)

with

Cj=θj(T )+Var

θj, [0, T ] . (1.6)

Next we describe the class of functionals that we work with in this paper. LetE0be

the space of all functionalsF:K→Cof the form

F (y)=fθ1, y

, . . . ,θn, y (1.7)

for some positive integern, wheref (λ1, . . . , λn)is an entire function of thencomplex variablesλ1, . . . , λnof exponential type; that is to say,

_f

λ1, . . . , λn ≤AFexp   BF

n

j=1 _λ_j

 (1.8)

for some positive constantsAF andBF.

To simplify the expressions, we use the following notations. Foru=(u1, . . . , un)∈Rn andλ=(λ1, . . . , λn)∈Cn, we write

u2₌ n

j=1

u2j, |u| = n

j=1

_u_j_, _|_λ_{| =}

n

j=1

_λ_j_, _d_u₌_du₁_···_du_n_,

fα u+βλ =fαu1+βλ1, . . . , αun+βλn ,

fθ, y =fθ1, y

, . . . ,θn, y .

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Hence (1.7) and (1.8) can be expressed alternatively as

F (y)=f_{θ, y} _, _{f (}_λ)_≤_A_F_exp_B_F_|_λ_|_, _(1.10)

respectively. In addition, we use the notation

Fj(y)=fj

_{θ, y} _, _(1.11)

wherefj(λ)=(∂/∂λj)f (λ1, . . . , λn)forj=1, . . . , n.

InSection 2, we show that ifF andGare elements ofE0, thenᏲα,βF (·),(F∗G)α(·), δF (·|w), andδF (y|·)are also elements ofE0. InSection 3, we examine all relationships

involving exactly two of the three concepts of “integral transform,” “convolution prod-uct,” and “ﬁrst variation,” while inSection 4, we examine all relationships involving all three of these concepts where each concept is used exactly once. For related work, see [2,5,7,9,10,11,13,14] and for a detailed survey of previous work, see [12].

Remark1.4. For anyF∈E₀and anyG∈E₀, we can always expressF by (1.7) and Gby

G(x)=gθ1, x, . . . ,θn, x (1.12)

using the same positive integern, wheref andgare entire functions of exponential type. For example, ifF∈E0is of the form

F (x)=rθ1, x

,θ2, x , (1.13)

andG∈E0is of the form

G(x)=sθ1, x

,θ3, x

,θ4, x , (1.14)

wherer (λ1, λ2)ands(λ1, λ3, λ4)are entire functions of exponential type, then we can

expressF andGby (1.7) and (1.12) withn=4 by choosingf (λ1, λ2, λ3, λ4)≡r (λ1, λ2)

andg(λ1, λ2, λ3, λ4)≡s(λ1, λ3, λ4). In addition, the positive constantsAF,BF,AG, and BGremain ﬁxed. Thus throughout this paper, we will always assume thatFandGbelong toE0and are given by (1.7) and (1.12), respectively.

Remark_1.5. _{We considered various other classes of functionals before deciding to}

work exclusively with the classE0 throughout this paper. One very natural class we

considered was L2(C)≡L2(C0[0, T ]), the space of all complex-valued functionals F

satisfying

C0[0,T ]

F (x)2m(dx) <∞. (1.15)

However in [8], Kim and Skoug showed thatL2(C)is not invariant under the action of

the integral transform operator. In fact, they showed that for everyβ∈Cwith|β|>1, there exists a functionalF∈L2(C)(the functionalFdepends onβ) withᏲα,β(F )∉L2(C)

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Another class of functionals we considered was

A=F∈L2(C):Ᏺα,β(F )∈L2(C)∀nonzeroα, β∈C

. (1.16)

But forF∈A, the ﬁrst variationδFofF may not exist; in fact, one needs some kind of a smoothness condition onF to even deﬁneδF.

As we will see inSection 2,E0is a very natural class of functionals in which to study

the relationships that exist among the integral transform, the convolution product, and the ﬁrst variation because forFandGinE0,Ᏺα,β(F )and(F∗G)αexist and belong to E0for all nonzero complex numbersαandβ, whileδF (y|w)exists and belongs toE0

for ally andw in K. In addition, E0is a very rich class of functionals. Note that if

E0is given by (1.7), then the entire functionf (λ1, . . . , λn)is bounded if and only if it is a constant function. Thus many of the functionals inE0 are unbounded, while for

example, all of the functionals considered in [11] are bounded. The so-called “tame functionals,” that is, functionals of the form

G(x)=gxt1 , . . . , xtm , 0< t1<···< tm≤T (1.17)

as well as functionals of the form (1.7), played a major role in the development of Wiener space integration theory. But functionals of the form (1.17) are inE0provided

g(λ1, . . . , λm)is an entire function of exponential growth. Included of course are all polynomials ofmcomplex variablesλ1, . . . , λm for all positive integersm, as well as such polynomials inx(t1), . . . , x(tm)multiplied by functionals like exp{

m

j=1ajxj(t)}, and so forth.

2. The integral transform, the convolution product, and the first variation of func-tionals inE0. In our ﬁrst theorem, we show that ifF is an element ofE0, then the

integral transform ofF exists and is an element ofE0.

Theorem2.1. LetF∈E₀be given by (1.7). Then the integral transformᏲ_α,βF exists, belongs toE0, and is given by the formula

Ᏺα,βF (y)=hθ, y (2.1)

fory∈K, where

h(λ)=(2π )−n/2

Rnf (α u+βλ)exp

−1₂u2

d u. (2.2)

Proof. For eachy∈K, using a well-known Wiener integration theorem, we obtain

Ᏺα,βF (y)=

C0[0,T ]

fα_{θ, x} ₊_β_{θ, y} _m(dx)

=(2π )−n/2

Rnf

α u+β_{θ, y} _exp₋1 2u

2

d u

=h_{θ, y} _,

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wherehis given by (2.2). By [6, Theorem 3.15],h(λ)is an entire function. Moreover, by inequality (1.8), we have

h(λ)≤(2π )−n/2

RnAFexp

BF|α u+βλ|− 1 2u

2_d_u

≤AᏲα,βFexp

BᏲα,βF|λ|

,

(2.4)

where

AᏲα,βF=AF 1 √ 2π Rexp

−u₂2+BF|αu|

du n

<∞ (2.5)

andBᏲα,βF=BF|β|. HenceᏲα,βF∈E0.

In our next theorem, we show that the convolution product of functionals fromE0is

an element ofE0.

Theorem2.2. LetF , G∈E₀be given by (1.7) and (1.12) with corresponding entire functionsf andg. Then the convolution(F∗G)αexists, belongs toE0, and is given by the formula

(F∗G)α(y)=kθ, y (2.6)

fory∈K, where

kλ =(2π )−n/2

Rnf

λ+α u √

2

g

λ−α u √ 2 exp −1 2u

2_d_u. _(2.7)

Proof. For eachy∈K, using a well-known Wiener integration theorem, we obtain

(F∗G)α(y)=

C0[0,T ]

F

_y₊_αx

√ 2

G

_y₋_αx

√ 2 m(dx) =

C0[0,T ]

f

_{θ, y} ₊_α_{θ, x} √

2

g

_{θ, y} ₋_α_{θ, x} √

2

m(dx)

=(2π )−n/2

Rnf

_{θ, y} ₊_α_u √

2

g

_{θ, y} ₋_α_u √

2

exp

−1₂u2

d u

=k_{θ, y} _,

(2.8)

wherekis given by (2.7). By [6, Theorem 3.15],k(λ)is an entire function and

k(λ)≤(2π )−n/2

RnAFAGexp

_B

F_√+BG 2

|_λ_|+|_α_||_u_| ₋1 2u

2

d u

=A(F∗G)αexp

B(F∗G)α|λ|

,

(2.9)

whereB(F∗G)α=(BF+BG)/ √

2 and

A(F∗G)α=AFAG 1 √ 2π Rexp

−u2

2 +B(F∗G)α|αu|

du n

<∞. (2.10)

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In Theorem 2.3, we ﬁxw∈K and considerδF (y|w)as a function of y, while in

Theorem 2.4, we ﬁxy∈Kand considerδF (y|w)as a function ofw.

Theorem2.3. LetF∈E₀be given by (1.7) and letw∈K. Then

δF (y|w)=p_{θ, y} _(2.11)

fory∈K, where

p(λ)= n

j=1

θj, w

fj(λ). (2.12)

Furthermore, as a function ofy∈K,δF (y|w)is an element ofE0.

Proof. Fory∈K,

δF (y|w)=_∂t∂ f_{θ, y} ₊_t_{θ, w} t=0

= n

j=1

θj, w

fj

_{θ, y} ₌_p_{θ, y} _, (2.13)

wherepis given by (2.12). Sincef (λ)is an entire function,fj(λ)and sop(λ)are entire functions. By the Cauchy integral formula, we have

fj

λ1, . . . , λj, . . . , λn = 1 2π i

|ζ−λj|=1

fλ1, . . . , ζ, . . . , λn

ζ−λj 2

dζ. (2.14)

By inequality (1.8), for anyζwith|ζ−λj| =1, we have

fλ1, . . . , ζ, . . . , λn

ζ−λj 2

≤AFexp

BFλ1+···+|ζ|+···+λn

≤AFexp

BF|λ|+BF

.

(2.15)

Hence

fj(λ)≤AFeBFexp

BF|λ|

, (2.16)

and so

p(λ)≤

n

j=1

θj, wfj(λ)≤AδF (·|w)exp

BδF (·|w)|λ|

, (2.17)

where

AδF (·|w)=AFeBFw∞ n

j=1

Cj<∞ (2.18)

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Theorem2.4. Lety∈Kand letF∈E₀be given by (1.7). Then

δF (y|w)=q_{θ, w} _(2.19)

forw∈K, where

q(λ)= n

j=1

λjfj

_{θ, y} _. _(2.20)

Furthermore, as a function ofw,δF (y|w)is an element ofE0.

Proof. _{Equations (}_2.19_{) and (}_2.20_{) are immediate from the ﬁrst part of the proof of} Theorem 2.3. Clearlyq(λ)is an entire function. Next, using (2.16) we obtain

q(λ)≤

n

j=1

λjfjθ, y

≤AFeBFexp

BFθ1, y+···+θn, y n

j=1 λj

< AFeBFexp

BFy∞C1+···+Cn

e|λ|

=AδF (y|·)expBδF (y|·)|λ|,

(2.21)

whereBδF (y|·)=1 and

AδF (y|·)=AFeBFexpBFy∞C1+···+Cn. (2.22)

Hence, as a function ofw,δF (y|w)∈E0.

We ﬁnish this section with some observations which we use later in this paper. First of all, (1.2) implies that

Ᏺα,βF

_y

√ 2

=Ᏺα,β/√2F (y) (2.23)

for ally∈K. Next, a direct calculation using (1.4), (1.2), (2.11), and (2.23) shows that

δᏲα,βF

_y

√ 2

_√w

2

=δᏲα,β/√2F (y|w)=

β √

2 n

j=1

θj, w

Ᏺα,β/√2Fj(y) (2.24)

for allyandwinK. Finally, by similar calculations, we obtain that

Ᏺα,βδF (·|w) y √

2

= √

2 β δᏲα,β/

√

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for allyandwinK, and for ally∈K,

Ᏺα,βF j(y)=βᏲα,β

Fj (y)=βᏲα,βFj(y). (2.26)

3. Relationships involving two concepts. In this section, we establish all of the various relationships involving exactly two of the three concepts of integral transform, convolution product, and ﬁrst variation for functionals belonging toE0. The seven

dis-tinct relationships, as well as alternative expressions for some of them, are given by (3.1), (3.2), (3.4), (3.7), (3.9), (3.11), and (3.13).

In view ofTheorem 2.1throughTheorem 2.4, all of the functionals that occur in this section are elements ofE0. For example, let F and Gbe any functionals inE0. Then

byTheorem 2.2, the functional(F∗G)αbelongs toE0, and hence byTheorem 2.1, the

functionalᏲα,β(F∗G)αalso belongs toE0. By similar arguments, all of the functionals

that arise in (3.1) through (3.14) and (3.16) through (3.20) exist and belong toE0.

Our ﬁrst formula (3.1) is useful because it permits one to calculateᏲα,β(F∗G)α without ever actually calculating(F∗G)α.

Formula 3.1. The integral transform of the convolution product of functionals

fromE0is given by the formula

Ᏺα,β(F∗G)α(y)=Ᏺα,βF

_y

√ 2

Ᏺα,βG

_y

√ 2

=Ᏺα,β/√2F (y)Ᏺα,β/√2G(y) (3.1)

for allyinK.

Proof. _{Formula 3.1}_{is a special case of [}₅_{, Theorem 3.1].}

Formula _3.2. _{The convolution product of the integral transform of functionals}

fromE0is given by the formula

Ᏺα,βF∗Ᏺα,βG α(y)

=(2π )−3n/2

R3nf

αr+√β

2θ, y + βα √

2u

·g

αs+√β 2

θ, y−βα√ 2u

exp

−u2+r₂2+s2

d u dr ds

(3.2)

for allyinK.

Proof. Using (1.3) and (1.2), we see that

Ᏺα,βF∗Ᏺα,βG α(y)

=

C0[0,T ]

Ᏺα,βF

_y₊_αx

√ 2

Ᏺα,βG

_y₋_αx

√ 2

m(dx)

=

C0[0,T ]

F

αz1+

β(y+αx) √

2

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·

C0[0,T ]

G

αz2+β(y√−αx)

2

mdz2

m(dx)

=

C₀3[0,T ] f

αθ, z 1

+√β

2θ, y + αβ √

2θ, x

·gαθ, z 2+

β √

2θ, y− αβ √

2θ, x

m(dx)mdz1 mdz2 .

(3.3)

Formula (3.2) now follows upon evaluating the above Wiener integrals.

Formula_3.3. _{The integral transform with respect to the ﬁrst argument of the}

vari-ation is given by the formula

Ᏺα,β

δF (·|w) (y)=1_βδᏲα,βF (y|w)= n

j=1

θj, w

Ᏺα,βFj(y) (3.4)

for allyandwinK.

Proof. By applyingTheorem 2.1to expression (2.11), we obtain

Ᏺα,β

δF (·|w) (y)=(2π )−n/2

Rnp

α u+β_{θ, y} _exp₋1 2u

2_d_u

=(2π )−n/2 n

j=1

θj, w Rnfj

α u+β_{θ, y} _exp₋1 2u

2

d u.

(3.5)

On the other hand, by applyingTheorem 2.3to expression (2.1) and then using (2.2), we obtain

1

βδᏲα,βF (y|w)= 1 β

n

j=1

θj, w

hj

_{θ, y}

=1 β

n

j=1

θj, w(2π )−n/2β

Rnfj

α u+βθ, y exp

−1 2u

2_d_u

=(2π )−n/2 n

j=1

θj, w

Rnfj

α u+β_{θ, y} _exp₋1 2u

2

d u,

(3.6)

and so (3.4) is established.

Formula_3.4. _{The Integral transform with respect to the second argument of the}

variation is given by the formula

Ᏺα,β

δF (y|·) (w)=βδF (y|w) (3.7)

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Proof. By applyingTheorem 2.1to expression (2.19), we obtain

Ᏺα,βδF (y|·) (w)=(2π )−n/2

Rnq

α u+β_{θ, w} _exp₋1 2u

2_d_u

=(2π )−n/2 n

j=1

Rn

αuj+β

θj, w fj

_{θ, y} _exp₋1 2u

2

d u

=β n

j=1

θj, w

fj

_{θ, y} ₌_{βδF (y}_|_w).

(3.8)

Formula3.5. The ﬁrst variation of the convolution product of functionals fromE₀ is given by the formula

δ(F∗G)α(y|w)= n

j=1

θj, w √

2

Fj∗G α(y)+

F∗Gj α(y)

(3.9)

for allyandwinK.

Proof. _{By applying}_{Theorem 2.3}_{to (}_2.6_{) and then using (}_2.7_{), we obtain}

δ(F∗G)α(y|w)

= n

j=1

θj, w

kj

_{θ, y}

=(2π )−n/2 n

j=1

θj, w √ 2 Rn fj

_{θ, y} ₊_α_u √

2

g

_{θ, y} ₋_α_u √

2

+f

_{θ, y} ₊_α_u √

2

gj

_{θ, y} ₋_α_u √ 2 exp −1 2u

2_d_u

= n

j=1

θj, w √

2

Fj∗G α(y)+

F∗Gj α(y)

.

(3.10)

Formula_3.6. _{The convolution product, with respect to the ﬁrst argument of the}

variation, of the variation of functionals fromE0is given by the formula

δF (·|w)∗δG(·|w) α(y)= n

j=1 n

l=1

θj, w

θl, w

Fj∗Gl α(y) (3.11)

for allyandwinK.

Proof. Applying the additive distribution properties of the convolution product to

the expressions

δF (y|w)= n

j=1

θj, w

Fj(y), δG(y|w)= n

l=1

θl, w

Gl(y) (3.12)

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Formula3.7. The convolution product, with respect to the second argument of the

variation, of the variation of functionals fromE0is given by the fromula

δF (y|·)∗δG(y|·) _α(w)=1₂δF (y|w)δG(y|w)−α₂2 n

j=1

Fj(y)Gj(y) (3.13)

for allyandwinK.

Proof. _{Upon applying}_{Theorem 2.2}_{to the expressions}

δF (y|w)= n

j=1

θj, w

fj

_{θ, y} _, _δG(y_|_w)₌ n

l=1

θl, w

gl

_{θ, w} _, _(3.14)

and using the fact that

Rnujulexp

−1 2u

2_d_u₌     

(2π )n/2 _if_j₌_l,

0 ifj=l, (3.15)

we obtain

δF (y|·)∗δG(y|·) _α(w)

=(2π )−n/2

Rn 

n

j=1

θj, w+αuj √

2 fj

_{θ, y}   · 

n

l=1

θl, w

−αul √

2 gl

_{θ, y}



exp

−1₂u2

d u

=1 2(2π )

−n/2 n

j=1 n

l=1

fjθ, y glθ, y

·

Rn

θj, w

+αuj θl, w

−αul exp

−1₂u2

d u

=1 2

n

j=1 n

l=1

θj, wθl, wfjθ, y glθ, y − α2

2 n

j=1

fjθ, y gjθ, y

=1 2 

n

j=1

θj, w fj _{θ, y}   

n

l=1

θl, w

gl

_{θ, y}  ₋α2

2 n

j=1

Fj(y)Gj(y)

=1

2δF (y|w)δG(y|w)− α2

2 n

j=1

Fj(y)Gj(y).

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Finally, lettingG=Fin (3.1), (3.9), (3.11), and (3.13) yields the formulas

Ᏺα,β(F∗F )α(y)=

Ᏺα,β/√2F (y) 2

, (3.17)

δ(F∗F )α(y|w)= #

2 n

j=1

θj, w

F∗Fj α(y), (3.18)

δF (·|w)∗δF (·|w) α(y)= n

j=1 n

l=1

θj, w

θl, w

Fj∗Fl α(y), (3.19)

δF (y|·)∗δF (y|·) _α(w)=1 2

δF (y|w)2−α2 2

n

j=1

Fj(y)2 (3.20)

for allyandwinK.

It is interesting to note that the left-hand side of each of the formulas (3.1), (3.2), (3.4), (3.7), (3.9), (3.11), (3.13), (3.17), (3.18), (3.19), and (3.20) involve exactly two of the operations of transform, convolution and ﬁrst variation, while each right-hand side involves at most one of these three operations.

4. Relationships involving three concepts. In this section, we examine all of the various relationships involving the integral transform, the convolution product, and the ﬁrst variation, where each concept is used exactly once. There are more than six possibilities since one can take the transform or the convolution with respect to either the ﬁrst or the second argument of the variation. However, in view of formula (3.4) and (3.7), there are some repetitions. To exhaust all possibilities, we need to take the variation of the expressions in (3.1) and (3.2), the convolution of the expressions in (3.4) and (3.7), and the transform of the expressions in formulas (3.9), (3.11), and (3.13). It turns out that there are ten distinct formulas, and these are given by (4.1) through (4.10) below. We omit the details of the calculations used to obtain (4.1) through (4.10) because the techniques needed are similar to those used above in Sections2and3.

Again, because of the theorems inSection 2, all of the functionals that arise in this section are automatically elements ofE0. As usual,F andGinE0are given by (1.7) and

(1.12), respectively.

Formula4.1. Taking the ﬁrst variation of the expressions in (3.1) or taking the transform of the expressions in (3.9) with respect to the ﬁrst argument of the variation and then using (2.23) and (2.24) yields the formula

δᏲα,β(F∗G)α(y|w)=βᏲα,βδ(F∗G)α(·|w)(y)

=Ᏺα,βF

y √

2

δᏲα,βG

y √

2 √w2

+δᏲα,βF

y √

2

w √

2

Ᏺα,βG

y √

2

=Ᏺα,β/√2F (y)δᏲα,β/√2G(y|w)

+δᏲα,β/√2F (y|w)Ᏺα,β/√2G(y)

(4.1)

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Formula4.2. Taking the ﬁrst variation of the expressions in (3.2) or replacingF withᏲα,βFandGwithᏲα,βGin (3.9) yields the formula

δᏲα,βF∗Ᏺα,βG α

y|w

=√β 2

n

j=1

θj, w

Ᏺα,βFj∗Ᏺα,βG α(y)+

Ᏺα,βF∗Ᏺα,βGj α(y)

(4.2)

for allyandwinK.

Formula 4.3. Taking the integral transform of the expressions in (3.9) with re-spect to the second argument of the variation yields the formula

Ᏺα,βδ(F∗G)α(y|·)(w)=βδ(F∗G)α(y|w)

=√β 2

n

j=1

θj, w

Fj∗G α(y)+

F∗Gj α(y)

(4.3)

for allyandwinK.

Formula4.4. Taking the integral transform of the expressions in (3.11) with respect to the ﬁrst argument of the variation and then using (3.1) and (2.25) yields the formula

Ᏺα,β

δF (·|w)∗δG(·|w) α(y)=Ᏺα,βδF (·|w)

_y

√ 2

Ᏺα,βδG(·|w)

_y

√ 2

=_β2₂δᏲα,β/√2F (y|w)δᏲα,β/√2G(y|w)

(4.4)

for allyandwinK.

Formula4.5. Taking the integral transform of the expressions in (3.11) with respect to the second argument of the variation yields the formula

C0[0,T ]

δF·|βw+αx ∗δG·|βw+αx _α(y)m(dx)

=β2_{δF (}_·|_w)_∗_δG(_·|_w)

α(y)+α2 n

j=1

Fj∗Gj α(y)

(4.5)

for allyandwinK.

Formula4.6. Taking the integral transform of the expressions in (3.13) with respect to the ﬁrst argument of the variation yields the formula

C0[0,T ]

δFβy+αx|· ∗δGβy+αx|· α(w)m(dx)

=1₂ n

j=1 n

l=1

θj, w

θl, w

Ᏺα,β

FjGl (y)− α2

2 n

j=1 Ᏺα,β

FjGj (y)

(4.6)

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Formula4.7. Taking the integral transform of the expressions in (3.13) with respect to the second argument of the variation yields the formula

Ᏺα,β

δF (y|·)∗δG(y|·) α(w)= β2

2δF (y|w)δG(y|w) (4.7)

for allyandwinK.

Formula_4.8. _{Taking the convolution product of the expressions in (}_3.4_{) with} re-spect to the ﬁrst argument of the variation yields the formula

Ᏺα,βδF (·|w)∗Ᏺα,βδG(·|w) α(y)= 1 β2

δᏲα,βF (·|w)∗δᏲα,βG(·|w) α(y)

= n

j=1 n

l=1

θj, w

θl, w

Ᏺα,βFj∗Ᏺα,βGl α(y) (4.8)

for allyandwinK.

Formula4.9. Taking the convolution product of the expressions in (3.4) with re-spect to the second argument of the variation, or replacingF withᏲα,βF andG with Ᏺα,βGin (3.13) and using (2.26) yields the formula

δᏲα,βF (y|·)∗δᏲα,βG(y|·) α(w)

=1₂δᏲα,βF (y|w)δᏲα,βG(y|w)− α2

2 n

j=1

Ᏺα,βF j(y)

Ᏺα,βG j(y)

=1₂δᏲα,βF (y|w)δᏲα,βG(y|w)− α2_β2

2 n

j=1

Ᏺα,βFj(y)Ᏺα,βGj(y)

(4.9)

for allyandwinK.

Formula 4.10. Taking the convolution product of the expressions in formula

(3.7) with respect to the second argument of the variation yields the formula

Ᏺα,βδF (y|·)∗Ᏺα,βδG(y|·) α(w)=β2

δF (y|·)∗δG(y|·) _α(w)

=β2 2



δF (y|w)δG(y|w)−α2 n

j=1

Fj(y)Gj(y)  

(4.10)

for allyandwinK.

For completeness, note that taking the convolution product of the expressions in (3.7) with respect to the ﬁrst argument of the variation, does not yield a new formula; we simply get formula (3.11) again.

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because they permit one to calculateᏲα,β(F∗G)α(y),(δF (y|·)∗δG(y|·))α(w),. . ., and (Ᏺα,βδF (y|·)∗Ᏺα,βδG(y|·))α(w)without actually having to calculate the con-volution products on the left-hand sides of formulas (3.1), (3.13),. . ., and (4.10). It is usually harder to calculate convolution products than transforms and ﬁrst variations.

5. Further results. It is well known, see for example [5,10], that for allF∈E0, all

y∈K, and alla,b, andcinC,

C0[0,T ]

F (aw+bx+cy)m(dw)

m(dx)

=

C0[0,T ]

F$#a2₊_b2_z₊_cy%_m(dz)

=

C0[0,T ]

F (aw+bx+cy)m(dx)

m(dw),

(5.1)

and that

Ᏺα,βᏲα,βF (y)=F (y)=Ᏺα,βᏲα,βF (y) (5.2)

providedββ=1 andα2₊_(βα₎2₌_{0. In particular, for all}_y_∈_K,

Ᏺα,βᏲiα/β,1/βF (y)=F (y)=Ᏺiα/β,1/βᏲα,βF (y) (5.3)

for all nonzero complex numbersαandβ.

If in (1.3) we replaceαwithiα/β, then (5.3) enables us to express the convolution product of the transforms ofF andGas a transform of the product ofF withG.

Theorem_5.1. _Let_α_and_β_{be nonzero complex numbers and let}_F _and_G_be

func-tionals fromE0given by (1.7) and (1.12), respectively. Then for ally∈K,

Ᏺα,βF∗Ᏺα,βG iα/β(y)=Ᏺα,β

F

_·

√ 2

G

_·

√ 2

(y)

=Ᏺα,β/√2(F G)(y).

(5.4)

Proof. Letα=iα/βandβ=1/β. Using (3.1), it follows that the formula

Ᏺα,βL1∗L2 α(y)=Ᏺα,βL1

_y

√ 2

Ᏺα,βL2

_y

√ 2

(5.5)

holds for allL1andL2inE0and ally∈K. LettingL1=Ᏺα,βF andL2=Ᏺα,βGin (5.5) and then using (5.3) yields the formula

Ᏺα,β

Ᏺα,βF∗Ᏺα,βG α(y)=Ᏺα,β

Ᏺα,βF y √

2

Ᏺα,β

Ᏺα,βG y √

2

=F

_y

√ 2

G

_y

√ 2

(5.6)

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Theorem5.2. Letα,β,F, andGbe as inTheorem 5.1. Then for allyandwinK,

δᏲα,βF∗Ᏺα,βG iα/β(y|w)= β √

2Ᏺα,β

δF (·|w)G(·)+F (·)δG(·|w) √y 2

. (5.7)

Proof. _{Using (}_5.4_{) and (}_2.25_{), we see that for all}_y_and_w_in_K,

δᏲα,βF∗Ᏺα,βG iα/β(y|w)=δᏲα,β F _· √ 2 G _· √ 2 (y|w)

=δᏲα,β/√2

F (·)G(·) (y|w)

=√β 2Ᏺα,β

δF (·|w)G(·)+F (·)δG·|w √y 2

.

(5.8)

Next, using (5.4), we obtain the following analogue ofFormula 4.8.

Theorem_5.3. _Let_F_,_G_,_α_{, and}_β_{be as in}_{Theorem 5.1}_{. Then for all}_y_and_w_in_K_,

δᏲα,βF (·|w)∗δᏲα,βG(·|w) iα/β(y)

=β2 n

l=1 n

j=1

θj, w

θl, w

Ᏺα,β/√2

FjGl (y).

(5.9)

Proof. Using (3.4), (5.4),Theorem 2.3, and (2.23), we obtain

δᏲα,βF (·|w)∗δᏲα,βG(·|w) iα/β(y) =β2_Ᏺ

α,βδF (·|w)∗Ᏺα,βδG(·|w) iα/β(y)

=β2_Ᏺ α,β δF _· √ 2 w δG _· √ 2 w (y)

=β2Ᏺα,β   

n

j=1

θj, w Fj _· √ 2   

n

l=1

θl, w Gl _· √ 2    (y)

=β2 n

l=1 n

j=1

θj, wθl, wᏲα,β/√2

FjGl (y)

(5.10)

for allyandwinK.

It is interesting to note that we can obtain analogues of Formulas4.9and4.10directly by use of (3.13) and (3.7) rather than usingTheorem 5.1as we did inTheorem 5.3to obtain an analogue ofFormula 4.8.

Theorem5.4. LetF,G,α, andβbe as inTheorem 5.1. Then for allyandwinK,

δᏲα,βF (y|·)∗δᏲα,βG(y|·) iα/β(w)

=1₂δᏲα,βF (y|w)δᏲα,βG(y|w)+ α2

2 n

j=1

Ᏺα,βFj(y)Ᏺα,βGj(y),

Ᏺα,βδF (y|·)∗Ᏺα,βδG(y|·) iα/β(w)

=β2

2δF (y|w)δG(y|w)+ α2

2 n

j=1

Fj(y)Gj(y).

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Example5.5. Next, we brieﬂy discuss the functionalsF (x)=n_j₌₁θ_j,x,G(x)= exp{F (x)}, H(x)= F (x)exp{F (x)}, M(x)= [F (x)]2 ₌_[n

j=1θj, x]2, and N(x) = n

j=1[θj, x]2, all of which are elements ofE0. The following formulas follow quite

readily for allyandwinK:

Ᏺα,βF (y)=βF (y), (5.12)

δF (y|w)=F (w), (5.13)

δᏲα,βF (y|w)=βF (w), (5.14)

Ᏺα,βG(y)=exp

_nα2

2 +βF (y)

, (5.15)

δG(y|w)=F (w)expF (y), (5.16)

δᏲα,βG(y|w)=βF (w)exp

_nα2

2 +βF (y)

, (5.17)

Ᏺα,βH(y)=

nα2+βF (y)exp

_nα2

2 +βF (y)

, (5.18)

δH(y|w)=1+F (y)F (w)expF (y), (5.19)

δᏲα,βH(y|w)=βF (w)

nα2+βF (y)+1exp

_nα2

2 +βF (y)

, (5.20)

Ᏺα,βM(y)=nα2+

βF (y)2, (5.21)

δM(y|w)=2F (w)F (y), (5.22)

δᏲα,βM(y|w)=2β2F (w)F (y), (5.23) Ᏺα,βN(y)=nα2+β2N(y), (5.24)

δN(y|w)=2 n

j=1

θj, w

θj, y

= n

j=1

Nj(y)Fj(w), (5.25)

δᏲα,βNy|w =β2δN(y|w)=β2 n

j=1

Nj(y)Fj(w). (5.26)

Finally, note that by using the various formulas in Sections3and4together with the formulas (5.12) through (5.26), we can immediately write down many additional formu-las involving the speciﬁc functionalsF,G,H,M, andNdeﬁned above inExample 5.5. For example, using (3.1), (5.15), and (5.21), we observe that

Ᏺα,β(M∗G)α(y)= *

nα2+β

2

2 F

2_(y)+_expnα2

2 +

β √

2F (y)

, (5.27)

and hence using (5.13), (5.16), and (5.22),

δᏲα,β(M∗G)α(y|w)= *

nα2₊β2

2F

2_(y)+_√β

2F (w)exp

_nα2

2 +

β √

2F (y)

+β2F (y)F (w)exp

_nα2

2 +

β √

2F (y)

.

(5.28)

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ncomplex variablesλ1, . . . , λnsuch that

fλ1, . . . , λn ≤AFexp   BF

n

j=1 λj1+σ

 

 (5.29)

for some positive constantsAF andBF. Note that ifσ =0, thenEσ =E0and for 0<

σ1< σ2<1,E0Eσ1Eσ2L2(C0[0, T ]).

A careful examination of the proofs of Theorems2.1, 2.2,2.3, and2.4shows that the conclusions of all four of these theorems hold for allF andGinEσ, 0≤σ <1. For example, to show that the conclusions ofTheorem 2.1hold forEσ, letF∈Eσbe given by (1.7) withf satisfying (5.29). Then proceeding as in the proof ofTheorem 2.1, we obtain thatᏲα,βF is given by (2.1) withhdeﬁned by (2.2) satisfying

hλ1, . . . , λn ≤AᏲα,βFexp   BᏲα,βF

n

j=1 λj

1+σ  

 (5.30)

with

AᏲα,βF=AF

1 √

2π

Rexp

−u2 2 +BF

2|αµ| 1+σ du

n

<∞, (5.31)

and withBᏲα,βF=BF(2|β|)

1+σ_{. Hence}_Ᏺ

α,βF exists and belongs toEσ.

Some possible extensions. _{It seems likely that using the functionals in}_E₀_(or_E_σ₎

as building blocks, one could show that the results established in this paper hold for larger classes of functionals.

For example, let{Fm}∞m=1be a sequence fromE0such that limm→∞Fm(y)exists for ally∈Kand letF (y)=limm→∞Fm(y). Now the condition

Fm(y)≤AexpBy∞ (5.32)

for ally∈Kand allm=1,2, . . .ensures the existence of the integral transformᏲα,βF since by the dominated convergence theorem,

lim

m→∞Ᏺα,βFm(y)=mlim→∞

C0[0,T ]

Fm(αx+βy)m(dx)

=

C0[0,T ]

F (αx+βy)m(dx)

=Ᏺα,βF (y)

(5.33)

for eachy∈K.Example 5.7shows thatF need not belong toEσfor anyσ∈[0,1). It seems as though ﬁnding appropriate conditions to put on the sequences{Fm}∞m=1

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Example5.7. Let{θ_j}∞_j

=1be a complete orthonormal sequence of functions inL2[0,

T ], each of bounded variation on[0, T ]. Form=1,2, . . .andy∈K, let

Fm(y)=exp   

m

j=1

θj, y

2j_C j

 

 (5.34)

withCjgiven by (1.6). ClearlyFm∈E0for eachm=1,2, . . . .

Also for eachm=1,2, . . .and eachy∈K,

Fm(y)≤exp

 

y∞

m

j=1

1 2j

 

≤expy∞. (5.35)

But limm→∞Fm(y)=exp{

_∞

j=1(θj, y/2jCj)} ≡F (y)is not an element ofE0(orEσ for 0≤σ <1) because it depends uponθm, yfor everym∈ {1,2, . . .}and so it cannot be written in the form (1.7) for any positive integern; recall that{θj}∞j=1is a complete

orthonormal set of functions inL2[0, T ].

Acknowledgment. _{The ﬁrst author wishes to express his gratitude to Professors}

D. Skoug and G. Johnson for their encouragement and valuable advice as well as the hospitality of the University of Nebraska, Lincoln.

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[2] R. H. Cameron and W. T. Martin,Fourier-Wiener transforms of analytic functionals, Duke Math. J.12(1945), 489–507.

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[4] ,Feynman integral of variations of functionals, Gaussian Random Fields (Nagoya, 1990), Ser. Probab. Statist., vol. 1, World Scientiﬁc Publishing, New Jersey, 1991, pp. 144–157.

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[11] C. Park, D. Skoug, and D. Storvick,Relationships among the first variation, the convolution product and the Fourier-Feynman transform, Rocky Mountain J. Math.28(1998), no. 4, 1447–1468.

[12] D. Skoug and D. Storvick,A survey of results involving transforms and convolutions in function space, Rocky Mountain J. Math.34(2004).

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Bong Jin Kim: Department of Mathematics, Daejin University, Pocheon 487-711, Korea E-mail address:[email protected]

Byoung Soo Kim: University College, Yonsei University, Seoul 120-749, Korea E-mail address:[email protected]

David Skoug: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0323, USA