# A Banach Algebra Similar to Cameron-Storvick’s One with Its Equivalent Spaces

Authors: Dong Hyun Cho
Journal: Journal of Function Spaces
 (2018)

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## Abstract

LetC[0,T] denote an analogue of a generalized Wiener space, that is, the space of continuous, real-valued functions on the interval [0,T]. In this paper, we introduce a Banach algebra on C[0,T] which generalizes Cameron-Storvick’s one, the space of generalized Fourier-Stieltjes transforms of the C-valued, and finite Borel measures on L2[0,T]. We also investigate properties of the Banach algebra on C[0,T] and equivalence between the Banach algebra and the Fresnel class which plays a significant role in Feynman integration theories and quantum mechanics.


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## Body

## 1. Introduction

LetC0[0,T] denote the Wiener space, that is, the space of continuous, real-valued functions x on the interval [0,T] with x(0)=0. The (generalized) Paley-Wiener-Zygmund (PWZ) stochastic integrals on the (generalized) Wiener space have been used in various papers, in particular, concerning Feynman integration theories [1–4]. In particular, the PWZ integral is used in the definition of Cameron-Storvick’s Banach algebra S of functions on C0[0,T] which is the space of generalized Fourier-Stieltjes transforms of the C-valued and finite Borel measures on L2[0,T] [1]. Johnson [4] showed that S is isometrically isomorphic to the Banach algebra of the Fresnel integrable functions given by Albeverio and Høegh-Krohn [5]. Further work for relationships between the Banach algebra S and the Fresnel class was studied by Chang et al. [6] on infinite dimensional Hilbert spaces, and the same work was done by Chang et al. [7] on the space Cα,β[0,T] which is a generalized Wiener space with mean function α and variance function β, where α and β are appropriate functions on [0,T]. We note that every Wiener path x in both C0[0,T] and Cα,β[0,T] starts at the origin; that is, x(0)=0.On the other hand, letC[0,T] denote an analogue of a generalized Wiener space, that is, the space of continuous, real-valued functions on the interval [0,T]. On the space C[0,T], Ryu [8, 9] introduced a finite measure wα,β;φ and investigated its properties, where α,β:[0,T]→R are continuous functions such that β is strictly increasing, and φ is arbitrary finite measure on the Borel class B(R) of R. On this space (C[0,T],wα,β;φ), the author [10] introduced an Itô type integral Iα,β and a generalized PWZ integral with their relation. The relation says that Iα,β is reduced to the generalized PWZ integral if φ is a probability measure on B(R) and the generalized PWZ integral exists. In this paper, we will introduce a Banach algebra Sα,β;φ on C[0,T] by using Iα,β, which generalizes Cameron-Storvick’s Banach algebra S with the mean function and the variance function determined by α and β, respectively. We also investigate properties of Sα,β;φ, and relationships between Sα,β;φ and the Fresnel class [5] which plays a significant role in Feynman integration theories and quantum mechanics. We note that every path in C[0,T] starts at arbitrary point so that C[0,T] generalizes both C0[0,T] and Cα,β[0,T].When the generalized PWZ integral onC[0,T] is defined, one of difficulties encountered is the existence of a complete orthonormal basis of functions in Lα,β2[0,T] such that these functions are of bounded variation and orthogonal in L0,β2[0,T], where L0,β2[0,T] and Lα,β2[0,T] are the L2-spaces with respect to the Lebesgue-Stieltjes measures induced by α and β [10]. In order to avoid this difficulty, we will use Iα,β instead of the generalized PWZ integral so that we can define the functions in Sα,β;φ regardless of the existence of the orthonormal basis of Lα,β2[0,T] satisfying the orthogonality in L0,β2[0,T] as described above.


## 2. An Analogue of a Generalized Wiener Space

In this section, we introduce an analogue of a generalized Wiener space with the Itô type integral as described in Section1.LetmL denote the Lebesgue measure on R. Let C[0,T] denote the space of continuous, real-valued functions on the interval [0,T]. Let α,β:[0,T]→R be two continuous functions, where β is strictly increasing. Let φ be a positive finite measure on B(R). For t→n=(t0,t1,…,tn) with 0=t0<t1<⋯<tn≤T, let Jt→n:C[0,T]→Rn+1 be the function given by (1)Jt→nx=xt0,xt1,…,xtn.For a rectangle ∏j=0nBj in B(Rn+1), the subset Jt→n-1(∏j=0nBj) of C[0,T] is called an interval I and let C be the set of all such intervals I. Define a premeasure mα,β;φ on C by (2)mα,β;φI=∫B0∫∏j=1nBjWnα,β,t→n,u→n,u0dmLnu→ndφu0,where for u→n=(u1,…,un)∈Rn and u0∈R, (3)Wnα,β,t→n,u→n,u0=1∏j=1n2πβtj-βtj-11/2×exp⁡-12∑j=1nuj-αtj-uj-1+αtj-12βtj-βtj-1.The Borel σ-algebra B(C[0,T]) of C[0,T] with the supremum norm coincides with the smallest σ-algebra generated by C and there exists a unique positive finite measure wα,β;φ on B(C[0,T]) with wα,β;φ(I)=mα,β;φ(I) for all I∈C. This measure wα,β;φ is called an analogue of a generalized Wiener measure on (C[0,T],B(C[0,T])) according to φ [8, 9].For further work, we give additional conditions forα and β. Let α and β be functions defined on [0,T] such that α is absolutely continuous and β′ is continuous, positive, and bounded away from 0. We observe that the functions α and β induce a Lebesgue-Stieltjes measure να,β on [0,T] by (4)να,βE=ναE+νβE,where να(E)=∫Edα(t) and νβ(E)=∫Edβ(t) for a Lebesgue measurable subset E of [0,T]. Define Lα,β2[0,T] to be the space of functions on [0,T] that are square integrable with respect to the measure να,β; that is, (5)Lα,β20,T=f:0,T⟶R∣∫0Tft2dνα,βt<∞.The space Lα,β2[0,T] is in fact a Hilbert space (as our notation suggests) and has the obvious inner product [11](6)f,gα,β=∫0Tftgtdνα,βtforf,g∈Lα,β20,T.LetS[0,T] denote the collection of all step functions on [0,T]. For f in Lα,β2[0,T], let {ϕn} be a sequence of the step functions in S[0,T] with limn→∞ϕn-fα,β=0. Define the Itô type integral Iα,β(f) of f by the L2(C[0,T])-limit(7)Iα,βfx=limn→∞⁡∫0Tϕntdxtfor all x∈C[0,T] for which this limit exists, where ∫0Tϕn(t)dx(t) denotes the Riemann-Stieltjes integral of ϕn with respect to x. We note that Iα,β(f)(x) exists for wα,β;φ a.e. x∈C[0,T] and the limit in (7) is independent of choice of the sequence {ϕn} in S[0,T]. Moreover, one can show that Iα,β is an injective, bounded linear operator from Lα,β2[0,T] into L2(C[0,T]). The following lemma is due to a result in [10].Lemma 1.
Iff∈Lα,β2[0,T] and x is absolutely continuous on [0,T] with x′∈L0,β2[0,T], then limn→∞∫0Tϕn(t)dx(t) exists and it is given by (8)limn→∞∫0Tϕntdxt=∫0Tftx′tdmLtfor any sequence {ϕn} of the step functions in S[0,T] with limn→∞ϕn-fα,β=0.Forx∈C[0,T] satisfying the assumption in Lemma 1, we redefine Iα,β(f)(x) as Iα,β(f)(x)=∫0Tf(t)x′(t)dmL(t). Nevertheless, we have (7) for wα,β;φ a.e. x∈C[0,T]. Also, Iα,β is still an injective, bounded linear operator from Lα,β2[0,T] into L2(C[0,T]).Throughout the remainder of this paper, unless otherwise specified, we assume thatφ(R)=1; that is, φ is a probability measure on B(R). We now have the following generalized PWZ theorem [10].Theorem 2.
Let{f1,…,fn} be a set of functions in Lα,β2[0,T] which are nonzero and orthogonal in L0,β2[0,T]. Then Iα,β(f1),…,Iα,β(fn) are independent random variables and each Iα,β(fj) has the normal distribution with the mean ∫0Tfj(t)dα(t) and the variance fj0,β2. Moreover, if f:Rn→C is Borel measurable, then (9)∫C0,TfIα,βf1x,…,Iα,βfnxdwα,β;φx=∗∏j=1n12πfj0,β21/2∫Rnfu→exp⁡-12∑j=1nuj-∫0Tfjtdαt2fj0,β2dmLnu→,where =∗ means that if either side exists, then both sides exist and they are equal.LetF:C[0,T]→C be a measurable function and suppose that the integral (10)JFλ≡∫C0,TFλ-1/2xdwα,β;φxexists as a finite number for all λ>0. If there exists a function JF∗(λ) analytic in C+≡{λ∈C:Reλ>0} such that JF∗(λ)=JF(λ) for all λ>0, then JF∗(λ) is defined to be a generalized analytic Wiener wα,β;φ-integral of F over C[0,T] with parameter λ and it is denoted by (11)Ewα,β;φanwλF=JF∗λfor λ∈C+. Let q be a nonzero real number. If Ewα,β;φanwλ[F] has a limit as λ approaches -iq through C+, then we call it a generalized analytic Feynman wα,β;φ-integral of F over C[0,T] with parameter q and it is denoted by (12)Ewα,β;φanfqF=limλ→-iq⁡Ewα,β;φanwλF.


## 3. A Banach Algebra with Its Applications

In this section, we introduce a Banach algebra which generalizes the Banach algebraS given by Cameron and Storvick [1]. To define it, we need the following lemmas.Lemma 3.
LetH0 be a separable, real Hilbert space with the inner product ·,·H0. Let A be the σ-algebra generated by the class of sets of the form(13)g∈H0:g,fH0<r,where f and r range over all elements in H0 and over all real numbers, respectively. Then we have A=B(H0), where B(H0) is the Borel σ-algebra of H0.Proof.
Since·,fH0 is continuous for each f∈H0, B(H0) contains all sets of the form given by (13) so that we have A⊆B(H0). To prove B(H0)⊆A, it suffices to show that for each r>0 and f0∈H0, the closed ball (14)Br,f0≡f∈H0:f-f0H0≤rbelongs to A since H0 is separable. Let D≡{fn:n∈N} be a dense subset of H0. For each n∈N, take gn∈H0 such that gn,fnH0=fnH0 and gnH0=1 which can be justified by Corollary 14.13 of [12] as an application of the Hahn-Banach theorem and the Riesz representation theorem. For each n∈N, let (15)Bn,r,f0=f∈H0:f-f0,gnH0≤r.Then for all positive integer n, we have (16)Bn,r,f0=f∈H0:f0,gnH0-r≤f,gnH0≤f0,gnH0+rwhich belongs to A. We will prove Br,f0=⋂n=1∞Bn,r,f0 so that Br,f0 also belongs to A. Indeed, if f∈Br,f0, then we have for all n∈N(17)f-f0,gnH0≤f-f0H0≤rby the Schwarz inequality, so that f∈⋂n=1∞Bn,r,f0; that is, Br,f0⊆⋂n=1∞Bn,r,f0. Conversely, let f∈⋂n=1∞Bn,r,f0 and let ϵ>0 arbitrary. Since D is a dense subset of H0, we can take fnϵ∈D with f-f0-fnϵH0<ϵ/2. Then we have by the Schwarz inequality (18)f-f0H0≤f-f0-fnϵH0+fnϵ,gnϵH0=f-f0-fnϵH0+fnϵ-f+f0,gnϵH0+f-f0,gnϵH0≤2f-f0-fnϵH0+f-f0,gnϵH0<ϵ+rsince f∈Bnϵ,r,f0. Since ϵ is arbitrary, we have f-f0H0≤r so that f∈Br,f0; that is, ⋂n=1∞Bn,r,f0⊆Br,f0. Now the proof is completed as desired.Remark 4.
( 1 ) By the Riesz representation theorem and Lemma 3, B(H0) is in fact the smallest σ-algebra satisfying that all bounded linear functionals on H0 are measurable.
( 2 ) Applying the method used in the proof of [13, Theorem 4.2, p. 74], we can also prove Lemma 3.Lemma 5.
( 1 ) We have L0,β2[0,T]=L2[0,T] as vector spaces, where L2[0,T] denotes the Lebesgue space. Moreover, the two norms ·0,β and ·mL are equivalent so that B(L0,β2[0,T])=B(L2[0,T]).
( 2 )  B(Lα,β2[0,T])=B(L0,β2[0,T]) if and only if Lα,β2[0,T]=L0,β2[0,T] as vector spaces. In this case, the two norms ·α,β and ·0,β are equivalent.Proof.
Sinceβ′ is bounded away from 0 and continuous so that it is bounded on 0,T, the two norms ·0,β and ·mL are equivalent which implies L0,β20,T=L20,T. Moreover, the identity operator from L0,β20,T,·0,β to (L2[0,T],·mL) is bicontinuous so that the topologies induced by ·0,β and ·mL are equal. We now have B(L0,β2[0,T])=B(L2[0,T]) which proves (1). If B(Lα,β2[0,T])=B(L0,β2[0,T]), then clearly we have Lα,β2[0,T]=L0,β2[0,T]. Conversely, if Lα,β2[0,T]=L0,β2[0,T], then by the open mapping theorem, the identity operator I:Lα,β20,T,·α,β→L0,β20,T,·0,β is a bounded operator and an open map since f0,β≤fα,β for all f∈Lα,β2[0,T]; that is, I is bicontinuous. Now the two norms ·α,β and ·0,β are equivalent, and the topologies induced by ·α,β and ·0,β are equal so that B(Lα,β2[0,T])=B(L0,β2[0,T]) which completes the proof of (2).Example 6.
( 1 ) It is not difficult to show that α is a constant function on [0,T] if and only if fα,0=0 for all f∈S[0,T] if and only if f,gα,0=0 for all f,g∈S[0,T]. In this case, f,gα,β=f,g0,β for all f,g∈L0,β2[0,T] so that Lα,β2[0,T]=L0,β2[0,T] isometrically which implies B(Lα,β2[0,T])=B(L0,β2[0, T]) by Lemma 5. In particular, if β(t)=t for all t∈[0,T], then Lα,β2[0,T]=L2[0,T] isometrically.
( 2 ) If for some constant c≥0, α(t)=cβ(t) for all t∈[0,T], which is the condition suggested by Yoo et al. [14], then f,gα,β=f,g0,(1+c)β so that Lα,β2[0,T]=L0,(1+c)β2[0,T] isometrically. Since L0,(1+c)β2[0,T]=L0,β2[0, T] as vector spaces, we have Lα,β2[0,T]=L0,β2[0,T] as vector spaces, but they need not be equal isometrically. In this case, ·α,β and ·0,β are equivalent and B(Lα,β2[0,T])=B(L0,β2[0,T]) by Lemma 5.
( 3 ) It is obvious that Lα,β2[0,T]⊆L0,β2[0,T]. Suppose that α′ is bounded on [0,T]. For f∈L0,β2[0,T], we have for some M>0(19)∫0Tft2dνα,βt=∫0Tft2α+β′tβ′tdνβt≤Mf0,β2<∞
sinceβ′ is bounded and bounded away from 0. We now have L0,β2[0,T]⊆Lα,β2[0,T] and hence Lα,β2[0,T]=L0,β2[0,T] as vector spaces, but they need not be equal isometrically. In this case, ·α,β and ·0,β are equivalent and B(Lα,β2[0,T])=B(L0,β2[0,T]) by Lemma 5.Throughout the remainder of this paper, we assumeLα,β2[0,T]=L0,β2[0, T] as sets, unless otherwise specified. In this case, we have Lα,β2[0,T]=L0,β2[0,T]=L2[0,T] as vector spaces and B≡B(Lα,β2[0,T])=B(L0,β2[0,T])=B(L2[0,T]) by Lemma 5, but they need not be equal isometrically as Hilbert spaces. We also note that Lα,β2[0,T] is separable since L0,β2[0,T] is separable [10] and the two norms ·α,β and ·0,β are equivalent by Lemma 5. Let M≡M(Lα,β2[0,T]) be the class of complex measures of finite variation on Lα,β2[0,T] with B as its σ-algebra of measurable sets. If μ∈M, then we set μ=varμ, the total variation of μ over Lα,β2[0,T]. We also note that M(Lα,β2[0,T])=M(L0,β2[0,T]) and they are Banach algebras under convolution and with the total variation norm, since Lα,β2[0,T] and L0,β2[0,T] are separable, real, infinite dimensional Hilbert spaces [15].We now have the following lemma.Lemma 7.
Forμ∈M, Iα,β(f)(x) is a μ×wα,β;φ-measurable function on Lα,β2[0,T]×C[0,T]. Moreover, for wα,β;φ a.e. x∈C[0,T], Iα,β(f)(x) exists for μ a.e. f∈Lα,β2[0,T].Proof.
Note that each term of the right-hand side of (7) is B×B(C[0,T])-measurable so that it is a μ×wα,β;φ-measurable function on Lα,β2[0,T]×C[0,T]. Since Iα,β(f)(x)=limn→∞∫0Tϕn(t)dx(t), Iα,β(f)(x) is also measurable with respect to the measure μ×wα,β;φ on Lα,β2[0,T]×C[0,T]. Moreover, we have that, for wα,β;φ a.e. x∈C[0,T], Iα,β(f)(x) exists for μ a.e. f∈Lα,β2[0,T].LetSα,β;φ be the space of functions of the form(20)Fμx=∫Lα,β20,Texp⁡iIα,βfxdμffor all x∈C[0,T] for which the integral exists, where μ∈M. We note that Fμ is well-defined for wα,β;φ a.e. x∈C[0,T] by Lemma 7 and it is an integrable function of x on C[0,T]. Moreover, Sα,β;φ is a linear space over C.We have the following uniqueness theorem.Theorem 8.
Forμ∈M and Fμ∈Sα,β;φ, let Fμ and μ be related by (20). Then Fμ is uniquely determined by μ; that is, there is a unique μ∈M such that (20) holds for wα,β;φ a.e. x∈C[0,T] so that there is an one-to-one correspondence between Sα,β;φ and M.Proof.
Suppose that there are two measuresμ1 and μ2 in M such that (20) holds with Fμ1(x)=Fμ2(x) for wα,β;φ a.e. x∈C[0,T]. Let μ3=μ1-μ2∈M. Then for wα,β;φ a.e. x∈C[0,T], we have (21)∫Lα,β20,Texp⁡iIα,βgxdμ3g=0.Let r1 be any real number and let f∈Lα,β2[0,T] with f0,β≠0. Let r=r1/f0,β2. If -∞≤a<r, then let ga,r(s)=χ[a,r](s), and if 0<δ<r-a/2, then let ga,r,δ be the trapezoidal approximation (22)ga,r,δs=s-aδfora≤s<a+δ1fora+δ≤s<r-δr-sδforr-δ≤s≤r0elsewhere,and let ga,r,δ∗(s)=ga,r,δ(s)exp{-1/2f0,β2s2} for s∈R. Then ga,r,δ∗(s) is a bounded continuous function of class L1(R) having a piecewise continuous derivative of bounded variation and vanishing outside a finite interval. Consequently, its Fourier transform F(ga,r,δ∗) is a bounded continuous function of class L1(R) and we have (23)Fga,r,δ∗u=12π1/2∫Rexp⁡-isuga,r,δ∗sdmLsforu∈R,ga,r,δ∗s=12π1/2∫Rexp⁡isuFga,r,δ∗udmLufors∈R.For g∈Lα,β2[0,T], let(24)θt=gt-g,f0,βf0,β2ftfor να,β a.e. t∈[0,T]. Then (24) holds for νβ a.e. t∈[0,T] which implies θ,f0,β=g,f0,β-g,f0,β=0 and (25)θ0,β2=g0,β2-g,f0,β2f0,β2.Suppose that θ≠0 in Lα,β2[0,T]. Then θ≠0 in L0,β2[0,T]. By Theorem 2, it is not difficult to show that (26)Gfx≡Fga,r,δ∗Iα,βfxexp⁡12f0,β2Iα,βfx-∫0Tftdαt2is an integrable function of x with respect to wα,β;φ over C[0,T]. Now we have by Theorem 2 and the linearity of Iα,β(27)∫C0,TGfxexp⁡iIα,βgxdwα,β;φx=∫C0,TFga,r,δ∗Iα,βfxexp⁡12f0,β2Iα,βfx-∫0Tftdαt2×exp⁡iIα,βθx+g,f0,βf0,β2Iα,βfxdwα,β;φx=12πf0,βθ0,β∫R2Fga,r,δ∗u1exp⁡-12θ0,β2u2-∫0Tθtdαt2+iu2+g,f0,βf0,β2u1dmL2u1,u2=12πf0,β21/2∫RFga,r,δ∗u1exp⁡-12g0,β2-g,f0,β2f0,β2+i∫0Tθtdαt+g,f0,βf0,β2u1dmLu1=1f0,βga,r,δ∗g,f0,βf0,β2exp⁡-12g0,β2-g,f0,β2f0,β2+i∫0Tθtdαt=1f0,βga,r,δg,f0,βf0,β2exp⁡-12g0,β2+i∫0Tθtdαtwhich still holds for θ=0 in Lα,β2[0,T] since θ=0 in Lα,β2[0,T] implies θ=0 in both Lα,02[0,T] and L0,β2[0,T]. We now have by the Fubini theorem(28)0=∫Lα,β20,T∫C0,TGfxexp⁡iIα,βgxdwα,β;φxdμ3g=1f0,β∫Lα,β20,Tga,r,δg,f0,βf0,β2dμfg,where μf is the measure on B defined by(29)dμfdμ3g=exp⁡-12g0,β2+i∫0Tθtdαtfor μ3 a.e. g∈Lα,β2[0,T]. Note that ga,r,δg,f0,β/f0,β2≤1 for all g and δ. Letting δ→0+, we have by the dominated convergence theorem (30)0=1f0,β∫Lα,β20,Tga,rg,f0,βf0,β2dμfg.Also, letting a→-∞, we have by the dominated convergence theorem again (31)0=1f0,β∫Lα,β20,Tg-∞,rg,f0,βf0,β2dμfgso that if (32)Er1=g∈Lα,β20,T:g,f0,β≤r1,then we have μf(Er1)=0. If f0,β=0, then Er1 is either ∅ or Lα,β2[0,T]. Let μ0 be the measure defined by (29) with replacing f by 1. If Er1=∅, then μ0(Er1)=0. If Er1=Lα,β2[0,T], then (33)μ0Er1=limn→∞⁡μ0g∈Lα,β20,T:g,10,β≤n=0by the above argument since 10,β>0. Solving (29) for μ3, we have for all f∈Lα,β2[0,T](34)μ3Er1=∫Er1exp⁡12g0,β2-i∫0Tθtdαtdμfg=0.Since Lα,β2[0,T]=L0,β2[0,T], B is generated by the sets of the form Er1 so that we have μ1(E)-μ2(E)=μ3(E)=0 for all E∈B, that is, μ1=μ2, which completes the proof.Remark 9.
A difference between the proof of Theorem8 and the proof of Theorem 2.1 in [1] is to use the additional condition Lα,β2[0,T]=L0,β2[0,T] as sets in Theorem 8. In order to apply Theorem 2 to the proof of Theorem 8, we need an orthonormalization process in the Hilbert space L0,β2[0,T] for the functions in the Hilbert space Lα,β2[0,T]. See (2) of Lemma 5.Example 10.
For some constantc≥0, let α(t)=cβ(t) for all t∈[0,T]. Then we have Lα,β2[0,T]=L0,(1+c)β2[0,T] isometrically. Replacing α and β by the zero function and (1+c)β, respectively, in the proof of Theorem 8, we can show that Fμ is uniquely determined by μ∈M.Corollary 11.
Suppose thatα is a constant function on [0,T]. Let F(·,y)∈Sα,β;φ  for each y∈Y, where Y is a measure space, and let F(x,y) be a measurable function of (x,y) on C[0,T]×Y. Let {μy} be the family of measures corresponding to F(·,y) so that for each y∈Y and μy∈M, (35)Fx,y=∫Lα,β20,Texp⁡iIα,βfxdμyffor wα,β;φ a.e. x∈C[0,T]. Then, for each E∈B, μy(E) is measurable as a function of y on Y.Proof.
This corollary follows from the fact that the method of proof of Theorem8 has provided a method for explicitly constructing μy in terms of F(·,y). We will use the same assumptions and notations in the proof of Theorem 8. Indeed, for f∈Lα,β2[0,T] with f0,β≠0, let (36)Ka,r,δfy=∫C0,TGfxFx,ydwα,β;φx.It is a measurable function of y by the assumption. Then we have (37)Ka,r,δfy=∫Lα,β20,T∫C0,TGfxexp⁡iIα,βgxdwα,β;φxdμyg=1f0,β∫Lα,β20,Tga,r,δg,f0,βf0,β2dζyg,where dζy/dμy(g)=exp-1/2g0,β2 for μy  a.e.  g∈Lα,β2[0,T]. Letting δ→0+, we have by the dominated convergence theorem (38)limδ→0+⁡Ka,r,δfy=1f0,β∫Lα,β20,Tga,rg,f0,βf0,β2dζygwhich is still a measurable function of y. Letting a→-∞, we have by the dominated convergence theorem again (39)lima→-∞⁡limδ→0+⁡Ka,r,δfy=1f0,β∫Lα,β20,Tg-∞,rg,f0,βf0,β2dζygso that if (40)Er1=g∈Lα,β20,T:g,f0,β≤r1,then ζy(Er1) is a measurable function of y on Y. If f0,β=0, Er1 is either ∅ or Lα,β2[0,T]. If Er1=∅, then 0=ζy(Er1) which is a measurable function of y. If Er1=Lα,β2[0,T], then we have (41)ζyEr1=limn→∞⁡ζyg∈Lα,β20,T:g,10,β≤nwhich is also a measurable function of y by the above argument. Since B is generated by the sets of the form Er1, ζy(E) is a measurable function of y on Y for all E∈B. Solving μy for ζy, we have for all E∈B(42)μyE=∫Eexp⁡12g0,β2dζygwhich is a measurable function of y on Y, since the integrand can be expressed by a limit of simple functions on Lα,β2[0,T].Definition 12.
Ifμ∈M and Fμ∈Sα,β;φ are related by (20), we define a norm Fμ of Fμ by Fμ=μ. We note that Fμ is well defined by Theorem 8.Using the same process used in the proof of Theorem 2.3 in [1], we can prove the following theorem.Theorem 13.
S α , β ; φ is a normed algebra. Moreover, the correspondence μ→Fμ given by (20) for μ∈M is an algebra isometric isomorphism between M and Sα,β;φ, so that Sα,β;φ is a Banach algebra.Remark 14.
( 1 ) One can show that Sα,β;φ is a Banach space without the isomorphism in Theorem 13. For more details, see the proof of Theorem 2.2 in [1].
( 2 ) If α(t)=0 and β(t)=t for all t∈[0,T], and φ is the Dirac measure concentrated at 0, then we can obtain all results of Section 2 in [1] from Theorems 8 and 13 so that Sα,β;φ generalizes the Banach algebra S introduced by Cameron and Storvick [1].


## 4. The Fresnel Class with Its Equivalent Spaces

In this section, we establish that the Fresnel class [4, 5] is isometrically isomorphic to Sα,β;φ. Some ideas of the results in this section follow from [4, 7], but the detailed proofs of the results in this section are quite different from those in [4, 7].LetH be the space of real-valued functions v on [0,T] which are absolutely continuous with v(0)=0 and v′/β′∈Lα,β2[0,T]. Define D:H→Lα,β2[0,T] by Dv=v′/β′ for v∈H, and an inner product ·,·H on H by (43)v1,v2H=Dv1,Dv20,β=∫0Tv1′tv2′tβ′tdmLtfor v1,v2∈H, which can be justified by the fact Lα,β2[0,T]=L0,β2[0,T] as vector spaces.We now have the following results.Lemma 15.
· , · H is well defined on H×H and (H,·,·H) is a real inner product space.Proof.
By the Minkowski inequality, it is clear thatH is a real linear space. Let v1,v2∈H. If Dv1(t)=z1(t) and Dv2(t)=z2(t) for να,β a.e. t∈[0,T], then Dv1(t)=z1(t) and Dv2(t)=z2(t) for νβ a.e. t∈[0,T] so that we have (44)v1,v2H=Dv1,Dv20,β=z1,z20,β,that is, ·,·H is well defined. It is obvious that ·,·H is a symmetric, bilinear, nonnegative definite form on H×H. Moreover, if v,vH=0 for v∈H, then v′t2/β′(t)=0 for mL a.e. t∈[0,T] so that v′(t)=0 for mL a.e. t∈[0,T] since β′ is bounded away from 0. Now we have for all t∈[0,T](45)vt=∫0tv′sdmLs+v0=0so that ·,·H is positive definite. Hence, the theorem is proved.Lemma 16.
DefineI:Lα,β2[0,T](=L0,β2[0,T])→H by (46)Ift=∫0tfsβ′sdmLsfor f∈Lα,β2[0,T] and for t∈[0,T]. Then D:H→Lα,β2[0,T](=L0,β2[0,T]) is a vector space isomorphism and I=D-1. Moreover, D and I are isometric isomorphisms between H and L0,β2[0,T] so that H is a separable, infinite dimensional Hilbert space, and both D:H→Lα,β2[0,T] and I:Lα,β2[0,T]→H are bounded linear operators.Proof.
Clearly,D and I are well defined and linear. Moreover, DIf=f for all f∈Lα,β2[0,T] and (IDv)(t)=v(t)-v(0)=v(t) for all t∈[0,T] if v∈H. These equalities tell us that IDv=v for all v∈H, which implies that D is a vector space isomorphism and I=D-1. For v1,v2∈H, we have v1,v2H=Dv1,Dv20,β by the definition of ·,·H so that D and I are isometric isomorphisms between H and L0,β2[0,T]. Now, H is a separable, infinite dimensional Hilbert space since L0,β2[0,T] is. Moreover, for f∈Lα,β2[0,T], we have (47)IfH=DIf0,β=f0,β≤fα,βso that I:Lα,β2[0,T]→H is bounded. Since I is onto, I=D-1 is an open map by the open mapping theorem so that D is bounded.Remark 17.
( 1 ) We have Lα,β2[0,T]=L0,β2[0,T] as vector spaces, but we note that it need not hold isometrically as Hilbert spaces since the inner products ·,·α,β and ·,·0,β need not be equal. Thus, D and I need not be isometric isomorphisms between H and Lα,β2[0,T] in Lemma 16.
( 2 ) Both D:H→Lα,β2[0,T] and I:Lα,β2[0,T]→H are bounded linear operators by Lemma 16 so that D:H→Lα,β2[0,T] and I:Lα,β2[0,T]→H are B(H)-B and B-B(H) measurable, respectively, where B(H) denotes the Borel σ-algebra of H.LetM(H) be the collection of C-valued, countably additive measures of finite variation on B(H). Define D:M(H)→M by Dσ=σ∘D-1 for σ∈M(H) and I:M→M(H) by Iμ=μ∘I-1 for μ∈M. We note that M=M(Lα,β2[0,T])=M(L0,β2[0,T]) with B as its σ-algebra of measurable sets and both D and I are well defined by the arguments in (2) of Remark 17. Moreover, M(H) is a Banach algebra under convolution and with the total variation norm, since H is a separable, infinite dimensional, real Hilbert space by Lemmas 15 and 16 [15].Lemma 18.
D is a Banach algebra isometric isomorphism between M(H) and M with D-1=I.Proof.
Clearly,D is linear and bijective with D-1=I. To prove that D preserves convolutions, let σ1,σ2∈M(H). For B∈B, we have (48)Dσ1∗Dσ2B=∫Lα,β20,T∫Lα,β20,TχBf+gdDσ1fdDσ2g.For each g∈Lα,β2[0,T], define Tg:Lα,β2[0,T]→Lα,β2[0,T] by Tg(f)=f+g for f∈Lα,β2[0,T]. Then Tg is continuous so that it is B-B measurable and Tg-1(B)=B-g∈B. Now we have by the change of variable theorem (49)Dσ1∗Dσ2B=∫Lα,β20,T∫Lα,β20,TχBfdDσ1∘Tg-1fdDσ2g=∫Lα,β20,TDσ1∘Tg-1BdDσ2g=∫Lα,β20,Tσ1D-1B-gdσ2∘D-1g=∫Hσ1D-1B-udσ2u.For each u∈H, define Su:H→H by Su(v)=v+u for v∈H. By the same argument as above, we have (50)Dσ1∗Dσ2B=∫Hσ1Su-1D-1Bdσ2u=∫H∫HχD-1Bvdσ1∘Su-1vdσ2u=∫H∫HχD-1Bv+udσ1vdσ2u=σ1∗σ2D-1B=Dσ1∗σ2B,that is, Dσ1*(Dσ2)=D(σ1∗σ2) so that D is an algebra isomorphism. To complete the proof, we have to show Dσ=σ for σ∈M(H), where · denotes the total variation norm on each space. Indeed, we have by the Riesz representation theorem and the change of variable theorem (51)Dσ=sup⁡∫Lα,β20,TvfdDσf:v∈CLα,β20,T,supf∈Lα,β20,T⁡vf=1=sup⁡∫Lα,β20,Tvfdσ∘D-1f:v∈CLα,β20,T,supf∈Lα,β20,T⁡vf=1=sup⁡∫Hv∘Dudσu:v∈CLα,β20,T,supf∈Lα,β20,T⁡vf=1,where C(Lα,β2[0,T]) denotes the set of all C-valued, bounded, and continuous functions on Lα,β2[0,T]. Let C(H) denote the set of all C-valued, bounded, and continuous functions on H. We now have that if v∈C(Lα,β2[0,T]), then v∘D∈C(H) and (52)sup⁡v∘Du:u∈H=sup⁡vf:f∈Lα,β20,Tsince D is a continuous isomorphism from H onto Lα,β2[0,T] by Lemma 16. Similarly, if g∈C(H), then g∘I∈C(Lα,β2[0,T]) and (53)sup⁡g∘If:f∈Lα,β20,T=sup⁡gu:u∈Hby Lemma 16. Thus, we have (54)Dσ=sup⁡∫Hgudσu:g∈CH,supu∈H⁡gu=1=σ,which completes the proof.LetF(σ) denote the Fourier transform of a measure σ. We note that H is a proper subset of C[0,T]. Now the following theorem holds.Theorem 19.
Forσ∈M(H), let μ=Dσ∈M. Then we have F(σ)=F(μ)∘D=FμH, where μ and Fμ are related by (20).Proof.
Sinceβ′ is continuous on [0,T], we can take M>0 such that β′(t)≤M for all t∈[0,T]. Then it follows that for all v∈H(55)∫0Tv′t2dβt≤M2Dv0,β2=M2vH2<∞.This means that v′∈L0,β2[0,T] for all v∈H. Thus, by Lemma 1, the integral of the right-hand side of (20) exists; that is, Fμ(v) exists. We now have by the change of variable theorem (56)FμHv=∫Lα,β20,Texp⁡i∫0Tftv′tdmLtdμf=∫Lα,β20,Texp⁡i∫0TftDvtdβtdσ∘D-1f=∫Hexp⁡i∫0TDutDvtdβtdσu=∫Hexp⁡iu,vHdσu=Fσvand since Lα,β2[0,T]=L0,β2[0,T] as vector spaces with the same Borel σ-algebra B, we have (57)FμHv=∫L0,β20,Texp⁡i∫0TftDvtdβtdμf=∫L0,β20,Texp⁡if,Dv0,βdμf=FμDv,which completes the proof.LetF(H) be the Fresnel class [4, 15] defined by (58)FH=Fσ:σ∈MH.Then the correspondence σ→F(σ) is injective and carries convolution to point-wise multiplication. Letting F(σ)=σ, we have that M(H) is isometrically isomorphic to F(H) and F(H) is a Banach algebra [15]. The Fresnel integral FI[F(σ)] is defined for F(σ)∈F(H) by the formula (59)FIFσ=∫Hexp⁡-i2hH2dσh.Now we have the following isomorphism theorem.Theorem 20.
The mapϕ:Sα,β;φ→F(H) defined by ϕ(Fμ)=F(σ)=FμH for μ∈M, where σ=Iμ∈M(H), is an isometric isomorphism as Banach algebras. The inverse ϕ-1 of ϕ is given by ϕ-1[F(σ)]=Fμ for σ∈M(H), where μ=Dσ∈M.Proof.
LetFμ∈Sα,β;φ with μ∈M and let σ=Iμ. By Theorem 13, the map ϕ1:Sα,β;φ→M defined by ϕ1(Fμ)=μ, is an isometric isomorphism between Banach algebras. By Lemma 18, the map I:M→M(H) is also an isometric isomorphism between Banach algebras. Let ϕ2:M(H)→F(H) be defined by ϕ2(σ)=F(σ) for σ∈M(H). By Proposition 2.3 of [15], ϕ2 also is an isometric isomorphism between Banach algebras, too. Now we have ϕ=ϕ2∘I∘ϕ1 and this theorem follows by Theorem 19.Theorem 21.
Forμ∈M and Fμ∈Sα,β;φ, let Fμ and μ be related by (20). Let σ=Iμ. Then we have the following: (1)
Ifλ>0, then(60)JFλ=∫Lα,β20,Texp⁡-12λf0,β2+iλ-1/2∫0Tftdαtdμf=∫Hexp⁡-12λvH2+iλ-1/2∫0TDvtdαtdσv.(2)
Ifλ>0 and α′∈Lα,β2[0,T] with α(0)=0 (or equivalently α∈H), then(61)JFλ=∫Hexp⁡-12λvH2+iλ-1/2v,αHdσv.(3)
If there existsM>0 satisfying(62)∫Lα,β20,Texp⁡Reiλ-1/2∫0Tftdαtdμf≤Mfor anyλ∈C+, then Ewα,β;φanwλ[Fμ] is given by the right-hand sides of (60). Moreover, if (62) holds for λ∈{λ∈C:Reλ≥0,λ≠0}, then for any nonzero real q, Ewα,β;φanfq[F] is given by the right-hand sides of (60) replacing λ by -iq. In addition, if α∈H, then Ewα,β;φanfq[F] is given by the right-hand side of (61) replacing λ by -iq.Proof.
Sinceμ=I-1σ=Dσ=σ∘D-1, we have for λ>0(63)JFλ=∫C0,TFμλ-1/2xdwα,β;φx=∫Lα,β20,T∫C0,Texp⁡iλ-1/2Iα,βfxdwα,β;φxdμf=∫Lα,β20,Texp⁡-12λf0,β2+iλ-1/2∫0Tftdαtdσ∘D-1f=∫Hexp⁡-12λvH2+iλ-1/2∫0TDvtdαtdσvby the Fubini theorem and the change of variable theorem, which proves (60). Moreover, it is not difficult to prove that α′∈Lα,β2[0,T] with α(0)=0 if and only if α∈H since β′ is bounded and bounded away from 0. Now if α′∈Lα,β2[0,T] with α(0)=0, then we have (64)JFλ=∫Hexp⁡-12λvH2+iλ-1/2∫0TDvtDαtdβtdσv=∫Hexp⁡-12λvH2+iλ-1/2v,αHdσv,which proves (61). If (62) holds, then we have (3) of this theorem by the definition of the generalized analytic Feynman wα,β;φ-integral and the dominated convergence theorem.Remark 22.
Applying(3) of Theorem 21, we can also obtain Theorem 3.1 of [2].LettingM=μ in (62) of Theorem 21, we now have the following corollary which is one of our main results.Corollary 23.
Forμ∈M and Fμ∈Sα,β;φ, let Fμ and μ be related by (20). If α(t)=0 for all t∈[0,T], then we have for any λ∈C+(65)Ewα,β;φanwλFμ=∫Lα,β20,Texp⁡-12λf0,β2dμf=∫Hexp⁡-12λvH2dσv,where σ=Iμ. Moreover, for any nonzero real q, Ewα,β;φanfq[F] is given by the right-hand sides of (65) replacing λ by -iq. In this case, we have Ewα,β;φanf1Fμ=FI[F(σ)].



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