# Generalized Analytic Fourier-Feynman Transform of Functionals in a Banach AlgebraℱA1,A2a,b

Authors: Jae Gil Choi; David Skoug; Seung Jun Chang
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We introduce the Fresnel type classℱA1,A2a,b. We also establish the existence of the generalized analytic Fourier-Feynman transform for functionals in the Banach algebra ℱA1,A2a,b.


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## Body

## 1. Introduction

LetH be a separable Hilbert space and let ℳ(H) be the space of all complex-valued Borel measures on H. The Fourier transform of σ in ℳ(H) is defined by
(1)f(σ)(h′)≡σ^(h′)=∫Hexp{i〈h,h′〉}dσ(h),h′∈H.
The set of all functions of the form (1) is denoted by ℱ(H) and is called the Fresnel class of H. Let (H,B,ν) be an abstract Wiener space. It is known [1, 2] that each functional of the form (1) can be extended to B uniquely by
(2)σ^(x)=∫Hexp{i(h,x)~}dσ(h),x∈B,
where (·,·)~ is a stochastic inner product between H and B. The Fresnel class ℱ(B) of B is the space of (equivalence classes of) all functionals of the form (2). There has been a tremendous amount of papers and books in the literature on the Fresnel integral theory and Fresnel classes ℱ(B) and ℱ(H) on abstract Wiener and Hilbert spaces. For an elementary introduction see [3, Chapter 20].Furthermore, in [1], Kallianpur and Bromley introduced a larger class ℱA1,A2 than the Fresnel class ℱ(B) and showed the existence of the analytic Feynman integral of functionals in ℱA1,A2 for a successful treatment of certain physical problems by means of a Feynman integral. The Fresnel class ℱA1,A2 of B2 is the space of (equivalence classes of) all functionals on B2 of the following form:
(3)F(x1,x2)=∫Hexp{∑j=12‍i(Aj1/2h,xj)~}dσ(h),
where A1 and A2 are bounded, nonnegative, and self-adjoint operators on H and σ∈ℳ(H).In this paper we study the functionalsF of the form (3) with (x1,x2) in a very general function space Ca,b2[0,T]≡Ca,b[0,T]×Ca,b[0,T]. The function space Ca,b[0,T], induced by generalized Brownian motion process, was introduced by Yeh [4, 5] and was used extensively in [6–13]. In this paper, we also construct a concrete theory of the generalized analytic Fourier-Feynman transform (GFFT) of functionals in a generalized Fresnel type class defined on Ca,b2[0,T]. Other work involving GFFT theories on Ca,b[0,T] include [6, 7, 9, 12, 13].The Wiener process used in [1, 2, 14–17] is stationary in time and is free of drift while the stochastic process used in this paper, as well as in [4, 6–13, 18], is nonstationary in time and is subject to a drift a(t).It turns out, as noted in Remark7 below, that including a drift term a(t) makes establishing the existence of the GFFT of functionals on Ca,b2[0,T] very difficult. However, when a(t)≡0 and b(t)=t on [0,T], the general function space Ca,b[0,T] reduces to the Wiener space C0[0,T].


## 2. Definitions and Preliminaries

Leta(t) be an absolutely continuous real-valued function on [0,T] with a(0)=0, a′(t)∈L2[0,T], and let b(t) be a strictly increasing, continuously differentiable real-valued function with b(0)=0 and b′(t)>0 for each t∈[0,T]. The generalized Brownian motion process Y determined by a(t) and b(t) is a Gaussian process with mean function a(t) and covariance function r(s,t)=min{b(s),b(t)}. For more details, see [6, 10, 12]. By Theorem 14.2 in [5], the probability measure μ induced by Y, taking a separable version, is supported by Ca,b[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, (Ca,b[0,T],ℬ(Ca,b[0,T]),μ) is the function space induced by Y where ℬ(Ca,b[0,T]) is the Borel σ-algebra of Ca,b[0,T]. We then complete this function space to obtain (Ca,b[0,T],𝒲(Ca,b[0,T]),μ), where 𝒲(Ca,b[0,T]) is the set of all Wiener measurable subsets of Ca,b[0,T].A subsetB of Ca,b[0,T] is said to be scale-invariant measurable provided ρB is 𝒲(Ca,b[0,T])-measurable for all ρ>0, and a scale-invariant measurable set N is said to be a scale-invariant null set provided μ(ρN)=0 for all ρ>0. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s-a.e.). A functional F is said to be scale-invariant measurable provided F is defined on a scale-invariant measurable set and F(ρ·) is 𝒲(Ca,b[0,T])-measurable for every ρ>0. If two functionals F and G defined on Ca,b[0,T] are equal s-a.e., we write F≈G.LetLa,b1,2[0,T] be the space of Lebesgue measurable functions on [0,T] given by
(4)La,b1,2[0,T]={v:∫0T|v(s)|2db(s)<∞,hhhhhhhhhhh∫0T|v(s)|d|a|(s)<∞},
where |a|(·) is the total variation function of a(·). Then La,b1,2[0,T] is a separable Hilbert space with inner product defined by
(5)(u,v)La,b1,2=∫0Tu(t)v(t)db(t)+(∫0Tu(t)da(t))(∫0T‍v(t)da(t)).
In particular, note that ∥u∥La,b1,2≡[(u,u)La,b1,2]1/2=0 if and only if u(t)=0 a.e. on [0,T].Let{ϕj}j=1∞ be a complete orthonormal set in La,b1,2[0,T], each of whose elements is of bounded variation on [0,T] such that
(6)∫0Tϕj(t)ϕk(t)db(t)={0,j≠k1,j=k.
Then for each v∈La,b1,2[0,T], the Paley-Wiener-Zygmund (PWZ) stochastic integral 〈v,x〉 is defined by the following formula:
(7)〈v,x〉=limn→∞∫0T‍∑j=1n(v,ϕj)a,bϕj(t)dx(t)
for all x∈Ca,b[0,T] for which the limit exists; one can show that for each v∈La,b1,2[0,T], the PWZ stochastic integral 〈v,x〉 exists for μ-a.e. x∈Ca,b[0,T], and if v is of bounded variation on [0,T], then the PWZ stochastic integral 〈v,x〉 equals the Riemann-Stieltjes integral ∫0T‍v(t)dx(t) for s-a.e. x∈Ca,b[0,T].Remark 1.
(1) For eachv∈La,b1,2[0,T], the PWZ stochastic integral 〈v,x〉 is a Gaussian random variable on Ca,b[0,T] with mean ∫0T‍v(s)da(s) and variance ∫0T‍v2(s)db(s).
(2) For allu, v∈La,b1,2[0,T],
(8)∫Ca,b[0,T]〈u,x〉〈v,x〉dμ(x)=∫0Tu(s)v(s)db(s)hhh+(∫0Tu(s)da(s))(∫0Tv(s)da(s)).
Hence, we see that for all u, v∈La,b1,2[0,T], ∫0T‍u(s)v(s)db(s)=0 if and only if 〈u,x〉 and 〈v,x〉 are independent random variables.The following Cameron-Martin subspace ofCa,b[0,T] plays an important role throughout this paper.Let(9)Ca,b′[0,T]={∫w∈Ca,b[0,T]:w(t)=∫0tz(s)db(s)hhhforsomez∈La,b1,2[0,T]∫}.
For w∈Ca,b′[0,T], let D:Ca,b′[0,T]→La,b1,2[0,T] be defined by the following formula:
(10)Dw(t)=dλwdλb(t),
where dλw/dλb denotes the Radon-Nikodym derivative of the signed measure λw induced by w, with respect to the Borel-Stieltjes measure λb induced by b. Then Ca,b′≡Ca,b′[0,T] with inner product
(11)(w1,w2)Ca,b′=∫Ca,b[0,T]〈Dw1,x〉〈Dw2,x〉dμ(x)
is a separable Hilbert space.Using (8), we observe that the linear operator given by (10) is an isometry. In fact, the inverse operator D-1:La,b1,2[0,T]→Ca,b′[0,T] is given by
(12)(D-1z)(t)=∫0tz(s)db(s).
Moreover, the triple (Ca,b′[0,T],Ca,b[0,T],μ) becomes an abstract Wiener space.Throughout this paper, forw∈Ca,b′[0,T], we will use the notation (w,x)~ instead of 〈Dw,x〉. We also use the following notations for w1, w2, w∈Ca,b′[0,T]:
(13)(w1,w2)b=∫0TDw1(t)Dw2(t)db(t),∥w∥b=(w,w)b.
Then Ca,b′[0,T] with the inner product given by (13) is also a separable Hilbert space. It is easy to see that the two norms ∥·∥Ca,b′ and ∥·∥b are equivalent. Furthermore, we have the following assertions.  (i)
a(·) is an element of Ca,b′[0,T].(ii)
For eachw∈Ca,b′[0,T], the random variable x↦(w,x)~ is Gaussian with mean (w,a)b and variance ∥w∥b2.(iii)
(w,αx)~=(αw,x)~=α(w,x)~ for any real number α, w∈Ca,b′[0,T] and x∈Ca,b[0,T].(iv)
Let{w1,…,wn} be a subset of Ca,b′[0,T] such that ∫0TDwi(t)Dwj(t)db(t)=δij, where δij is the Kronecker delta. Then the random variables (wi,x)~’s are independent.In this paper, we adopt as much as possible the definitions and notations used in [7, 9, 12, 13] for the definitions of the generalized analytic Feynman integral and the GFFT of functionals on Ca,b[0,T].The following integration formula is used several times in this paper:(14)∫ℝexp{-αu2+βu}du=παexp{β24α}
for complex numbers α and β with Re(α)>0.


## 3. The GFFT of Functionals in a Banach AlgebraℱA1,A2a,b

Letℳ(Ca,b′[0,T]) be the space of complex-valued, countably additive (and hence finite) Borel measures on Ca,b′[0,T]. ℳ(Ca,b′[0,T]) is a Banach algebra under the total variation norm and with convolution as multiplication.We define the Fresnel type classℱ(Ca,b[0,T]) of functionals on Ca,b[0,T] as the space of all stochastic Fourier transforms of elements of ℳ(Ca,b′[0,T]); that is, F∈ℱ(Ca,b[0,T]) if and only if there exists a measure f in ℳ(Ca,b′[0,T]) such that
(15)F(x)=∫Ca,b′[0,T]exp{i(w,x)~}df(w)
for s-a.e. x∈Ca,b[0,T]. More precisely, since we will identify functionals which coincide s-a.e. on Ca,b[0,T], ℱ(Ca,b[0,T]) can be regarded as the space of all s-equivalence classes of functionals of the form (15).The Fresnel type classℱ(Ca,b[0,T]) is a Banach algebra with norm
(16)∥F∥=∥f∥=∫Ca,b′[0,T]d|f|(w).
In fact, the correspondence f↦F is injective, carries convolution into pointwise multiplication and is a Banach algebra isomorphism where f and F are related by (15).Remark 2.
The Banach algebraℱ(Ca,b[0,T]) contains several interesting functions which arise naturally in quantum mechanics. Let ℳ(ℝ) be the class of ℂ-valued countably additive measures on ℬ(ℝ), the Borel class of ℝ. For ν∈ℳ(ℝ), the Fourier transform ν^ of ν is a complex-valued function defined on ℝ by the following formula:
(17)ν^(u)=∫ℝexp{iuv}dν(v).
Let𝒢 be the set of all complex-valued functions on [0,T]×ℝ of the form θ(s,u)=σ^s(u), where {σs:0≤s≤T} is a family from ℳ(ℝ) satisfying the following two conditions:(i)
for everyE∈ℬ(ℝ), σs(E) is Borel measurable in s,(ii)
∫0T‍∥σs∥db(s)<+∞.
Letθ∈𝒢 and let H be given by
(18)H(x)=exp{∫0Tθ(t,x(t))dt}
for s-a.e. x∈Ca,b[0,T]. Then, using the methods similar to those used in [18], we can show that the function θ(t,u) is Borel-measurable and that θ(t,x(t)), ∫0Tθ(t,x(t))dt, and H(x) are elements of ℱ(Ca,b[0,T]). These facts are relevant to quantum mechanics where exponential functions play a prominent role.LetA be a nonnegative self-adjoint operator on Ca,b′[0,T] and f any complex measure on Ca,b′[0,T]. Then the functional
(19)F(x)=∫Ca,b′[0,T]exp{i(A1/2w,x)~}df(w)
belongs to ℱ(Ca,b[0,T]) because it can be rewritten as ∫Ca,b′[0,T]exp{i(w,x)~}dfA(w) for fA=f∘(A1/2)-1. Let A be self-adjoint but not nonnegative. Then A has the form
(20)A=A+-A-,
where both A+ and A- are bounded, nonnegative, and self-adjoint operators.In this section we will extend the ideas of [1] to obtain expressions of the generalized analytic Feynman integral and the GFFT of functionals of the form (19) when A is no longer required to be nonnegative. To do this, we will introduce definitions and notations analogous to those in [7, 12, 13].Let𝒲(Ca,b2[0,T]) denote the class of all Wiener measurable subsets of the product function space Ca,b2[0,T]. A subset B of Ca,b2[0,T] is said to be scale-invariant measurable provided {(ρ1x1,ρ2x2):(x1,x2)∈B} is 𝒲(Ca,b2[0,T])-measurable for every ρ1>0 and ρ2>0, and a scale-invariant measurable subset N of Ca,b2[0,T] is said to be scale-invariant null provided (μ×μ)({(ρ1x1,ρ2x2):(x1,x2)∈N})=0 for every ρ1>0 and ρ2>0. A property that holds except on a scale-invariant null set is said to hold s-a.e. on Ca,b2[0,T]. A functional F on Ca,b2[0,T] is said to be scale-invariant measurable provided F is defined on a scale-invariant measurable set and F(ρ1·,ρ2·) is 𝒲(Ca,b2[0,T])-measurable for every ρ1>0 and ρ2>0. If two functionals F and G defined on Ca,b2[0,T] are equal s-a.e., then we write F≈G.We denote the product function space integral of a𝒲(Ca,b2[0,T])-measurable functional F by
(21)E[F]≡Ex→[F(x1,x2)]=∫Ca,b2[0,T]F(x1,x2)d(μ×μ)(x1,x2)
whenever the integral exists.Throughout this paper, letℂ, ℂ+ and ℂ~+ denote the set of complex numbers, complex numbers with positive real part, and nonzero complex numbers with nonnegative real part, respectively. Furthermore, for all λ∈ℂ~+, λ-1/2(orλ1/2) is always chosen to have positive real part. We also assume that every functional F on Ca,b2[0,T] we consider is s-a.e. defined and is scale-invariant measurable.Definition 3.
Letℂ+2≡ℂ+×ℂ+ and let ℂ~+2≡ℂ~+×ℂ~+. Let F:Ca,b2[0,T]→ℂ be such that for each λ1>0 and λ2>0, the function space integral
(22)J(λ1,λ2)=∫Ca,b2[0,T]F(λ1-1/2x1,λ2-1/2x2)d(μ×μ)(x1,x2)
exists. If there exists a function J*(λ1,λ2) analytic in ℂ+2 such that J*(λ1,λ2)=J(λ1,λ2) for all λ1>0 and λ2>0, then J*(λ1,λ2) is defined to be the analytic function space integral of F over Ca,b2[0,T] with parameter λ→=(λ1,λ2), and for λ→∈ℂ+2 we write
(23)Eanλ→[F]≡Ex→anλ→[F(x1,x2)]≡Ex1,x2an(λ1,λ2)[F(x1,x2)]=J*(λ1,λ2).
Let q1 and q2 be nonzero real numbers. Let F be a functional such that Eanλ→[F] exists for all λ→∈ℂ+2. If the following limit exists, we call it the generalized analytic Feynman integral of F with parameter q→=(q1,q2), and we write
(24)Eanfq→[F]≡Ex→anfq→[F(x1,x2)]≡Ex1,x2anf(q1,q2)[F(x1,x2)]=limλ→→-iq→Eanλ→[F],
where λ→=(λ1,λ2)→-iq→=(-iq1,-iq2) through values in ℂ+2.Definition 4.
Letq1 and q2 be nonzero real numbers. For λ→=(λ1,λ2)∈ℂ+2 and (y1,y2)∈Ca,b2[0,T], let
(25)Tλ→(F)(y1,y2)≡T(λ1,λ2)(F)(y1,y2)=Ex→anλ→[F(y1+x1,y2+x2)].
For p∈(1,2], we define the Lp analytic GFFT, Tq→(p)(F) of F, by the formula (λ→∈ℂ+2)(26)Tq→(p)(F)(y1,y2)≡T(q1,q2)(p)(F)(y1,y2)=limλ→→-iq→Tλ→(F)(y1,y2)
if it exists; that is, for each ρ1>0 and ρ2>0,
(27)limλ→→-iq→∫Ca,b2[0,T]‍|Tq→(p)Tλ→(F)(ρ1y1,ρ2y2)hhhhhhhhhhhh-Tq→(p)(F)(ρ1y1,ρ2y2)|p′d(μ×μ)(y1,y2)=0,
where 1/p+1/p′=1. We define the L1 analytic GFFT, Tq→(1)(F) of F, by the formula (λ→∈ℂ+2)(28)Tq→(1)(F)(y1,y2)=limλ→→-iq→Tλ→(F)(y1,y2)
if it exists.We note that for1≤p≤2, Tq→(p)(F) is defined only s-a.e. We also note that if Tq→(p)(F) exists and if F≈G, then Tq→(p)(G) exists and Tq→(p)(G)≈Tq→(p)(F). Moreover, from Definition 4, we see that for q1, q2∈ℝ-{0}(29)Ex→anfq→[F(x1,x2)]=Tq→(1)(F)(0,0).Next we give the definition of the generalized Fresnel type classℱA1,A2a,b.Definition 5.
LetA1 and A2 be bounded, nonnegative, and self-adjoint operators on Ca,b′[0,T]. The generalized Fresnel type class ℱA1,A2a,b of functionals on Ca,b2[0,T] is defined as the space of all functionals F on Ca,b2[0,T] of the following form:
(30)F(x1,x2)=∫Ca,b′[0,T]‍exp{∑j=12‍i(Aj1/2w,xj)~}df(w)
for some f∈ℳ(Ca,b′[0,T]). More precisely, since we identify functionals which coincide s-a.e. on Ca,b2[0,T], ℱA1,A2a,b can be regarded as the space of all s-equivalence classes of functionals of the form (30).Remark 6.
(1) In Definition5, let A1 be the identity operator on Ca,b′[0,T] and A2≡0. Then ℱA1,A2a,b is essentially the Fresnel type class ℱ(Ca,b[0,T]), and for p∈[1,2] and nonzero real numbers q1 and q2,
(31)T(q1,q2)(p)(F)(y1,y2)=Tq1(p)(F0)(y1),
if it exists, where F0(x1)=F(x1,x2) for all (x1,x2)∈Ca,b2[0,T] and Tq1(p)(F0) means the Lp analytic GFFT on Ca,b[0,T]; see [6, 12].
(2) The mapf↦F defined by (30) sets up an algebra isomorphism between ℳ(Ca,b′[0,T]) and ℱA1,A2a,b if Ran(A1+A2) is dense in Ca,b′[0,T], where Ran indicates the range of an operator. In this case ℱA1,A2a,b becomes a Banach algebra under the norm ∥F∥=∥f∥. For more details see [1].Remark 7.
LetF be given by (30). In evaluating Ex→[F(λ1-1/2x1,λ2-1/2x2)] and T(λ1,λ2)(F)(y1,y2)=Ex→[F(y1+λ1-1/2x1,y2+λ2-1/2x2)] for λ1>0 and λ2>0, the expression
(32)ψ(λ→;A→;w)≡ψ(λ1,λ2;A1,A2;w)=exp{∑j=12‍[-(Ajw,w)b2λj+iλj-1/2(Aj1/2w,a)b]}
occurs. Clearly, for λj>0, j∈{1,2}, |ψ(λ→;A→;w)|≤1 for all w∈Ca,b′[0,T]. But for λ→∈ℂ~+2, |ψ(λ→;A→;w)| is not necessarily bounded by 1.
Note that for eachλj∈ℂ~+ with λj=αj+iβj, j∈{1,2},
(33)λj1/2=αj2+βj2+αj2+iβj|βj|αj2+βj2-αj2,λj-1/2=αj2+βj2+αj2(αj2+βj2)-iβj|βj|αj2+βj2-αj2(αj2+βj2).
Hence, for λj∈ℂ~+ with λj=αj+iβj, j∈{1,2},
(34)|ψ(λ→;A→;w)|=exp{αj2+βj2-αj2(αj2+βj2)∑j=12‍[αj2+βj2-αj2(αj2+βj2)-αj2(αj2+βj2)(Ajw,w)bhhhhhhhhhhh+βj|βj|αj2+βj2-αj2(αj2+βj2)(Aj1/2w,a)b]}.
The right hand side of (34) is an unbounded function of w for w∈Ca,b′[0,T]. Thus Eanλ→[F], Eanfq→[F], Tλ→(F), and Tq→(p)(F) might not exist. Thus throughout this paper we will need to put additional restrictions on the complex measure f corresponding to F in order to obtain our results for the GFFT and the generalized analytic Feynman integral of F.In view of Remark7, we clearly need to impose additional restrictions on the functionals F in ℱA1,A2a,b.For a positive real numberq0, let
(35)Γq0={αj2+βj2-αj2(αj2+βj2)λ→=(λ1,λ2)∈ℂ~+2∣λj=αj+iβj,hhhh|Im(λj-1/2)|=αj2+βj2-αj2(αj2+βj2)hhhh<12q0,j=1,2αj2+βj2-αj2(αj2+βj2)}
and let
(36)k(q0;A→;w)≡k(q0;A1,A2;w)=exp{∑j=12‍(2q0)-1/2∥Aj1/2w∥b∥a∥b}.
Then, for all λ→=(λ1,λ2)∈Γq0(37)|ψ(λ→;A→;w)|≤exp{∑j=12‍αj2+βj2-αj2(αj2+βj2)|(Aj1/2w,a)b|}≤exp{∑j=12‍αj2+βj2-αj2(αj2+βj2)∥Aj1/2w∥b∥a∥b}<k(q0;A→;w).We note that for all realqj with |qj|>q0, j∈{1,2},
(38)(-iqj)-1/2=1|2qj|+sign(qj)i|2qj|
and (-iq1,-iq2)∈Γq0.For the existence of the GFFT ofF, we define a subclass ℱA1,A2q0 of ℱA1,A2a,b by F∈ℱA1,A2q0 if and only if
(39)∫Ca,b′[0,T]‍k(q0;A→;w)d|f|(w)<+∞,
where f and F are related by (30) and k is given by (36).Remark 8.
Note that in casea(t)≡0 and b(t)=t on [0,T], the function space Ca,b[0,T] reduces to the classical Wiener space C0[0,T] and (w,a)b=0 for all w∈Ca,b′[0,T]=C0′[0,T]. Hence, for all λ→∈ℂ~+2, |ψ(λ→;A→;w)|≤1 and for any positive real number q0, ℱA1,A2q0=ℱA1,A2, the Kallianpur and Bromley’s class introduced in Section 1.Theorem 9.
Letq0 be a positive real number and let F be an element of ℱA1,A2q0. Then for any nonzero real numbers q1 and q2 with |qj|>q0, j∈{1,2}, the L1 analytic GFFT of F, Tq→(1)(F) exists and is given by the following formula:
(40)Tq→(1)(F)(y1,y2)=∫Ca,b′[0,T]‍exp{∑j=12‍i(Aj1/2w,yj)~}ψ(-iq→;A→;w)df(w)
for s-a.e. (y1,y2)∈Ca,b2[0,T], where ψ is given by (32).Proof.
We first note that forj∈{1,2}, the PWZ stochastic integral (Aj1/2w,x)~ is a Gaussian random variable with mean (Aj1/2w,a)b and variance ∥Aj1/2w∥b2=(Ajw,w)b. Hence, using (30), the Fubini theorem, the change of variables theorem and (14), we have that for all λ1>0 and λ2>0,
(41)J(y1,y2;λ1,λ2)≡Ex→[F(y1+λ1-1/2x1,y2+λ2-1/2x2)]=∫Ca,b′[0,T]‍exp{∑j=12‍i(Aj1/2w,yj)~}hhhhhh×(∏j=12‍Exj[exp{iλj-1/2(Aj1/2w,xj)~}])df(w)=∫Ca,b′[0,T]‍exp{∑j=12‍i(Aj1/2w,yj)~}hhhhhhh×[-[uj-(Aj1/2w,a)b]22(Ajw,w)b}duj∏j=12‍∏j=12‍(2π(Ajw,w)b)-1/2hhhhhhhhhh×∫ℝexp{[uj-(Aj1/2w,a)b]22(Ajw,w)biλj-1/2ujhhhhhhhhhhhhhhhhhhh-[uj-(Aj1/2w,a)b]22(Ajw,w)b}duj∏j=12‍]df(w)=∫Ca,b′[0,T]‍exp{∑j=12‍i(Aj1/2w,yj)~}ψ(λ→;A→;w)df(w).
Let
(42)Tλ→(F)(y1,y2)=∫Ca,b′[0,T]‍exp{∑j=12‍i(Aj1/2w,yj)~}ψ(λ→;A→;w)df(w)
for each λ→∈ℂ+2. Clearly,
(43)Tλ→(F)(y1,y2)=J(y1,y2;λ1,λ2)
for all λ1>0 and λ2>0. Let Γq0 be given by (35). Then for all λ→∈Int(Γq0)(44)|Tλ→(F)(y1,y2)|<∫Ca,b′[0,T]‍k(q0;A→;w)d|f|(w)<+∞.
Using this fact and the dominated convergence theorem, we see that Tλ→(F)(y1,y2) is a continuous function of λ→=(λ1,λ2) on Int(Γq0). For each w∈Ca,b′[0,T], ψ(λ→;A→;w) is an analytic function of λ→ throughout the domain Int(Γq0) so that ∫Δ‍ψ(λ→;A→;w)dλ→=0 for every rectifiable simple closed curve Δ in Int(Γq0). By (42), the Fubini theorem and the Morera theorem, we see that Tλ→(F)(y1,y2) is an analytic function of λ→ throughout the domain Int(Γq0). Finally, using (28) with the dominated convergence theorem, we obtain the desired result.Theorem 10.
Letq0 and F be as in Theorem 9. Then for all p∈(1,2] and all nonzero real numbers q1 and q2 with |qj|>q0, j∈{1,2}, the Lp analytic GFFT of F, Tq→(p)(F) exists and is given by the right hand side of (40) for s-a.e. (y1,y2)∈Ca,b2[0,T].Proof.
LetΓq0 be given by (35). It was shown in the proof of Theorem 9 that Tλ→(F)(y1,y2) is an analytic function of λ→ throughout the domain Int(Γq0). In view of Definition 4, it will suffice to show that for each ρ1>0 and ρ2>0,
(45)limλ→→-iq→∫Ca,b2[0,T]‍|Tq→(p)Tλ→(F)(ρ1y1,ρ2y2)hhhhhhhhhhhh-Tq→(p)(F)(ρ1y1,ρ2y2)|p′×d(μ×μ)(y1,y2)=0.
Fixingp∈(1,2] and using the inequalities (37) and (39), we obtain that for all ρj>0, j∈{1,2} and all λ→∈Γq0,
(46)|Tλ→(F)(ρ1y1,ρ2y2)-Tq→(p)(F)(ρ1y1,ρ2y2)|p′≤|∫Ca,b′[0,T]‍exp{∑j=12‍iρj(Aj1/2w,yj)~}hhhhh×[ψ(λ→;A→;w)-ψ(-iq→;A→;w)]df(w){∑j=12‍iρj(Aj1/2w,yj)~}|p′≤(∫Ca,b′[0,T]‍[|ψ(λ→;A→;w)|+|ψ(-iq→;A→;w)|]d|f|(w))p′≤(2∫Ca,b′[0,T]‍k(q0;A→;w)d|f|(w))p′<+∞.
Hence, by the dominated convergence theorem, we see that for each p∈(1,2] and each ρ1>0 and ρ2>0,
(47)limλ→→-iq→∫Ca,b2[0,T]‍|Tq→(p)Tλ→(F)(ρ1y1,ρ2y2)hhhhhhhhhhhh-Tq→(p)(F)(ρ1y1,ρ2y2)|p′d(μ×μ)(y1,y2)=limλ→→-iq→∫Ca,b2[0,T]‍|∫Ca,b′[0,T]‍exp{∑j=12‍i(Aj1/2w,ρjyj)~}hhhhhhhhhhhhhhhhhhhhh×ψ(λ→;A→;w)df(w)hhhhhhhhhhhhhhhh-∫Ca,b′[0,T]‍exp{∑j=12‍i(Aj1/2w,ρjyj)~}hhhhhhhhhhhhhhhhhhhhhh×ψ(-iq→;A→;w)df(w)∑j=12‍|p′hhhhhhhhhhhhhhhh×d(μ×μ)(y1,y2)=∫Ca,b2[0,T]‍|∫Ca,b′[0,T]‍exp{∑j=12‍i(Aj1/2w,ρjyj)~}hhhhhhhhhhhh×limλ→→-iq→[ψ(λ→;A→;w)hhhhhhhhhhhhhhhhhhhh-ψ(-iq→;A→;w)]df(w)∑j=12‍|p′hhhhhhh×d(μ×μ)(y1,y2)=0
which concludes the proof of Theorem 10.Remark 11.
(
1
) In view of Theorems 9 and 10, we see that for each p∈[1,2], the Lp analytic GFFT of F, Tq→(p)(F) is given by the right hand side of (40) for q0, q1, q2, and F as in Theorem 9, and for s-a.e. (y1,y2)∈Ca,b2[0,T],
(48)Tq→(p)(F)(y1,y2)=Ex→anfq→[F(y1+x1,y2+x2)],p∈[1,2].
In particular, using this fact and (29), we have that for all p∈[1,2],
(49)Tq→(p)(F)(0,0)=Ex→anfq→[F(x1,x2)].
(
2
) For nonzero real numbers q1 and q2 with |qj|>q0, j∈{1,2}, define a set function fq→A→:ℬ(Ca,b′[0,T])→ℂ by
(50)fq→A→(B)=∫B‍ψ(-iq→;A→;w)df(w),B∈ℬ(Ca,b′[0,T]),
where f and F are related by (30) and ℬ(Ca,b′[0,T]) is the Borel σ-algebra of Ca,b′[0,T]. Then it is obvious that fq→A→ belongs to ℳ(Ca,b′[0,T]) and for all p∈[1,2], Tq→(p)(F) can be expressed as
(51)Tq→(p)(F)(y1,y2)=∫Ca,b′[0,T]‍exp{∑j=12‍i(Aj1/2w,yj)~}dfq→A→(w)
for s-a.e. (y1,y2)∈Ca,b2[0,T]. Hence, Tq→(p)(F) belongs to ℱA1,A2a,b for all p∈[1,2].


## 4. Relationships between the GFFT and the Function Space Integral of Functionals inℱA1,A2a,b

In this section we establish a relationship between the GFFT and the function space integral of functionals in the Fresnel type classℱA1,A2a,b.Throughout this section, for convenience, we use the following notation: for givenλ∈ℂ~+ and n=1,2,…, let
(52)Gn(λ,x)=exp{[1-λ2]∑k=1n‍[(ek,x)~]2hhhhhhhh+(λ1/2-1)∑k=1n‍(ek,a)Ca,b′(ek,x)~},
where {en}n=1∞ is a complete orthonormal set in (Ca,b′[0,T],∥·∥b).To obtain our main results, Theorems14 and 17 below, we state a fundamental integration formula for the function space Ca,b[0,T].Let{e1,…,en} be an orthonormal set in (Ca,b′[0,T],∥·∥b), let k:ℝn→ℂ be a Lebesgue measurable function, and let K:Ca,b[0,T]→ℂ be given by
(53)K(x)=k((e1,x)~,…,(en,x)~).
Then
(54)E[K]=∫Ca,b[0,T]‍k((e1,x)~,…,(en,x)~)dμ(x)=(2π)-n/2∫ℝn‍k(u1,…,un)hhhhhhhh×exp{-∑j=1n‍[uj-(ej,a)b]22}du1…dun
in the sense that if either side of (54) exists, both sides exist and equality holds.We also need the following lemma to obtain our main theorem in this section.Lemma 12.
Let{e1,…,en} be an orthonormal subset of (Ca,b′[0,T],∥·∥b) and let w be an element of Ca,b′[0,T]. Then for each λ∈ℂ+, the function space integral
(55)Ex[Gn(λ,x)exp{i(w,x)~}]
exists and is given by the formula
(56)Ex[Gn(λ,x)exp{i(w,x)~}]=λ-n/2exp{+i(en+1,a)b[∥w∥b2-∑k=1n‍(ek,w)b2]1/2[λ-12λ]∑k=1n‍(ek,w)b2-12∥w∥b2hhhhhhhhhhhhhh+iλ-1/2∑k=1n‍(ek,a)b(ek,w)bhhhhhhhhhhhhhh+i(en+1,a)b[∥w∥b2-∑k=1n‍(ek,w)b2]1/2},
where Gn is given by (52) above and
(57)en+1=[∥w∥b2-∑j=1n‍(ej,w)b2]-1/2{w-∑j=1n‍(ej,w)bej}.Proof (Outline).
Using the Gram-Schmidt process, for anyw∈Ca,b′[0,T], we can write w=∑k=1n+1‍ckek, where {e1,…,en,en+1} is an orthonormal set in (Ca,b′[0,T],∥·∥b) and
(58)ck={(ek,w)b,k=1,…,n[∥w∥b2-∑j=1n‍(ej,w)b2]1/2,k=n+1.
Then using (52), (54), the Fubini theorem, and (14), it follows that (56) holds for all λ∈ℂ+.The following remark will be very useful in the proof of our main theorem in this section.Remark 13.
Letq0 be a positive real number and let Γq0 be given by (35). For real numbers q1 and q2 with |qj|>q0, j∈{1,2}, let {λ→n}n=1∞={(λ1,n,λ2,n)}n=1∞ be a sequence in ℂ+2 such that
(59)λ→n=(λ1,n,λ2,n)⟶-iq→=(-iq1,-iq2).
Let λj,n=αj,n+iβj,n for j∈{1,2} and n∈ℕ. Then for j∈{1,2}, Re(λj,n)=αj,n>0 and
(60)λj,n-1=(αj,n+iβj,n)-1=αj,n-iβj,nαj,n2+βj,n2
for each n∈ℕ. Since |Im((-iqj)-1/2)|=1/2|qj|<1/2q0 for j∈{1,2}, there exists a sufficiently large L∈ℕ such that for any n≥L, λ1,n and λ2,n are in Int(Γq0) and
(61)δ(q1,q2)≡sup({|Im(λ2,n-1/2)||Im(λ1,n-1/2)|:n≥L}hhhhhh∪{|Im(λ2,n-1/2)|:n≥L}hhhhhh∪{|Im((-iq1)-1/2)|,hhhhhhhhh|Im((-iq2)-1/2)|}{|Im(λ1,n-(1/2))|:n≥L})<12q0.
Thus, there exists a positive real number ε>1 such that δ(q1,q2)<1/(ε2q0).
Let{en}n=1∞ be a complete orthonormal set in (Ca,b′[0,T],∥·∥b). Using Parseval's identity, it follows that
(62)(g1,g2)b=∑n=1∞‍(en,g1)b(en,g2)b
for all g1, g2∈Ca,b′[0,T]. In addition for each g∈Ca,b′[0,T],
(63)∥g∥b2-∑k=1n‍(ek,g)b2=∑k=n+1∞‍(ek,g)b2≥0
for every n∈ℕ.
Since(64)(g,a)b=∑n=1∞‍(en,g)b(en,a)b
and for ε>1(65)-ε∥g∥b∥a∥b<-∥g∥b∥a∥b≤(g,a)b≤∥g∥b∥a∥b<ε∥g∥b∥a∥b,
there exists a sufficiently large Kj∈ℕ such that for any n≥Kj(66)|∑k=1n‍(ek,Aj1/2w)b(ek,a)b|<ε∥Aj1/2w∥b∥a∥b
for j∈{1,2}.
Using these and a long and tedious calculation we can show that for everyn≥max{L,K1,K2},
(67)|exp{∑j=12‍([λj,n-12λj,n]∑k=1n‍(ek,Aj1/2w)b2-12∥Aj1/2w∥b2hhhhhhhhh+iλj,n-1/2∑k=1n‍(ek,Aj1/2w)b(ek,a)b+i(en+1,a)bhhhhhhhhh×[∥Aj1/2w∥b2-∑k=1n‍(ek,Aj1/2w)b2]1/2)}|<k(q0;A→;w),
where k(q0;A→;w) is given by (36).In our next theorem, forF∈ℱA1,A2a,b, we express the GFFT of F as the limit of a sequence of function space integrals on Ca,b2[0,T].Theorem 14.
Letq0 and F be as in Theorem 10. Let {en}n=1∞ be a complete orthonormal set in (Ca,b′[0,T],∥·∥b) and let {(λ1,n,λ2,n)}n=1∞ be a sequence in ℂ+2 such that λj,n→-iqj where qj is a real number with |qj|>q0, j∈{1,2}. Then for p∈[1,2] and for s-a.e. (y1,y2)∈Ca,b2[0,T],
(68)Tq→(p)(F)(y1,y2)=limn→∞λ1,nn/2λ2,nn/2×Ex→[Gn(λ1,n,x1)Gn(λ2,n,x2)F(y1+x1,y2+x2)],
where Gn is given by (52).Proof.
By Theorems9 and 10, we know that for each p∈[1,2], the Lp analytic GFFT of F, Tq→(p)(F) exists and is given by the right hand side of (40). Thus, it suffices to show that
(69)Tq→(1)(F)(y1,y2)=Ex→anfq→[F(y1+x1,y2+x2)]=limn→∞λ1,nn/2λ2,nn/2×Ex→[Gn(λ1,n,x1)Gn(λ2,n,x2)hhhh×F(y1+x1,y2+x2)]·
Using (30), the Fubini theorem and (56) with λ and w replaced with λj,n and Aj1/2w, j∈{1,2}, respectively, we see that
(70)λ1,nn/2λ2,nn/2Ex→[Gn(λ1,n,x1)Gn(λ2,n,x2)F(y1+x1,y2+x2)]=λ1,nn/2λ2,nn/2∫Ca,b′[0,T]‍exp{∑j=12‍i(Aj1/2w,yj)~}×(∏j=12‍Exj[Gn(λj,n-1/2,xj)hhhhhhh×exp{i(Aj1/2w,xj)~}]∏j=12‍)df(w)=∫Ca,b′[0,T]‍exp{∑j=12‍(-∑k=1n‍(ek,Aj1/2w)b2]1/2i(Aj1/2w,yj)~hhhhhhhhhhhhhhhhhh+[λj,n-12λj,n]∑k=1n‍(ek,Aj1/2w)b2hhhhhhhhhhhhhhhhhh-12∥Aj1/2w∥b2hhhhhhhhhhhhhhhhhh+iλj,n-1/2∑k=1n‍(ek,a)b(ek,Aj1/2w)bhhhhhhhhhhhhhhhhhh+i(en+1,a)b[∑k=1n‍∥Aj1/2w∥b2hhhhhhhhhhhhhhhhhhhhhhhh-∑k=1n‍(ek,Aj1/2w)b2]1/2)}df(w).
But, by Remark 13 we see that the last expression of (70) is dominated by (39) on the region Γq0 given by (35) for all but a finite number of values of n. Next using the dominated convergence theorem, Parseval’s relation and (40), we obtain the desired result.Corollary 15.
Letq0, F, {en}n=1∞, {(λ1,n,λ2,n)}n=1∞ and (q1,q2) be as in Theorem 14. Then
(71)Ex→anfq→[F(x1,x2)]=limn→∞λ1,nn/2λ2,nn/2×Ex→[Gn(λ1,n,x1)Gn(λ2,n,x2)F(x1,x2)],
where Gn is given by (52).Corollary 16.
Letq0, F and {en}n=1∞ be as in Theorem 14 and let Γq0 be given by (35). Let λ→=(λ1,λ2)∈Int(Γq0) and {(λ1,n,λ2,n)}n=1∞ be a sequence in ℂ+2 such that λj,n→λj, j∈{1,2}. Then
(72)Ex→anλ→[F(x1,x2)]=limn→∞λ1,nn/2λ2,nn/2×Ex→[Gn(λ1,n,x1)Gn(λ2,n,x2)F(x1,x2)],
where Gn is given by (52).Our another result, namely, a change of scale formula for function space integrals now follows from Corollary16 above.Theorem 17.
LetF∈ℱA1,A2a,b and let {en}n=1∞ be a complete orthonormal set in (Ca,b′[0,T],∥·∥b). Then for any ρ1>0 and ρ2>0,
(73)Ex→[F(ρ1x1,ρ2x2)]=limn→∞ρ1-nρ2-n×Ex→[Gn(ρ1-2,x1)Gn(ρ2-2,x2)F(x1,x2)],
where Gn is given by (52).Proof.
Simply chooseλj=ρj-2 for j∈{1,2} and λj,n=ρj-2 for j∈{1,2} and n∈ℕ in (72).Remark 18.
Of course, if we choosea(t)≡0, b(t)=t, A1=I (identity operator), and A2=0 (zero operator), then the function space Ca,b[0,T] reduces to the classical Wiener space C0[0,T] and the generalized Fresnel type class ℱA1,A2a,b reduces to the Fresnel class ℱ(C0[0,T]). It is known that ℱ(C0[0,T]) forms a Banach algebra over the complex field. In this case, we have the relationships between the analytic Feynman integral and the Wiener integral on classical Wiener space as discussed in [14, 15].
In recent paper [19], Yoo et al. have studied a change of scale formula for function space integral of the functionals in the Banach algebra 𝒮(La,b2[0,T]); the Banach algebra 𝒮(La,b2[0,T]) is introduced in [12].


## 5. Functionals inℱA1,A2a,b

In this section, we prove a theorem ensuring that various functionals are inℱA1,A2a,b.Theorem 19.
LetA1 and A2 be bounded, nonnegative, and self-adjoint operators on Ca,b′[0,T]. Let (Y,𝒴,γ) be a σ-finite measure space and let φl:Y→Ca,b′[0,T] be 𝒴–ℬ(Ca,b′[0,T]) measurable for l∈{1,…,d}. Let θ:Y×ℝd→ℂ be given by θ(η;·)=ν^η(·), where νη∈ℳ(ℝd) for every η∈Y and where the family {νη:η∈Y} satisfies(i)
νη(E) is a 𝒴-measurable function of η for every E∈ℬ(ℝd);(ii)
∥νη∥∈L1(Y,𝒴,γ).
Under these hypothesis, the functionalF:Ca,b2[0,T]→ℂ given by
(74)F(x1,x2)=∫Y‍θ(η;∑j=12‍(Aj1/2φ1(η),xj)~,hhhhhhhh…,∑j=12‍(Aj1/2φd(η),xj)~)dγ(η)
belongs to ℱA1,A2a,b and satisfies the inequality
(75)∥F∥≤∫Y‍∥νη∥dγ(η).Proof.
Using the techniques similar to those used in [20], we can show that ∥νη∥ is measurable as a function of η, that θ is 𝒴-measurable, and that the integrand in (74) is a measurable function of η for every (x1,x2)∈Ca,b2[0,T].
We define a measureτ on 𝒴×ℬ(ℝd) by
(76)τ(E)=∫Y‍νη(E(η))dγ(η),forE∈𝒴×ℬ(ℝd).
Then by the first assertion of Theorem 3.1 in [17], τ satisfies ∥τ∥≤∫Y‍∥νη∥dγ(η). Now let Φ:Y×ℝd→Ca,b′[0,T] be defined by Φ(η;v1,…,vd)=∑l=1d‍vlφl(η). Then Φ is 𝒴×ℬ(ℝd)–ℬ(Ca,b′[0,T])-measurable on the hypothesis for φl, l∈{1,…,d}. Let σ=τ∘Φ-1. Then clearly σ∈ℳ(Ca,b′[0,T]) and satisfies ∥σ∥≤∥τ∥.
From the change of variables theorem and the second assertion of Theorem 3.1 in [17], it follows that for a.e. (x1,x2)∈Ca,b2[0,T] and for every ρ1>0 and ρ2>0,
(77)F(ρ1x1,ρ2x2)=∫Y‍ν^η(∑j=12‍(Aj1/2φ1(η),ρjxj)~,hhhhhhhhhh…,∑j=12‍(Aj1/2φd(η),ρjxj)~)dγ(η)=∫Y‍[∑j=12‍∫ℝd‍exp{i∑l=1d‍vl[∑j=12‍(Aj1/2φl(η),hhhhhhhhhhhhhhhhhhhhhhhρjxjAj1/2φl(η))~∑j=12‍(Aj1/2φl(η),]∑l=1d‍vl[∑j=12‍(Aj1/2φl(η),}dνηhhhhhhh×(v1,…,vd)∑j=12‍]dγ(η)=∫Y×ℝd‍exp{i∑l=1d‍vl[∑j=12‍(Aj1/2φl(η),ρjxj)~]}dτ×(η;v1,…,vd)=∫Y×ℝd‍exp{∑j=12‍i(Aj1/2Φ(η;v1,…,vd),ρjxj)~}dτ×(η;v1,…,vd)=∫Ca,b′[0,T]‍exp{∑j=12‍i(Aj1/2w,ρjxj)~}dτ∘Φ-1(w)=∫Ca,b′[0,T]‍exp{∑j=12‍i(Aj1/2w,ρjxj)~}dσ(w).
Thus, the functional F given by (74) belongs to ℱA1,A2a,b and satisfies the inequality
(78)∥F∥=∥σ∥≤∥τ∥≤∫Y‍∥νη∥dγ(η).As mentioned in (2) of Remark6, ℱA1,A2a,b is a Banach algebra if Ran(A1+A2) is dense in Ca,b′[0,T]. In this case, many analytic functionals of F can be formed. The following corollary is relevant to Feynman integration theories and quantum mechanics where exponential functions play an important role.Corollary 20.
LetA1 and A2 be bounded, nonnegative and self-adjoint operators on Ca,b′[0,T] such that Ran(A1+A2) is dense in Ca,b′[0,T]. Let F be given by (74) with θ as in Theorem 19, and let β:ℂ→ℂ be an entire function. Then (β∘F)(x1,x2) is in ℱA1,A2a,b. In particular, exp{F(x1,x2)}∈ℱA1,A2a,b.Corollary 21.
LetA1 and A2 be bounded, nonnegative, and self-adjoint operators on Ca,b′[0,T], and let {g1,…,gd} be a finite subset of Ca,b′[0,T]. Given β=ν^ where ν∈ℳ(ℝd), define F:Ca,b2[0,T]→ℂ by
(79)F(x1,x2)=β(∑j=12‍(Aj1/2g1,xj)~,…,∑j=12‍(Aj1/2gd,xj)~).
Then F is an element of ℱA1,A2a,b.Proof.
Let(Y,𝒴,γ) be a probability space, and for l∈{1,…,d}, let φl(η)≡gl. Take θ(η;·)=β(·)=ν^(·). Then for all ρ1>0 and ρ2>0 and for a.e. (x1,x2)∈Ca,b2[0,T],
(80)∫Y‍θ(η;∑j=12‍(Aj1/2φ1(η),ρjxj)~,hhhhhh…,∑j=12‍(Aj1/2φd(η),ρjxj)~)dγ(η)=∫Y‍β(∑j=12‍(Aj1/2g1,ρjxj)~,hhhhhhhhh…,∑j=12‍(Aj1/2gd,ρjxj)~)dγ(η)=β(∑j=12‍(Aj1/2g1,ρjxj)~,…,∑j=12‍(Aj1/2gd,ρjxj)~)=F(ρ1x1,ρ2x2).
Hence, F∈ℱA1,A2a,b.Remark 22.
Letd=1 and let (Y,𝒴,γ)=([0,T],ℬ([0,T]),mL) in Theorem 19 where mL denotes the Lebesgue measure on [0,T]. Then Theorems 4.6, 4.7, and 4.9 in [18] follow from the results in this section by letting A1 be the identity operator and letting A2≡0 on Ca,b′[0,T]. The function θ studied in [18] (and mentioned in Remark 2 above) is interpreted as the potential energy in quantum mechanics.



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*Source: 954098-2013-11-18.xml*