Generalized Analytic Fourier-Feynman Transform of Functionals in a Banach Algebra ℱ_𝐴^𝑎,𝑏_1,𝐴2

Abstract

We introduce the Fresnel type class ℱ_𝐴^𝑎,𝑏_1,𝐴2. We also establish the existence of the generalized analytic Fourier-Feynman transform for functionals in the Banach algebra ℱ_𝐴^𝑎,𝑏_1,𝐴2.

1. Introduction

Let 𝐻 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 𝑓(𝜎)ℎ^′≡𝜎ℎ^′=_𝐻exp𝑖ℎ,ℎ^′𝑑𝜎(ℎ),ℎ^′∈𝐻. (1) 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 𝜎(𝑥)=_𝐻exp{𝑖(ℎ,𝑥)^∼}𝑑𝜎(ℎ),𝑥∈𝐵, (2) 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 ℱ_𝐴1,𝐴2 than the Fresnel class ℱ(𝐵) and showed the existence of the analytic Feynman integral of functionals in ℱ_𝐴1,𝐴2 for a successful treatment of certain physical problems by means of a Feynman integral. The Fresnel class ℱ_𝐴1,𝐴2 of 𝐵² is the space of (equivalence classes of) all functionals on 𝐵² of the following form: 𝐹𝑥₁,𝑥₂=_𝐻exp²_𝑗=1𝑖𝐴_𝑗^1/2ℎ,𝑥_𝑗^∼𝑑𝜎(ℎ), (3) where 𝐴₁ and 𝐴₂ are bounded, nonnegative, and self-adjoint operators on 𝐻 and 𝜎∈ℳ(𝐻).

In this paper we study the functionals 𝐹 of the form (3) with (𝑥₁,𝑥₂) in a very general function space 𝐶²_𝑎,𝑏[0,𝑇]≡𝐶_𝑎,𝑏[0,𝑇]×𝐶_𝑎,𝑏[0,𝑇]. The function space 𝐶_𝑎,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 𝐶²_𝑎,𝑏[0,𝑇]. Other work involving GFFT theories on 𝐶_𝑎,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 Remark 7 below, that including a drift term 𝑎(𝑡) makes establishing the existence of the GFFT of functionals on 𝐶²_𝑎,𝑏[0,𝑇] very difficult. However, when 𝑎(𝑡)≡0 and 𝑏(𝑡)=𝑡 on [0,𝑇], the general function space 𝐶_𝑎,𝑏[0,𝑇] reduces to the Wiener space 𝐶₀[0,𝑇].

2. Definitions and Preliminaries

Let 𝑎(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 𝐶_𝑎,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, (𝐶_𝑎,𝑏[0,𝑇],ℬ(𝐶_𝑎,𝑏[0,𝑇]),𝜇) is the function space induced by 𝑌 where ℬ(𝐶_𝑎,𝑏[0,𝑇]) is the Borel 𝜎-algebra of 𝐶_𝑎,𝑏[0,𝑇]. We then complete this function space to obtain (𝐶_𝑎,𝑏[0,𝑇],𝒲(𝐶_𝑎,𝑏[0,𝑇]),𝜇), where 𝒲(𝐶_𝑎,𝑏[0,𝑇]) is the set of all Wiener measurable subsets of 𝐶_𝑎,𝑏[0,𝑇].

A subset 𝐵 of 𝐶_𝑎,𝑏[0,𝑇] is said to be scale-invariant measurable provided 𝜌𝐵 is 𝒲(𝐶_𝑎,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(𝐶_𝑎,𝑏[0,𝑇])-measurable for every 𝜌>0. If two functionals 𝐹 and 𝐺 defined on 𝐶_𝑎,𝑏[0,𝑇] are equal s-a.e., we write 𝐹≈𝐺.

Let 𝐿^1,2_𝑎,𝑏[0,𝑇] be the space of Lebesgue measurable functions on [0,𝑇] given by 𝐿^1,2_𝑎,𝑏[0,𝑇]=𝑣∶^𝑇₀|𝑣(𝑠)|²𝑑𝑏(𝑠)<∞,_𝑇0|𝑣(𝑠)|𝑑|𝑎|(𝑠)<∞, (4) where |𝑎|(⋅) is the total variation function of 𝑎(⋅). Then 𝐿^1,2_𝑎,𝑏[0,𝑇] is a separable Hilbert space with inner product defined by (𝑢,𝑣)_𝐿^1,2_𝑎,𝑏=^𝑇₀𝑢(𝑡)𝑣(𝑡)(𝑡)+𝑑𝑏_𝑇0𝑢(𝑡)𝑑𝑎(𝑡)^𝑇₀𝑣(𝑡)𝑑𝑎(𝑡). (5) In particular, note that ‖𝑢‖_𝐿^1,2_𝑎,𝑏≡[(𝑢,𝑢)_𝐿^1,2_𝑎,𝑏]^1/2=0 if and only if 𝑢(𝑡)=0 a.e. on [0,𝑇].

Let {𝜙_𝑗}^∞_𝑗=1 be a complete orthonormal set in 𝐿^1,2_𝑎,𝑏[0,𝑇], each of whose elements is of bounded variation on [0,𝑇] such that ^𝑇₀𝜙_𝑗(𝑡)𝜙_𝑘(𝑡)𝑑𝑏(𝑡)=0,𝑗≠𝑘1,𝑗=𝑘. (6) Then for each 𝑣∈𝐿^1,2_𝑎,𝑏[0,𝑇], the Paley-Wiener-Zygmund (PWZ) stochastic integral ⟨𝑣,𝑥⟩ is defined by the following formula: ⟨𝑣,𝑥⟩=lim_𝑛→∞^𝑇₀^𝑛_𝑗=1𝑣,𝜙_{𝑗𝑎,𝑏}𝜙_𝑗(𝑡)𝑑𝑥(𝑡) (7) for all 𝑥∈𝐶_𝑎,𝑏[0,𝑇] for which the limit exists; one can show that for each 𝑣∈𝐿^1,2_𝑎,𝑏[0,𝑇], the PWZ stochastic integral ⟨𝑣,𝑥⟩ exists for 𝜇-a.e. 𝑥∈𝐶_𝑎,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 ∫^𝑇₀𝑣(𝑡)𝑑𝑥(𝑡) for s-a.e. 𝑥∈𝐶_𝑎,𝑏[0,𝑇].

Remark 1. (1) For each 𝑣∈𝐿^1,2_𝑎,𝑏[0,𝑇], the PWZ stochastic integral ⟨𝑣,𝑥⟩ is a Gaussian random variable on 𝐶_𝑎,𝑏[0,𝑇] with mean ∫^𝑇₀𝑣(𝑠)𝑑𝑎(𝑠) and variance ∫^𝑇₀𝑣²(𝑠)𝑑𝑏(𝑠).
(2) For all 𝑢, 𝑣∈𝐿^1,2_𝑎,𝑏[0,𝑇], _{𝐶𝑎,𝑏[0,𝑇]}⟨⟩(𝑥)=𝑢,𝑥⟩⟨𝑣,𝑥𝑑𝜇_𝑇0𝑢(𝑠)𝑣(𝑠)(𝑠)+𝑑𝑏_𝑇0𝑢(𝑠)𝑑𝑎(𝑠)^𝑇₀𝑣(𝑠)𝑑𝑎(𝑠). (8) Hence, we see that for all 𝑢, 𝑣∈𝐿^1,2_𝑎,𝑏[0,𝑇], ∫^𝑇₀𝑢(𝑠)𝑣(𝑠)𝑑𝑏(𝑠)=0 if and only if ⟨𝑢,𝑥⟩ and ⟨𝑣,𝑥⟩ are independent random variables.

The following Cameron-Martin subspace of 𝐶_𝑎,𝑏[0,𝑇] plays an important role throughout this paper.

Let 𝐶^′_𝑎,𝑏[0,𝑇]=𝑤∈𝐶_𝑎,𝑏[0,𝑇]∶𝑤(𝑡)=^𝑡₀𝑧(𝑠)𝑑𝑏(𝑠)forsome𝑧∈𝐿^1,2_𝑎,𝑏[0,𝑇]. (9) For 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇], let 𝐷∶𝐶^′_𝑎,𝑏[0,𝑇]→𝐿^1,2_𝑎,𝑏[0,𝑇] be defined by the following formula: 𝐷𝑤(𝑡)=𝑑𝜆_𝑤𝑑𝜆_𝑏(𝑡), (10) where 𝑑𝜆_𝑤/𝑑𝜆_𝑏 denotes the Radon-Nikodym derivative of the signed measure 𝜆_𝑤 induced by 𝑤, with respect to the Borel-Stieltjes measure 𝜆_𝑏 induced by 𝑏. Then 𝐶^′_𝑎,𝑏≡𝐶^′_𝑎,𝑏[0,𝑇] with inner product 𝑤₁,𝑤_{2𝐶′𝑎,𝑏}=_{𝐶𝑎,𝑏[0,𝑇]}𝐷𝑤₁,𝑥𝐷𝑤₂,𝑥𝑑𝜇(𝑥) (11) 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∶𝐿^1,2_𝑎,𝑏[0,𝑇]→𝐶^′_𝑎,𝑏[0,𝑇] is given by 𝐷⁻¹𝑧(𝑡)=^𝑡₀𝑧(𝑠)𝑑𝑏(𝑠). (12) Moreover, the triple (𝐶^′_𝑎,𝑏[0,𝑇],𝐶_𝑎,𝑏[0,𝑇],𝜇) becomes an abstract Wiener space.

Throughout this paper, for 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇], we will use the notation (𝑤,𝑥)^∼ instead of ⟨𝐷𝑤,𝑥⟩. We also use the following notations for 𝑤₁, 𝑤₂, 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇]: 𝑤₁,𝑤_2𝑏=^𝑇₀𝐷𝑤₁(𝑡)𝐷𝑤₂(𝑡)(𝑡),‖𝑤‖𝑑𝑏_𝑏=(𝑤,𝑤)_𝑏. (13) Then 𝐶^′_𝑎,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 ‖⋅‖_{𝐶′𝑎,𝑏} and ‖⋅‖_𝑏 are equivalent. Furthermore, we have the following assertions. (i)𝑎(⋅) is an element of 𝐶^′_𝑎,𝑏[0,𝑇]. (ii)For each 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇], the random variable 𝑥↦(𝑤,𝑥)^∼ is Gaussian with mean (𝑤,𝑎)_𝑏 and variance ‖𝑤‖²_𝑏. (iii)(𝑤,𝛼𝑥)^∼=(𝛼𝑤,𝑥)^∼=𝛼(𝑤,𝑥)^∼ for any real number 𝛼, 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇] and 𝑥∈𝐶_𝑎,𝑏[0,𝑇].(iv)Let {𝑤₁,…,𝑤_𝑛} be a subset of 𝐶^′_𝑎,𝑏[0,𝑇] such that ∫^𝑇₀𝐷𝑤_𝑖(𝑡)𝐷𝑤_𝑗(𝑡)𝑑𝑏(𝑡)=𝛿_𝑖𝑗, where 𝛿_𝑖𝑗 is the Kronecker delta. Then the random variables (𝑤_𝑖,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 𝐶_𝑎,𝑏[0,𝑇].

The following integration formula is used several times in this paper: _ℝexp−𝛼𝑢²+𝛽𝑢𝑑𝑢=𝜋𝛼𝛽exp²4𝛼 (14) for complex numbers 𝛼 and 𝛽 with Re(𝛼)>0.

3. The GFFT of Functionals in a Banach Algebra ℱ_𝐴^𝑎,𝑏_1,𝐴2

Let ℳ(𝐶^′_𝑎,𝑏[0,𝑇]) be the space of complex-valued, countably additive (and hence finite) Borel measures on 𝐶^′_𝑎,𝑏[0,𝑇]. ℳ(𝐶^′_𝑎,𝑏[0,𝑇]) is a Banach algebra under the total variation norm and with convolution as multiplication.

We define the Fresnel type class ℱ(𝐶_𝑎,𝑏[0,𝑇]) of functionals on 𝐶_𝑎,𝑏[0,𝑇] as the space of all stochastic Fourier transforms of elements of ℳ(𝐶^′_𝑎,𝑏[0,𝑇]); that is, 𝐹∈ℱ(𝐶_𝑎,𝑏[0,𝑇]) if and only if there exists a measure 𝑓 in ℳ(𝐶^′_𝑎,𝑏[0,𝑇]) such that 𝐹(𝑥)=_{𝐶′𝑎,𝑏[0,𝑇]}exp{𝑖(𝑤,𝑥)^∼}𝑑𝑓(𝑤) (15) for s-a.e. 𝑥∈𝐶_𝑎,𝑏[0,𝑇]. More precisely, since we will identify functionals which coincide s-a.e. on 𝐶_𝑎,𝑏[0,𝑇], ℱ(𝐶_𝑎,𝑏[0,𝑇]) can be regarded as the space of all 𝑠-equivalence classes of functionals of the form (15).

The Fresnel type class ℱ(𝐶_𝑎,𝑏[0,𝑇]) is a Banach algebra with norm ‖𝐹‖=𝑓=_{𝐶′𝑎,𝑏[0,𝑇]}𝑑𝑓(𝑤). (16) 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 algebra ℱ(𝐶_𝑎,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: 𝜈(𝑢)=_ℝexp{𝑖𝑢𝑣}𝑑𝜈(𝑣). (17)
Let 𝒢 be the set of all complex-valued functions on [0,𝑇]×ℝ of the form 𝜃(𝑠,𝑢)=𝜎_𝑠(𝑢), where {𝜎_𝑠∶0≤𝑠≤𝑇} is a family from ℳ(ℝ) satisfying the following two conditions:(i)for every 𝐸∈ℬ(ℝ), 𝜎_𝑠(𝐸) is Borel measurable in 𝑠, (ii)∫^𝑇₀‖𝜎_𝑠‖𝑑𝑏(𝑠)<+∞.
Let 𝜃∈𝒢 and let 𝐻 be given by 𝐻(𝑥)=exp^𝑇₀𝜃(𝑡,𝑥(𝑡))𝑑𝑡 (18) for s-a.e. 𝑥∈𝐶_𝑎,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)), ∫^𝑇₀𝜃(𝑡,𝑥(𝑡))𝑑𝑡, and 𝐻(𝑥) are elements of ℱ(𝐶_𝑎,𝑏[0,𝑇]). These facts are relevant to quantum mechanics where exponential functions play a prominent role.

Let 𝐴 be a nonnegative self-adjoint operator on 𝐶^′_𝑎,𝑏[0,𝑇] and 𝑓 any complex measure on 𝐶^′_𝑎,𝑏[0,𝑇]. Then the functional 𝐹(𝑥)=_{𝐶′𝑎,𝑏[0,𝑇]}exp𝑖𝐴^1/2𝑤,𝑥^∼𝑑𝑓(𝑤) (19) belongs to ℱ(𝐶_𝑎,𝑏[0,𝑇]) because it can be rewritten as ∫_{𝐶′𝑎,𝑏[0,𝑇]}exp{𝑖(𝑤,𝑥)^∼}𝑑𝑓_𝐴(𝑤) for 𝑓_𝐴=𝑓∘(𝐴^1/2)⁻¹. Let 𝐴 be self-adjoint but not nonnegative. Then 𝐴 has the form 𝐴=𝐴⁺−𝐴⁻, (20) 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].

Let 𝒲(𝐶²_𝑎,𝑏[0,𝑇]) denote the class of all Wiener measurable subsets of the product function space 𝐶²_𝑎,𝑏[0,𝑇]. A subset 𝐵 of 𝐶²_𝑎,𝑏[0,𝑇] is said to be scale-invariant measurable provided {(𝜌₁𝑥₁,𝜌₂𝑥₂)∶(𝑥₁,𝑥₂)∈𝐵} is 𝒲(𝐶²_𝑎,𝑏[0,𝑇])-measurable for every 𝜌₁>0 and 𝜌₂>0, and a scale-invariant measurable subset 𝑁 of 𝐶²_𝑎,𝑏[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 𝐶²_𝑎,𝑏[0,𝑇]. A functional 𝐹 on 𝐶²_𝑎,𝑏[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(𝐶²_𝑎,𝑏[0,𝑇])-measurable for every 𝜌₁>0 and 𝜌₂>0. If two functionals 𝐹 and 𝐺 defined on 𝐶²_𝑎,𝑏[0,𝑇] are equal s-a.e., then we write 𝐹≈𝐺.

We denote the product function space integral of a 𝒲(𝐶²_𝑎,𝑏[0,𝑇])-measurable functional 𝐹 by 𝐸[𝐹]≡𝐸_𝑥𝐹𝑥₁,𝑥₂=_{𝐶2𝑎,𝑏[0,𝑇]}𝐹𝑥₁,𝑥₂𝑑𝜇×𝜇𝑥₁,𝑥₂ (21) whenever the integral exists.

Throughout this paper, let ℂ, 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(or𝜆^1/2) is always chosen to have positive real part. We also assume that every functional 𝐹 on 𝐶²_𝑎,𝑏[0,𝑇] we consider is s-a.e. defined and is scale-invariant measurable.

Definition 3. Let ℂ²₊≡ℂ₊×ℂ₊ and let ℂ²₊≡ℂ₊×ℂ₊. Let 𝐹∶𝐶²_𝑎,𝑏[0,𝑇]→ℂ be such that for each 𝜆₁>0 and 𝜆₂>0, the function space integral 𝐽𝜆₁,𝜆₂=_{𝐶2𝑎,𝑏[0,𝑇]}𝐹𝜆₁^−1/2𝑥₁,𝜆₂^−1/2𝑥₂𝑑𝜇×𝜇𝑥₁,𝑥₂ (22) exists. If there exists a function 𝐽^∗(𝜆₁,𝜆₂) analytic in ℂ²₊ 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 𝐶²_𝑎,𝑏[0,𝑇] with parameter 𝜆=(𝜆₁,𝜆₂), and for 𝜆∈ℂ²₊ we write 𝐸^an𝜆[𝐹]≡𝐸^an𝜆_𝑥𝐹𝑥₁,𝑥₂≡𝐸^{an12)(𝜆,𝜆}_𝑥1,𝑥2𝐹𝑥₁,𝑥₂=𝐽^∗𝜆₁,𝜆₂. (23) Let 𝑞₁ and 𝑞₂ be nonzero real numbers. Let 𝐹 be a functional such that 𝐸^an𝜆[𝐹] exists for all 𝜆∈ℂ²₊. If the following limit exists, we call it the generalized analytic Feynman integral of 𝐹 with parameter 𝑞=(𝑞₁,𝑞₂), and we write 𝐸^anf𝑞[𝐹]≡𝐸^anf𝑞_𝑥𝐹𝑥₁,𝑥₂≡𝐸^{anf12)(𝑞,𝑞}_𝑥1,𝑥2𝐹𝑥₁,𝑥₂=lim_{𝑞𝜆→−𝑖}𝐸^an𝜆[𝐹], (24) where 𝜆=(𝜆₁,𝜆₂)→−𝑖𝑞=(−𝑖𝑞₁,−𝑖𝑞₂) through values in ℂ²₊.

Definition 4. Let 𝑞₁ and 𝑞₂ be nonzero real numbers. For 𝜆=(𝜆₁,𝜆₂)∈ℂ²₊ and (𝑦₁,𝑦₂)∈𝐶²_𝑎,𝑏[0,𝑇], let 𝑇_𝜆(𝐹)𝑦₁,𝑦₂≡𝑇_{(𝜆1,𝜆2)}(𝐹)𝑦₁,𝑦₂=𝐸^an𝜆_𝑥𝐹𝑦₁+𝑥₁,𝑦₂+𝑥₂. (25) For 𝑝∈(1,2], we define the 𝐿_𝑝 analytic GFFT, 𝑇_𝑞^(𝑝)(𝐹) of 𝐹, by the formula (𝜆∈ℂ²₊)𝑇_𝑞^(𝑝)(𝐹)𝑦₁,𝑦₂≡𝑇_𝑞^(𝑝)_1,𝑞2(𝐹)𝑦₁,𝑦₂=lim_{𝑞𝜆→−𝑖}𝑇_𝜆(𝐹)𝑦₁,𝑦₂ (26) if it exists; that is, for each 𝜌₁>0 and 𝜌₂>0, lim_{𝑞𝜆→−𝑖𝐶2𝑎,𝑏[0,𝑇]}𝑇_𝜆(𝐹)𝜌₁𝑦₁,𝜌₂𝑦₂−𝑇_𝑞^(𝑝)(𝐹)𝜌₁𝑦₁,𝜌₂𝑦₂^𝑝′𝑑𝜇×𝜇𝑦₁,𝑦₂=0, (27) where 1/𝑝+1/𝑝^′=1. We define the 𝐿₁ analytic GFFT, 𝑇_𝑞⁽¹⁾(𝐹) of 𝐹, by the formula (𝜆∈ℂ²₊)𝑇_𝑞⁽¹⁾(𝐹)𝑦₁,𝑦₂=lim_{𝑞𝜆→−𝑖}𝑇_𝜆(𝐹)𝑦₁,𝑦₂ (28) if it exists.

We note that for 1≤𝑝≤2, 𝑇_𝑞^(𝑝)(𝐹) is defined only s-a.e. We also note that if 𝑇_𝑞^(𝑝)(𝐹) exists and if 𝐹≈𝐺, then 𝑇_𝑞^(𝑝)(𝐺) exists and 𝑇_𝑞^(𝑝)(𝐺)≈𝑇_𝑞^(𝑝)(𝐹). Moreover, from Definition 4, we see that for 𝑞₁, 𝑞₂∈ℝ−{0}𝐸^anf𝑞_𝑥𝐹𝑥₁,𝑥₂=𝑇_𝑞⁽¹⁾(𝐹)(0,0). (29)

Next we give the definition of the generalized Fresnel type class ℱ_𝐴^𝑎,𝑏_1,𝐴2.

Definition 5. Let 𝐴₁ and 𝐴₂ be bounded, nonnegative, and self-adjoint operators on 𝐶^′_𝑎,𝑏[0,𝑇]. The generalized Fresnel type class ℱ_𝐴^𝑎,𝑏_1,𝐴2 of functionals on 𝐶²_𝑎,𝑏[0,𝑇] is defined as the space of all functionals 𝐹 on 𝐶²_𝑎,𝑏[0,𝑇] of the following form: 𝐹𝑥₁,𝑥₂=_{𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝐴_𝑗^1/2𝑤,𝑥_𝑗^∼𝑑𝑓(𝑤) (30) for some 𝑓∈ℳ(𝐶^′_𝑎,𝑏[0,𝑇]). More precisely, since we identify functionals which coincide s-a.e. on 𝐶²_𝑎,𝑏[0,𝑇], ℱ_𝐴^𝑎,𝑏_1,𝐴2 can be regarded as the space of all 𝑠-equivalence classes of functionals of the form (30).

Remark 6. (1) In Definition 5, let 𝐴₁ be the identity operator on 𝐶^′_𝑎,𝑏[0,𝑇] and 𝐴₂≡0. Then ℱ_𝐴^𝑎,𝑏_1,𝐴2 is essentially the Fresnel type class ℱ(𝐶_𝑎,𝑏[0,𝑇]), and for 𝑝∈[1,2] and nonzero real numbers 𝑞₁ and 𝑞₂, 𝑇_𝑞^(𝑝)_1,𝑞2(𝐹)𝑦₁,𝑦₂=𝑇_𝑞^(𝑝)₁𝐹₀𝑦₁, (31) if it exists, where 𝐹₀(𝑥₁)=𝐹(𝑥₁,𝑥₂) for all (𝑥₁,𝑥₂)∈𝐶²_𝑎,𝑏[0,𝑇] and 𝑇_𝑞^(𝑝)₁(𝐹₀) means the 𝐿_𝑝 analytic GFFT on 𝐶_𝑎,𝑏[0,𝑇]; see [6, 12].
(2) The map 𝑓↦𝐹 defined by (30) sets up an algebra isomorphism between ℳ(𝐶^′_𝑎,𝑏[0,𝑇]) and ℱ_𝐴^𝑎,𝑏_1,𝐴2 if Ran(𝐴₁+𝐴₂) is dense in 𝐶^′_𝑎,𝑏[0,𝑇], where Ran indicates the range of an operator. In this case ℱ_𝐴^𝑎,𝑏_1,𝐴2 becomes a Banach algebra under the norm ‖𝐹‖=‖𝑓‖. For more details see [1].

Remark 7. Let 𝐹 be given by (30). In evaluating 𝐸_𝑥[𝐹(𝜆₁^−1/2𝑥₁,𝜆₂^−1/2𝑥₂)] and 𝑇_{(𝜆1,𝜆2)}(𝐹)(𝑦₁,𝑦₂)=𝐸_𝑥[𝐹(𝑦₁+𝜆₁^−1/2𝑥₁,𝑦₂+𝜆₂^−1/2𝑥₂)] for 𝜆₁>0 and 𝜆₂>0, the expression 𝜓𝜆𝜆;𝐴;𝑤≡𝜓₁,𝜆₂;𝐴₁,𝐴₂;𝑤=exp²_𝑗=1−𝐴_𝑗𝑤,𝑤_𝑏2𝜆_𝑗+𝑖𝜆_𝑗^−1/2𝐴_𝑗^1/2𝑤,𝑎_𝑏 (32) occurs. Clearly, for 𝜆_𝑗>0, 𝑗∈{1,2}, |𝜓(𝜆;𝐴;𝑤)|≤1 for all 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇]. But for 𝜆∈ℂ²₊, |𝜓(𝜆;𝐴;𝑤)| is not necessarily bounded by 1.
Note that for each 𝜆_𝑗∈ℂ₊ with 𝜆_𝑗=𝛼_𝑗+𝑖𝛽_𝑗, 𝑗∈{1,2}, 𝜆_𝑗^1/2=𝛼²_𝑗+𝛽²_𝑗+𝛼_𝑗2𝛽+𝑖^𝑗𝛽_𝑗𝛼²_𝑗+𝛽²_𝑗−𝛼_𝑗2,𝜆_𝑗^−1/2=𝛼²_𝑗+𝛽²_𝑗+𝛼_𝑗2𝛼²_𝑗+𝛽²_𝑗𝛽−𝑖^𝑗𝛽_𝑗𝛼²_𝑗+𝛽²_𝑗−𝛼_𝑗2𝛼²_𝑗+𝛽²_𝑗. (33) Hence, for 𝜆_𝑗∈ℂ₊ with 𝜆_𝑗=𝛼_𝑗+𝑖𝛽_𝑗, 𝑗∈{1,2}, 𝜓𝜆;𝐴;𝑤=exp²_𝑗=1−𝛼_𝑗2𝛼²_𝑗+𝛽²_𝑗𝐴_𝑗𝑤,𝑤_𝑏+𝛽_𝑗𝛽_𝑗𝛼²_𝑗+𝛽²_𝑗−𝛼_𝑗2𝛼²_𝑗+𝛽²_𝑗𝐴_𝑗^1/2𝑤,𝑎_𝑏. (34)
The right hand side of (34) is an unbounded function of 𝑤 for 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇]. Thus 𝐸^an𝜆[𝐹], 𝐸^anf𝑞[𝐹], 𝑇_𝜆(𝐹), and 𝑇_𝑞^(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 Remark 7, we clearly need to impose additional restrictions on the functionals 𝐹 in ℱ_𝐴^𝑎,𝑏_1,𝐴2.

For a positive real number 𝑞₀, let Γ_𝑞0=𝜆=𝜆₁,𝜆₂∈ℂ²₊∣𝜆_𝑗=𝛼_𝑗+𝑖𝛽_𝑗,Im𝜆_𝑗^−1/2=𝛼²_𝑗+𝛽²_𝑗−𝛼_𝑗2𝛼²_𝑗+𝛽²_𝑗<12𝑞₀,𝑗=1,2 (35) and let 𝑘𝑞₀;𝐴;𝑤≡𝑘𝑞₀;𝐴₁,𝐴₂;𝑤=exp²_𝑗=12𝑞₀^−1/2𝐴_𝑗^1/2𝑤_𝑏‖𝑎‖_𝑏. (36) Then, for all 𝜆=(𝜆₁,𝜆₂)∈Γ_𝑞0𝜓𝜆;𝐴;𝑤≤exp²_𝑗=1𝛼²_𝑗+𝛽²_𝑗−𝛼_𝑗2𝛼²_𝑗+𝛽²_𝑗𝐴_𝑗^1/2𝑤,𝑎_𝑏≤exp²_𝑗=1𝛼²_𝑗+𝛽²_𝑗−𝛼_𝑗2𝛼²_𝑗+𝛽²_𝑗𝐴_𝑗^1/2𝑤_𝑏‖𝑎‖_𝑏<𝑘𝑞₀;𝐴;𝑤. (37)

We note that for all real 𝑞_𝑗 with |𝑞_𝑗|>𝑞₀, 𝑗∈{1,2}, −𝑖𝑞_𝑗^−1/2=12𝑞_𝑗+sign𝑞_𝑗𝑖2𝑞_𝑗 (38) and (−𝑖𝑞₁,−𝑖𝑞₂)∈Γ_𝑞0.

For the existence of the GFFT of 𝐹, we define a subclass ℱ^𝑞0_𝐴1,𝐴2 of ℱ_𝐴^𝑎,𝑏_1,𝐴2 by 𝐹∈ℱ^𝑞0_𝐴1,𝐴2 if and only if _{𝐶′𝑎,𝑏[0,𝑇]}𝑘𝑞₀;𝐴;𝑤𝑑𝑓(𝑤)<+∞, (39) where 𝑓 and 𝐹 are related by (30) and 𝑘 is given by (36).

Remark 8. Note that in case 𝑎(𝑡)≡0 and 𝑏(𝑡)=𝑡 on [0,𝑇], the function space 𝐶_𝑎,𝑏[0,𝑇] reduces to the classical Wiener space 𝐶₀[0,𝑇] and (𝑤,𝑎)_𝑏=0 for all 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇]=𝐶^′₀[0,𝑇]. Hence, for all 𝜆∈ℂ²₊, |𝜓(𝜆;𝐴;𝑤)|≤1 and for any positive real number 𝑞₀, ℱ^𝑞0_𝐴1,𝐴2=ℱ_𝐴1,𝐴2, the Kallianpur and Bromley’s class introduced in Section 1.

Theorem 9. Let 𝑞₀ be a positive real number and let 𝐹 be an element of ℱ^𝑞0_𝐴1,𝐴2. Then for any nonzero real numbers 𝑞₁ and 𝑞₂ with |𝑞_𝑗|>𝑞₀, 𝑗∈{1,2}, the 𝐿₁ analytic GFFT of 𝐹, 𝑇_𝑞⁽¹⁾(𝐹) exists and is given by the following formula: 𝑇_𝑞⁽¹⁾(𝐹)𝑦₁,𝑦₂=_{𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝐴_𝑗^1/2𝑤,𝑦_𝑗^∼𝜓−𝑖𝑞;𝐴;𝑤𝑑𝑓(𝑤) (40) for s-a.e. (𝑦₁,𝑦₂)∈𝐶²_𝑎,𝑏[0,𝑇], where 𝜓 is given by (32).

Proof. We first note that for 𝑗∈{1,2}, the PWZ stochastic integral (𝐴_𝑗^1/2𝑤,𝑥)^∼ is a Gaussian random variable with mean (𝐴_𝑗^1/2𝑤,𝑎)_𝑏 and variance ‖𝐴_𝑗^1/2𝑤‖²_𝑏=(𝐴_𝑗𝑤,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, 𝐽𝑦₁,𝑦₂;𝜆₁,𝜆₂≡𝐸_𝑥𝐹𝑦₁+𝜆₁^−1/2𝑥₁,𝑦₂+𝜆₂^−1/2𝑥₂=_{𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝐴_𝑗^1/2𝑤,𝑦_𝑗^∼ײ_𝑗=1𝐸_𝑥𝑗exp𝑖𝜆_𝑗^−1/2𝐴_𝑗^1/2𝑤,𝑥_𝑗^∼(𝑤)=𝑑𝑓_{𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝐴_𝑗^1/2𝑤,𝑦_𝑗^∼ײ_𝑗=12𝜋𝐴_𝑗𝑤,𝑤_𝑏^−1/2×_ℝexp𝑖𝜆_𝑗^−1/2𝑢_𝑗−𝑢_𝑗−𝐴_𝑗^1/2𝑤,𝑎_𝑏²2𝐴_𝑗𝑤,𝑤_𝑏𝑑𝑢_𝑗(𝑤)=𝑑𝑓_{𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝐴_𝑗^1/2𝑤,𝑦_𝑗^∼𝜓𝜆;𝐴;𝑤𝑑𝑓(𝑤). (41) Let 𝑇_𝜆(𝐹)𝑦₁,𝑦₂=_{𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝐴_𝑗^1/2𝑤,𝑦_𝑗^∼𝜓𝜆;𝐴;𝑤𝑑𝑓(𝑤) (42) for each 𝜆∈ℂ²₊. Clearly, 𝑇_𝜆(𝐹)𝑦₁,𝑦₂=𝐽𝑦₁,𝑦₂;𝜆₁,𝜆₂ (43) for all 𝜆₁>0 and 𝜆₂>0. Let Γ_𝑞0 be given by (35). Then for all 𝜆∈Int(Γ_𝑞0)𝑇_𝜆(𝐹)𝑦₁,𝑦₂<_{𝐶′𝑎,𝑏[0,𝑇]}𝑘𝑞₀;𝐴;𝑤𝑑𝑓(𝑤)<+∞. (44) Using this fact and the dominated convergence theorem, we see that 𝑇_𝜆(𝐹)(𝑦₁,𝑦₂) is a continuous function of 𝜆=(𝜆₁,𝜆₂) on Int(Γ_𝑞0). For each 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇], 𝜓(𝜆;𝐴;𝑤) is an analytic function of 𝜆 throughout the domain Int(Γ_𝑞0) so that ∫_Δ𝜓(𝜆;𝐴;𝑤)𝑑𝜆=0 for every rectifiable simple closed curve Δ in Int(Γ_𝑞0). By (42), the Fubini theorem and the Morera theorem, we see that 𝑇_𝜆(𝐹)(𝑦₁,𝑦₂) is an analytic function of 𝜆 throughout the domain Int(Γ_𝑞0). Finally, using (28) with the dominated convergence theorem, we obtain the desired result.

Theorem 10. Let 𝑞₀ and 𝐹 be as in Theorem 9. Then for all 𝑝∈(1,2] and all nonzero real numbers 𝑞₁ and 𝑞₂ with |𝑞_𝑗|>𝑞₀, 𝑗∈{1,2}, the 𝐿_𝑝 analytic GFFT of 𝐹, 𝑇_𝑞^(𝑝)(𝐹) exists and is given by the right hand side of (40) for s-a.e. (𝑦₁,𝑦₂)∈𝐶²_𝑎,𝑏[0,𝑇].

Proof. Let Γ_𝑞0 be given by (35). It was shown in the proof of Theorem 9 that 𝑇_𝜆(𝐹)(𝑦₁,𝑦₂) is an analytic function of 𝜆 throughout the domain Int(Γ_𝑞0). In view of Definition 4, it will suffice to show that for each 𝜌₁>0 and 𝜌₂>0, lim_{𝑞𝜆→−𝑖𝐶2𝑎,𝑏[0,𝑇]}𝑇_𝜆(𝐹)𝜌₁𝑦₁,𝜌₂𝑦₂−𝑇_𝑞^(𝑝)(𝐹)𝜌₁𝑦₁,𝜌₂𝑦₂^𝑝′×𝑑𝜇×𝜇𝑦₁,𝑦₂=0. (45)
Fixing 𝑝∈(1,2] and using the inequalities (37) and (39), we obtain that for all 𝜌_𝑗>0, 𝑗∈{1,2} and all 𝜆∈Γ_𝑞0, 𝑇_𝜆(𝐹)𝜌₁𝑦₁,𝜌₂𝑦₂−𝑇_𝑞^(𝑝)(𝐹)𝜌₁𝑦₁,𝜌₂𝑦₂^𝑝′≤_{𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝜌_𝑗𝐴_𝑗^1/2𝑤,𝑦_𝑗^∼×𝜓𝜆;𝐴;𝑤−𝜓−𝑖𝑞;𝐴;𝑤𝑑𝑓(𝑤)^𝑝′≤_{𝐶′𝑎,𝑏[0,𝑇]}𝜓𝜆;𝐴;𝑤+𝜓−𝑖𝑞;𝐴;𝑤𝑑𝑓(𝑤)^𝑝′≤2_{𝐶′𝑎,𝑏[0,𝑇]}𝑘𝑞₀;𝐴;𝑤𝑑𝑓(𝑤)^𝑝′<+∞. (46) Hence, by the dominated convergence theorem, we see that for each 𝑝∈(1,2] and each 𝜌₁>0 and 𝜌₂>0, lim_{𝑞𝜆→−𝑖𝐶2𝑎,𝑏[0,𝑇]}𝑇_𝜆(𝐹)𝜌₁𝑦₁,𝜌₂𝑦₂−𝑇_𝑞^(𝑝)(𝐹)𝜌₁𝑦₁,𝜌₂𝑦₂^𝑝′𝑑𝜇×𝜇𝑦₁,𝑦₂=lim_{𝑞𝜆→−𝑖𝐶2𝑎,𝑏[0,𝑇]𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝐴_𝑗^1/2𝑤,𝜌_𝑗𝑦_𝑗^∼×𝜓𝜆;𝐴;𝑤𝑑𝑓(𝑤)−_{𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝐴_𝑗^1/2𝑤,𝜌_𝑗𝑦_𝑗^∼×𝜓−𝑖𝑞;𝐴;𝑤𝑑𝑓(𝑤)^𝑝′×𝑑𝜇×𝜇𝑦₁,𝑦₂=_{𝐶2𝑎,𝑏[0,𝑇]𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝐴_𝑗^1/2𝑤,𝜌_𝑗𝑦_𝑗^∼×lim_{𝑞𝜆→−𝑖}𝜓(𝑤)𝜆;𝐴;𝑤−𝜓−𝑖𝑞;𝐴;𝑤𝑑𝑓^𝑝′×𝑑𝜇×𝜇𝑦₁,𝑦₂=0 (47) 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 𝐿_𝑝 analytic GFFT of 𝐹, 𝑇_𝑞^(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)∈𝐶²_𝑎,𝑏[0,𝑇], 𝑇_𝑞^(𝑝)(𝐹)𝑦₁,𝑦₂=𝐸^anf𝑞_𝑥𝐹𝑦₁+𝑥₁,𝑦₂+𝑥₂,𝑝∈[1,2]. (48) In particular, using this fact and (29), we have that for all 𝑝∈[1,2], 𝑇_𝑞^(𝑝)(𝐹)(0,0)=𝐸^anf𝑞_𝑥𝐹𝑥₁,𝑥₂. (49)
(2) For nonzero real numbers 𝑞₁ and 𝑞₂ with |𝑞_𝑗|>𝑞₀, 𝑗∈{1,2}, define a set function 𝑓^𝐴_𝑞∶ℬ(𝐶^′_𝑎,𝑏[0,𝑇])→ℂ by 𝑓^𝐴_𝑞(𝐵)=_𝐵𝜓−𝑖𝑞;𝐴;𝑤𝑑𝑓(𝑤),𝐵∈ℬ𝐶^′_𝑎,𝑏[0,𝑇], (50) where 𝑓 and 𝐹 are related by (30) and ℬ(𝐶^′_𝑎,𝑏[0,𝑇]) is the Borel 𝜎-algebra of 𝐶^′_𝑎,𝑏[0,𝑇]. Then it is obvious that 𝑓^𝐴_𝑞 belongs to ℳ(𝐶^′_𝑎,𝑏[0,𝑇]) and for all 𝑝∈[1,2], 𝑇_𝑞^(𝑝)(𝐹) can be expressed as 𝑇_𝑞^(𝑝)(𝐹)𝑦₁,𝑦₂=_{𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝐴_𝑗^1/2𝑤,𝑦_𝑗^∼𝑑𝑓^𝐴_𝑞(𝑤) (51) for s-a.e. (𝑦₁,𝑦₂)∈𝐶²_𝑎,𝑏[0,𝑇]. Hence, 𝑇_𝑞^(𝑝)(𝐹) belongs to ℱ_𝐴^𝑎,𝑏_1,𝐴2 for all 𝑝∈[1,2].

4. Relationships between the GFFT and the Function Space Integral of Functionals in ℱ_𝐴^𝑎,𝑏_1,𝐴2

In this section we establish a relationship between the GFFT and the function space integral of functionals in the Fresnel type class ℱ_𝐴^𝑎,𝑏_1,𝐴2.

Throughout this section, for convenience, we use the following notation: for given 𝜆∈ℂ₊ and 𝑛=1,2,…, let 𝐺_𝑛(𝜆,𝑥)=exp1−𝜆2^𝑛_𝑘=1𝑒_𝑘,𝑥^∼2+𝜆^1/2−1^𝑛_𝑘=1𝑒_𝑘,𝑎_{𝐶′𝑎,𝑏}𝑒_𝑘,𝑥^∼, (52) where {𝑒_𝑛}^∞_𝑛=1 is a complete orthonormal set in (𝐶^′_𝑎,𝑏[0,𝑇],‖⋅‖_𝑏).

To obtain our main results, Theorems 14 and 17 below, we state a fundamental integration formula for the function space 𝐶_𝑎,𝑏[0,𝑇].

Let {𝑒₁,…,𝑒_𝑛} be an orthonormal set in (𝐶^′_𝑎,𝑏[0,𝑇],‖⋅‖_𝑏), let 𝑘∶ℝ^𝑛→ℂ be a Lebesgue measurable function, and let 𝐾∶𝐶_𝑎,𝑏[0,𝑇]→ℂ be given by 𝐾(𝑥)=𝑘𝑒₁,𝑥^∼,…,𝑒_𝑛,𝑥^∼. (53) Then 𝐸[𝐾]=_{𝐶𝑎,𝑏[0,𝑇]}𝑘𝑒₁,𝑥^∼,…,𝑒_𝑛,𝑥^∼(𝑥)=()𝑑𝜇2𝜋^−𝑛/2_ℝ𝑛𝑘𝑢₁,…,𝑢_𝑛×exp−^𝑛_𝑗=1𝑢_𝑗−𝑒_𝑗,𝑎_𝑏²2𝑑𝑢₁…𝑑𝑢_𝑛 (54) 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,…,𝑒_𝑛} be an orthonormal subset of (𝐶^′_𝑎,𝑏[0,𝑇],‖⋅‖_𝑏) and let 𝑤 be an element of 𝐶^′_𝑎,𝑏[0,𝑇]. Then for each 𝜆∈ℂ₊, the function space integral 𝐸_𝑥𝐺_𝑛(𝜆,𝑥)exp{𝑖(𝑤,𝑥)^∼} (55) exists and is given by the formula 𝐸_𝑥𝐺_𝑛(𝜆,𝑥)exp{𝑖(𝑤,𝑥)^∼}=𝜆^−𝑛/2exp𝜆−12𝜆^𝑛_𝑘=1𝑒_𝑘,𝑤²_𝑏−12‖𝑤‖²_𝑏+𝑖𝜆^𝑛−1/2_𝑘=1𝑒_𝑘,𝑎_𝑏𝑒_𝑘,𝑤_𝑏+𝑖𝑒_𝑛+1,𝑎_𝑏‖𝑤‖²_𝑏−^𝑛_𝑘=1𝑒_𝑘,𝑤²_𝑏^1/2, (56) where 𝐺_𝑛 is given by (52) above and 𝑒_𝑛+1=‖𝑤‖²_𝑏−^𝑛_𝑗=1𝑒_𝑗,𝑤²_𝑏^−1/2𝑤−^𝑛_𝑗=1𝑒_𝑗,𝑤_𝑏𝑒_𝑗. (57)

Proof (Outline). Using the Gram-Schmidt process, for any 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇], we can write 𝑤=∑^𝑛+1_𝑘=1𝑐_𝑘𝑒_𝑘, where {𝑒₁,…,𝑒_𝑛,𝑒_𝑛+1} is an orthonormal set in (𝐶^′_𝑎,𝑏[0,𝑇],‖⋅‖_𝑏) and 𝑐_𝑘=𝑒_𝑘,𝑤_𝑏𝑤‖,𝑘=1,…,𝑛‖²_𝑏−∑^𝑛_𝑗=1𝑒_𝑗,𝑤²_𝑏^1/2,𝑘=𝑛+1. (58) 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. Let 𝑞₀ be a positive real number and let Γ_𝑞0 be given by (35). For real numbers 𝑞₁ and 𝑞₂ with |𝑞_𝑗|>𝑞₀, 𝑗∈{1,2}, let {𝜆_𝑛}^∞_𝑛=1={(𝜆_1,𝑛,𝜆_2,𝑛)}^∞_𝑛=1 be a sequence in ℂ²₊ such that 𝜆_𝑛=𝜆_1,𝑛,𝜆_2,𝑛⟶−𝑖𝑞=−𝑖𝑞₁,−𝑖𝑞₂. (59) Let 𝜆_𝑗,𝑛=𝛼_𝑗,𝑛+𝑖𝛽_𝑗,𝑛 for 𝑗∈{1,2} and 𝑛∈ℕ. Then for 𝑗∈{1,2}, Re(𝜆_𝑗,𝑛)=𝛼_𝑗,𝑛>0 and 𝜆⁻¹_𝑗,𝑛=𝛼_𝑗,𝑛+𝑖𝛽_𝑗,𝑛⁻¹=𝛼_𝑗,𝑛−𝑖𝛽_𝑗,𝑛𝛼²_𝑗,𝑛+𝛽²_𝑗,𝑛 (60) for each 𝑛∈ℕ. Since |Im((−𝑖𝑞_𝑗)^−1/2)|=1/2|𝑞_𝑗|<1/2𝑞₀ for 𝑗∈{1,2}, there exists a sufficiently large 𝐿∈ℕ such that for any 𝑛≥𝐿, 𝜆_1,𝑛 and 𝜆_2,𝑛 are in Int(Γ_𝑞0) and 𝛿𝑞₁,𝑞₂≡supIm𝜆^−1/2_1,𝑛∪𝜆∶𝑛≥𝐿Im^−1/2_2,𝑛∪∶𝑛≥𝐿Im−𝑖𝑞₁^−1/2,Im−𝑖𝑞₂^−1/2<12𝑞₀. (61) Thus, there exists a positive real number 𝜀>1 such that 𝛿(𝑞₁,𝑞₂)<1/(𝜀2𝑞₀).
Let {𝑒_𝑛}^∞_𝑛=1 be a complete orthonormal set in (𝐶^′_𝑎,𝑏[0,𝑇],‖⋅‖_𝑏). Using Parseval's identity, it follows that 𝑔₁,𝑔_2𝑏=^∞_𝑛=1𝑒_𝑛,𝑔_1𝑏𝑒_𝑛,𝑔_2𝑏 (62) for all 𝑔₁, 𝑔₂∈𝐶^′_𝑎,𝑏[0,𝑇]. In addition for each 𝑔∈𝐶^′_𝑎,𝑏[0,𝑇], 𝑔²_𝑏−^𝑛_𝑘=1𝑒_𝑘,𝑔²_𝑏=^∞_𝑘=𝑛+1𝑒_𝑘,𝑔²_𝑏≥0 (63) for every 𝑛∈ℕ.
Since 𝑔,𝑎_𝑏=^∞_𝑛=1𝑒_𝑛,𝑔_𝑏𝑒_𝑛,𝑎_𝑏 (64) and for 𝜀>1−𝜀𝑔_𝑏‖𝑎‖_𝑏<−𝑔_𝑏‖𝑎‖_𝑏≤𝑔,𝑎_𝑏≤𝑔_𝑏‖𝑎‖_𝑏<𝜀𝑔_𝑏‖𝑎‖_𝑏, (65) there exists a sufficiently large 𝐾_𝑗∈ℕ such that for any 𝑛≥𝐾_𝑗^𝑛_𝑘=1𝑒_𝑘,𝐴_𝑗^1/2𝑤_𝑏𝑒_𝑘,𝑎_𝑏<𝜀𝐴_𝑗^1/2𝑤_𝑏‖𝑎‖_𝑏 (66) for 𝑗∈{1,2}.
Using these and a long and tedious calculation we can show that for every 𝑛≥max{𝐿,𝐾₁,𝐾₂}, exp²_𝑗=1𝜆_𝑗,𝑛−12𝜆_𝑗,𝑛^𝑛_𝑘=1𝑒_𝑘,𝐴_𝑗^1/2𝑤²_𝑏−12𝐴_𝑗^1/2𝑤²_𝑏+𝑖𝜆^𝑛−1/2_{𝑗,𝑛𝑘=1}𝑒_𝑘,𝐴_𝑗^1/2𝑤_𝑏𝑒_𝑘,𝑎_𝑏+𝑖𝑒_𝑛+1,𝑎_𝑏×𝐴_𝑗^1/2𝑤²_𝑏−^𝑛_𝑘=1𝑒_𝑘,𝐴_𝑗^1/2𝑤²_𝑏^1/2<𝑘𝑞₀;𝐴;𝑤, (67) where 𝑘(𝑞₀;𝐴;𝑤) is given by (36).

In our next theorem, for 𝐹∈ℱ_𝐴^𝑎,𝑏_1,𝐴2, we express the GFFT of 𝐹 as the limit of a sequence of function space integrals on 𝐶²_𝑎,𝑏[0,𝑇].

Theorem 14. Let 𝑞₀ and 𝐹 be as in Theorem 10. Let {𝑒_𝑛}^∞_𝑛=1 be a complete orthonormal set in (𝐶^′_𝑎,𝑏[0,𝑇],‖⋅‖_𝑏) and let {(𝜆_1,𝑛,𝜆_2,𝑛)}^∞_𝑛=1 be a sequence in ℂ²₊ such that 𝜆_𝑗,𝑛→−𝑖𝑞_𝑗 where 𝑞_𝑗 is a real number with |𝑞_𝑗|>𝑞₀, 𝑗∈{1,2}. Then for 𝑝∈[1,2] and for s-a.e. (𝑦₁,𝑦₂)∈𝐶²_𝑎,𝑏[0,𝑇], 𝑇_𝑞^(𝑝)(𝐹)𝑦₁,𝑦₂=lim_𝑛→∞𝜆^𝑛/2_1,𝑛𝜆^𝑛/2_2,𝑛×𝐸_𝑥𝐺_𝑛𝜆_1,𝑛,𝑥₁𝐺_𝑛𝜆_2,𝑛,𝑥₂𝐹𝑦₁+𝑥₁,𝑦₂+𝑥₂, (68) where 𝐺_𝑛 is given by (52).

Proof. By Theorems 9 and 10, we know that for each 𝑝∈[1,2], the 𝐿_𝑝 analytic GFFT of 𝐹, 𝑇_𝑞^(𝑝)(𝐹) exists and is given by the right hand side of (40). Thus, it suffices to show that 𝑇_𝑞⁽¹⁾(𝐹)𝑦₁,𝑦₂=𝐸^anf𝑞_𝑥𝐹𝑦₁+𝑥₁,𝑦₂+𝑥₂=lim_𝑛→∞𝜆^𝑛/2_1,𝑛𝜆^𝑛/2_2,𝑛×𝐸_𝑥𝐺_𝑛𝜆_1,𝑛,𝑥₁𝐺_𝑛𝜆_2,𝑛,𝑥₂×𝐹𝑦₁+𝑥₁,𝑦₂+𝑥₂⋅ (69)
Using (30), the Fubini theorem and (56) with 𝜆 and 𝑤 replaced with 𝜆_𝑗,𝑛 and 𝐴_𝑗^1/2𝑤, 𝑗∈{1,2}, respectively, we see that 𝜆^𝑛/2_1,𝑛𝜆^𝑛/2_2,𝑛𝐸_𝑥𝐺_𝑛𝜆_1,𝑛,𝑥₁𝐺_𝑛𝜆_2,𝑛,𝑥₂𝐹𝑦₁+𝑥₁,𝑦₂+𝑥₂=𝜆^𝑛/2_1,𝑛𝜆^𝑛/2_{2,𝑛𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝐴_𝑗^1/2𝑤,𝑦_𝑗^∼ײ_𝑗=1𝐸_𝑥𝑗𝐺_𝑛𝜆^−1/2_𝑗,𝑛,𝑥_𝑗×exp𝑖𝐴_𝑗^1/2𝑤,𝑥_𝑗^∼(𝑤)=𝑑𝑓_{𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝐴_𝑗^1/2𝑤,𝑦_𝑗^∼+𝜆_𝑗,𝑛−12𝜆_𝑗,𝑛^𝑛_𝑘=1𝑒_𝑘,𝐴_𝑗^1/2𝑤²_𝑏−12𝐴_𝑗^1/2𝑤²_𝑏+𝑖𝜆^𝑛−1/2_{𝑗,𝑛𝑘=1}𝑒_𝑘,𝑎_𝑏𝑒_𝑘,𝐴_𝑗^1/2𝑤_𝑏+𝑖𝑒_𝑛+1,𝑎_𝑏𝐴_𝑗^1/2𝑤²_𝑏−^𝑛_𝑘=1𝑒_𝑘,𝐴_𝑗^1/2𝑤²_𝑏^1/2𝑑𝑓(𝑤). (70) But, by Remark 13 we see that the last expression of (70) is dominated by (39) on the region Γ_𝑞0 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. Let 𝑞₀, 𝐹, {𝑒_𝑛}^∞_𝑛=1, {(𝜆_1,𝑛,𝜆_2,𝑛)}^∞_𝑛=1 and (𝑞₁,𝑞₂) be as in Theorem 14. Then 𝐸^anf𝑞_𝑥𝐹𝑥₁,𝑥₂=lim_𝑛→∞𝜆^𝑛/2_1,𝑛𝜆^𝑛/2_2,𝑛×𝐸_𝑥𝐺_𝑛𝜆_1,𝑛,𝑥₁𝐺_𝑛𝜆_2,𝑛,𝑥₂𝐹𝑥₁,𝑥₂, (71) where 𝐺_𝑛 is given by (52).

Corollary 16. Let 𝑞₀, 𝐹 and {𝑒_𝑛}^∞_𝑛=1 be as in Theorem 14 and let Γ_𝑞0 be given by (35). Let 𝜆=(𝜆₁,𝜆₂)∈Int(Γ_𝑞0) and {(𝜆_1,𝑛,𝜆_2,𝑛)}^∞_𝑛=1 be a sequence in ℂ²₊ such that 𝜆_𝑗,𝑛→𝜆_𝑗, 𝑗∈{1,2}. Then 𝐸^an𝜆_𝑥𝐹𝑥₁,𝑥₂=lim_𝑛→∞𝜆^𝑛/2_1,𝑛𝜆^𝑛/2_2,𝑛×𝐸_𝑥𝐺_𝑛𝜆_1,𝑛,𝑥₁𝐺_𝑛𝜆_2,𝑛,𝑥₂𝐹𝑥₁,𝑥₂, (72) where 𝐺_𝑛 is given by (52).

Our another result, namely, a change of scale formula for function space integrals now follows from Corollary 16 above.

Theorem 17. Let 𝐹∈ℱ_𝐴^𝑎,𝑏_1,𝐴2 and let {𝑒_𝑛}^∞_𝑛=1 be a complete orthonormal set in (𝐶^′_𝑎,𝑏[0,𝑇],‖⋅‖_𝑏). Then for any 𝜌₁>0 and 𝜌₂>0, 𝐸_𝑥𝐹𝜌₁𝑥₁,𝜌₂𝑥₂=lim_𝑛→∞𝜌₁^−𝑛𝜌₂^−𝑛×𝐸_𝑥𝐺_𝑛𝜌₁⁻²,𝑥₁𝐺_𝑛𝜌₂⁻²,𝑥₂𝐹𝑥₁,𝑥₂, (73) where 𝐺_𝑛 is given by (52).

Proof. Simply choose 𝜆_𝑗=𝜌_𝑗⁻² for 𝑗∈{1,2} and 𝜆_𝑗,𝑛=𝜌_𝑗⁻² for 𝑗∈{1,2} and 𝑛∈ℕ in (72).

Remark 18. Of course, if we choose 𝑎(𝑡)≡0, 𝑏(𝑡)=𝑡, 𝐴₁=𝐼 (identity operator), and 𝐴₂=0 (zero operator), then the function space 𝐶_𝑎,𝑏[0,𝑇] reduces to the classical Wiener space 𝐶₀[0,𝑇] and the generalized Fresnel type class ℱ_𝐴^𝑎,𝑏_1,𝐴2 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(𝐿²_𝑎,𝑏[0,𝑇]); the Banach algebra 𝒮(𝐿²_𝑎,𝑏[0,𝑇]) is introduced in [12].

5. Functionals in ℱ_𝐴^𝑎,𝑏_1,𝐴2

In this section, we prove a theorem ensuring that various functionals are in ℱ_𝐴^𝑎,𝑏_1,𝐴2.

Theorem 19. Let 𝐴₁ and 𝐴₂ be bounded, nonnegative, and self-adjoint operators on 𝐶^′_𝑎,𝑏[0,𝑇]. Let (𝑌,𝒴,𝛾) be a 𝜎-finite measure space and let 𝜑_𝑙∶𝑌→𝐶^′_𝑎,𝑏[0,𝑇] be 𝒴--ℬ(𝐶^′_𝑎,𝑏[0,𝑇]) measurable for 𝑙∈{1,…,𝑑}. Let 𝜃∶𝑌×ℝ^𝑑→ℂ be given by 𝜃(𝜂;⋅)=𝜈_𝜂(⋅), where 𝜈_𝜂∈ℳ(ℝ^𝑑) for every 𝜂∈𝑌 and where the family {𝜈_𝜂∶𝜂∈𝑌} satisfies(i)𝜈_𝜂(𝐸) is a 𝒴-measurable function of 𝜂 for every 𝐸∈ℬ(ℝ^𝑑);(ii)‖𝜈_𝜂‖∈𝐿¹(𝑌,𝒴,𝛾).
Under these hypothesis, the functional 𝐹∶𝐶²_𝑎,𝑏[0,𝑇]→ℂ given by 𝐹𝑥₁,𝑥₂=_𝑌𝜃𝜂;²_𝑗=1𝐴_𝑗^1/2𝜑₁𝜂,𝑥_𝑗^∼,…,²_𝑗=1𝐴_𝑗^1/2𝜑_𝑑𝜂,𝑥_𝑗^∼𝑑𝛾𝜂 (74) belongs to ℱ_𝐴^𝑎,𝑏_1,𝐴2 and satisfies the inequality ‖𝐹‖≤_𝑌𝜈_𝜂𝑑𝛾𝜂. (75)

Proof. Using the techniques similar to those used in [20], we can show that ‖𝜈_𝜂‖ 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)∈𝐶²_𝑎,𝑏[0,𝑇].
We define a measure 𝜏 on 𝒴×ℬ(ℝ^𝑑) by 𝜏(𝐸)=_𝑌𝜈_𝜂𝐸^(𝜂)𝑑𝛾𝜂,for𝐸∈𝒴×ℬℝ^𝑑. (76) Then by the first assertion of Theorem 3.1 in [17], 𝜏 satisfies ‖𝜏‖≤∫_𝑌‖𝜈_𝜂‖𝑑𝛾(𝜂). Now let Φ∶𝑌×ℝ^𝑑→𝐶^′_𝑎,𝑏[0,𝑇] be defined by Φ(𝜂;𝑣₁,…,𝑣_𝑑)=∑^𝑑_𝑙=1𝑣_𝑙𝜑_𝑙(𝜂). Then Φ is 𝒴×ℬ(ℝ^𝑑)--ℬ(𝐶^′_𝑎,𝑏[0,𝑇])-measurable on the hypothesis for 𝜑_𝑙, 𝑙∈{1,…,𝑑}. Let 𝜎=𝜏∘Φ⁻¹. Then clearly 𝜎∈ℳ(𝐶^′_𝑎,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)∈𝐶²_𝑎,𝑏[0,𝑇] and for every 𝜌₁>0 and 𝜌₂>0, 𝐹𝜌₁𝑥₁,𝜌₂𝑥₂=_𝑌𝜈_𝜂²_𝑗=1𝐴_𝑗^1/2𝜑₁𝜂,𝜌_𝑗𝑥_𝑗^∼,…,²_𝑗=1𝐴_𝑗^1/2𝜑_𝑑𝜂,𝜌_𝑗𝑥_𝑗^∼𝜂=𝑑𝛾_𝑌ℝ𝑑exp𝑖^𝑑_𝑙=1𝑣_𝑙²_𝑗=1𝐴_𝑗^1/2𝜑_𝑙𝜂,𝜌_𝑗𝑥_𝑗^∼𝑑𝜈_𝜂×𝑣₁,…,𝑣_𝑑𝜂=𝑑𝛾_{𝑌×ℝ𝑑}exp𝑖^𝑑_𝑙=1𝑣_𝑙²_𝑗=1𝐴_𝑗^1/2𝜑_𝑙𝜂,𝜌_𝑗𝑥_𝑗^∼×𝑑𝜏𝜂;𝑣₁,…,𝑣_𝑑=_{𝑌×ℝ𝑑}exp²_𝑗=1𝑖𝐴_𝑗^1/2Φ𝜂;𝑣₁,…,𝑣_𝑑,𝜌_𝑗𝑥_𝑗^∼×𝑑𝜏𝜂;𝑣₁,…,𝑣_𝑑=_{𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝐴_𝑗^1/2𝑤,𝜌_𝑗𝑥_𝑗^∼𝑑𝜏∘Φ⁻¹(𝑤)=_{𝐶′𝑎,𝑏[0,𝑇]}exp²_𝑗=1𝑖𝐴_𝑗^1/2𝑤,𝜌_𝑗𝑥_𝑗^∼𝑑𝜎(𝑤). (77) Thus, the functional 𝐹 given by (74) belongs to ℱ_𝐴^𝑎,𝑏_1,𝐴2 and satisfies the inequality ‖𝐹‖=‖𝜎‖≤‖𝜏‖≤_𝑌𝜈_𝜂𝑑𝛾𝜂. (78)

As mentioned in (2) of Remark 6, ℱ_𝐴^𝑎,𝑏_1,𝐴2 is a Banach algebra if Ran(𝐴₁+𝐴₂) is dense in 𝐶^′_𝑎,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. Let 𝐴₁ and 𝐴₂ be bounded, nonnegative and self-adjoint operators on 𝐶^′_𝑎,𝑏[0,𝑇] such that Ran(𝐴₁+𝐴₂) is dense in 𝐶^′_𝑎,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 ℱ_𝐴^𝑎,𝑏_1,𝐴2. In particular, exp{𝐹(𝑥₁,𝑥₂)}∈ℱ_𝐴^𝑎,𝑏_1,𝐴2.

Corollary 21. Let 𝐴₁ and 𝐴₂ be bounded, nonnegative, and self-adjoint operators on 𝐶^′_𝑎,𝑏[0,𝑇], and let {𝑔₁,…,𝑔_𝑑} be a finite subset of 𝐶^′_𝑎,𝑏[0,𝑇]. Given 𝛽=𝜈 where 𝜈∈ℳ(ℝ^𝑑), define 𝐹∶𝐶²_𝑎,𝑏[0,𝑇]→ℂ by 𝐹𝑥₁,𝑥₂=𝛽²_𝑗=1𝐴_𝑗^1/2𝑔₁,𝑥_𝑗^∼,…,²_𝑗=1𝐴_𝑗^1/2𝑔_𝑑,𝑥_𝑗^∼. (79) Then 𝐹 is an element of ℱ_𝐴^𝑎,𝑏_1,𝐴2.

Proof. Let (𝑌,𝒴,𝛾) be a probability space, and for 𝑙∈{1,…,𝑑}, let 𝜑_𝑙(𝜂)≡𝑔_𝑙. Take 𝜃(𝜂;⋅)=𝛽(⋅)=𝜈(⋅). Then for all 𝜌₁>0 and 𝜌₂>0 and for a.e. (𝑥₁,𝑥₂)∈𝐶²_𝑎,𝑏[0,𝑇], _𝑌𝜃𝜂;²_𝑗=1𝐴_𝑗^1/2𝜑₁𝜂,𝜌_𝑗𝑥_𝑗^∼,…,²_𝑗=1𝐴_𝑗^1/2𝜑_𝑑𝜂,𝜌_𝑗𝑥_𝑗^∼𝜂=𝑑𝛾_𝑌𝛽²_𝑗=1𝐴_𝑗^1/2𝑔₁,𝜌_𝑗𝑥_𝑗^∼,…,²_𝑗=1𝐴_𝑗^1/2𝑔_𝑑,𝜌_𝑗𝑥_𝑗^∼𝑑𝛾𝜂=𝛽²_𝑗=1𝐴_𝑗^1/2𝑔₁,𝜌_𝑗𝑥_𝑗^∼,…,²_𝑗=1𝐴_𝑗^1/2𝑔_𝑑,𝜌_𝑗𝑥_𝑗^∼=𝐹𝜌₁𝑥₁,𝜌₂𝑥₂. (80) Hence, 𝐹∈ℱ_𝐴^𝑎,𝑏_1,𝐴2.

Remark 22. Let 𝑑=1 and let (𝑌,𝒴,𝛾)=([0,𝑇],ℬ([0,𝑇]),𝑚_𝐿) in Theorem 19 where 𝑚_𝐿 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 𝐶^′_𝑎,𝑏[0,𝑇]. The function 𝜃 studied in [18] (and mentioned in Remark 2 above) is interpreted as the potential energy in quantum mechanics.

Acknowledgments

The authors would like to express their gratitude to the referees for their valuable comments and suggestions which have improved the original paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2011-0014552).