A Rotation of Admixable Operators on Abstract Wiener Space with Applications

Abstract

We investigate certain rotation properties of the abstract Wiener measure. To determine our rotation property for the Wiener measure, we introduce the concept of an admixable operator via an algebraic structure on abstract Wiener space. As for applications, we define the analytic Fourier-Feynman transform and the convolution product associated with the admixable operators and proceed to establish the relationships between this transform and the corresponding convolution product.

1. Introduction and Preliminaries

Let (𝐶₀[0,𝑇],ℳ,𝑚_𝑤) denote one-parameter Wiener space. Bearman's rotation theorem [1] for Wiener measure has played an important role in various research areas in mathematics and physics involving Wiener integration theory. Bearman's theorem was further developed by Cameron and Storvick [2] and by Johnson and Skoug [3] in their studies of Wiener integral equations. Recently, in [4], using results in [5], Chang et al. obtained results involving a very general multiple Fourier-Feynman transform on Wiener space.

Let 𝐻 be a real separable infinite-dimensional Hilbert space with inner product ⟨⋅,⋅⟩ and norm |⋅|=⟨⋅,⋅⟩. Let ‖⋅‖₀ be a measurable norm on 𝐻 with respect to the Gaussian cylinder set measure 𝜈₀ on 𝐻. Let 𝐵 denote the completion of 𝐻 with respect to ‖⋅‖₀. Let 𝑖 denote the natural injection from 𝐻 to 𝐵. The adjoint operator 𝑖^∗ of 𝑖 is one to one and maps 𝐵^∗ continuously onto a dense subset of 𝐻^∗, where 𝐵^∗ and 𝐻^∗ are topological duals of 𝐵 and 𝐻, respectively. By identifying 𝐻^∗ with 𝐻 and 𝐵^∗ with 𝑖^∗𝐵^∗, we have a triple 𝐵^∗⊂𝐻^∗≈𝐻⊂𝐵 and ⟨𝑥,𝑦⟩=(𝑥,𝑦) for all 𝑥 in 𝐻 and 𝑦 in 𝐵^∗, where (⋅,⋅) denotes the natural dual pairing between 𝐵 and 𝐵^∗. By the well-known result of Gross [6], 𝜈₀∘𝑖⁻¹ provides a unique countably additive extension, 𝜈, to the Borel 𝜎-algebra ℬ(𝐵) of 𝐵. 𝜈 is a probability measure on the Borel 𝜎-algebra ℬ(𝐵) of 𝐵 which satisfies _𝐵𝑖(𝑥)−1exp𝑦,𝑥𝑑𝜈=exp2𝑦²forevery𝑦∈𝐵^∗. (1) The triple (𝐵,𝐻,𝜈) is called an abstract Wiener space. For more details, see [6–9].

Let {𝑒_𝑗}^∞_𝑗=1 be a complete orthonormal set in 𝐻 such that 𝑒_𝑗's are in 𝐵^∗. For each ℎ∈𝐻 and 𝑥∈𝐵, we define the stochastic inner product (ℎ,𝑥)^∼ by (ℎ,𝑥)^∼=lim^𝑛_{𝑛→∞𝑗=1}ℎ,𝑒_𝑗𝑒_𝑗,𝑥,ifthelimitexists,0,otherwise. (2) For every ℎ (≠0) in 𝐻, (ℎ,𝑥)^∼ exists for 𝜈-a.e. 𝑥∈𝐵, and it is a Gaussian random variable on 𝐵 with mean zero and variance |ℎ|²; that is, (1) holds with 𝑦∈𝐵^∗ replaced with ℎ∈𝐻. In fact, the stochastic inner product (ℎ,𝑥)^∼ is essentially independent of the choice of the complete orthonormal set used in its definition. Also, if both ℎ and 𝑥 are in 𝐻, then Parseval's identity gives (ℎ,𝑥)^∼=⟨ℎ,𝑥⟩. Furthermore, (ℎ,𝜆𝑥)^∼=(𝜆ℎ,𝑥)^∼=𝜆(ℎ,𝑥)^∼ for any 𝜆∈ℝ, ℎ∈𝐻, and 𝑥∈𝐵. We also see that, if {ℎ₁,…,ℎ_𝑛} is an orthonormal set in 𝐻, then the random variables (ℎ_𝑗,𝑥)^∼’s are independent.

Let 𝒲(𝐵) be the class of 𝜈-measurable subsets of 𝐵. A subset 𝐸 of 𝐵 is said to be scale-invariant measurable [3, 7] provided 𝜌𝐸 is 𝒲(𝐵)-measurable for every 𝜌>0, and a scale-invariant measurable subset 𝑁 of 𝐵 is said to be scale-invariant null provided 𝜈(𝜌𝑁)=0 for every 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 on 𝐵 is said to be scale-invariant measurable provided 𝐹 is defined on a scale-invariant measurable set and 𝐹(𝜌⋅) is 𝒲(𝐵)-measurable for every 𝜌>0. If two functionals 𝐹 and 𝐺 on 𝐵 are equal 𝑠-a.e., that is, for each 𝜌>0, 𝜈({𝑥∈𝐵∶𝐹(𝜌𝑥)≠𝐺(𝜌𝑥)})=0, then we write 𝐹≈𝐺.

Next, we introduce the concept of an admixable operator on 𝐵.

Definition 1. Let ⊙ be an operation between 𝐻 and 𝐵^∗ which satisfies the conditions: (1)𝐵^∗×𝐵^∗∋(𝑔₁,𝑔₂)↦𝑔₁⊙𝑔₂=𝑔₂⊙𝑔₁∈𝐵^∗. (2)𝐻×𝐵^∗∋(ℎ,𝑔)↦ℎ⊙𝑔=𝑔⊙ℎ∈𝐻. (3) If ℎ⊙𝑔=0 for ℎ∈𝐻 and 𝑔∈𝐵^∗, then ℎ=0 or 𝑔=0. (4) For every ℎ∈𝐻 and every 𝑔₁,𝑔₂∈𝐵^∗, ℎ⊙𝑔₁⊙𝑔₂=ℎ⊙𝑔₁⊙𝑔₂. (3)(5) For every ℎ₁,ℎ₂∈𝐻 and every 𝑔∈𝐵^∗, ℎ₁+ℎ₂⊙𝑔=ℎ₁⊙𝑔+ℎ₂⊙𝑔. (4)(6) For every 𝑔₁,𝑔₂∈𝐵^∗, there exists 𝑔₃∈𝐵^∗ such that 𝑔₁^⊙2+𝑔₂^⊙2=𝑔₃^⊙2, (5) where 𝑔^⊙2=𝑔⊙𝑔. In this case, we write 𝑔₃=^⊙𝑔₁^⊙2+𝑔₂^⊙2. (7) For every ℎ₁,ℎ₂∈𝐻 and every 𝑔∈𝐵^∗, ⟨ℎ₁,ℎ₂⊙𝑔⟩=⟨ℎ₁⊙𝑔,ℎ₂⟩. (6)
Given 𝑔∈𝐵^∗, let 𝐴_𝑔∶𝐵→𝐵 be a linear operator associated with 𝑔. The operator 𝐴_𝑔 is said to be 𝑔^⊙-admixable provided (ℎ,𝐴_𝑔𝑥)^∼=(ℎ⊙𝑔,𝑥)^∼ for all ℎ∈𝐻.

For a finite subset 𝒱={𝑣₁,…,𝑣_𝑚} of 𝐻, let 𝑋_𝒱∶𝐵→ℝ^𝑚 be the random vector given by 𝑋_𝒱(𝑥)=𝑣₁,𝑥^∼,…,𝑣_𝑚,𝑥^∼≡(𝑣,𝑥)^∼. (7) A functional 𝐹 is called a cylinder-type functional on 𝐵 if there exists a linearly independent subset 𝒱={𝑣₁,…,𝑣_𝑚} of 𝐻 such that 𝐹(𝑥)=𝜓𝑋_𝒱(𝑥),𝑥∈𝐵, (8) where 𝜓 is a complex-valued Lebesgue measurable function on ℝ^𝑚. It is easy to show that, for the given cylinder-type functional 𝐹 of the form (8), there exists an orthogonal subset ℋ={ℎ₁,…,ℎ_𝑛} of 𝐻 such that 𝐹 is expressed as 𝐹(𝑥)≡𝑓𝑋_ℋ(𝑥)=𝑓ℎ₁,𝑥^∼,…,ℎ_𝑛,𝑥^∼,𝑥∈𝐵, (9) where 𝑓 is a complex-valued Lebesgue measurable function on ℝ^𝑛. Thus, we lose no generality in assuming that every cylinder-type functional on 𝐵 is of the form (9).

Lemma 2 (Chung, [7]). Let (𝐵,𝐻,𝜈) be an abstract Wiener space, and let ℋ={ℎ₁,…,ℎ_𝑛} be an orthogonal set in 𝐻. Let 𝑓 be a complex-valued function defined on ℝ^𝑛. Then, for the cylinder-type functional 𝐹 given by (9) on 𝐵,(i)𝐹 is ℬ(𝐵)-measurable if and only if 𝑓 is Borel measurable on ℝ^𝑛, (ii)𝐹 is 𝒲(𝐵)-measurable if and only if 𝑓 is Lebesgue measurable on ℝ^𝑛.

For 𝑔∈𝐵^∗, let 𝐴_𝑔 be the 𝑔^⊙-admixable operator on 𝐵. In this case, for any orthogonal subset ℋ of 𝐻, 𝑋_ℋ𝐴_𝑔𝑥=ℎ₁,𝐴_𝑔𝑥^∼,…,ℎ_𝑛,𝐴_𝑔𝑥^∼=ℎ₁⊙𝑔,𝑥^∼,…,ℎ_𝑛⊙𝑔,𝑥^∼. (10)

The seminal results by Bearman in [1] are summarized as follows (see [2]): if 𝐹(𝑎²+𝑏²𝑥) is Wiener integrable on 𝐶₀[0,𝑇] for 𝑎,𝑏∈ℝ, then 𝐹(𝑎𝑥₁+𝑏𝑥₂) is integrable on (𝐶₀[0,𝑇])² and _{(𝐶0[0,𝑇])2}𝐹𝑎𝑥₁+𝑏𝑥₂𝑑𝑚_𝑤×𝑚_𝑤𝑥₁,𝑥₂=_{𝐶0[0,𝑇]}𝐹𝑎²+𝑏²𝑥𝑑𝑚_𝑤(𝑥). (11)

The main purpose of this paper is to establish a rotation property for the abstract Wiener integral, _𝐵2𝐹𝐴_𝑔1𝑥₁+𝐴_𝑔2𝑥₂𝑑(𝜈×𝜈)𝑥₁,𝑥₂=_𝐵𝐹𝐴_𝑔𝑥𝑑𝜈(𝑥), (12) where 𝐹 is given by (9) and 𝐴_𝑔 is determined by 𝐴_𝑔1 and 𝐴_𝑔2.

2. A Typical Example of an Abstract Wiener Space

The classical Wiener space 𝐶₀[0,𝑇], which is one of the most important examples of abstract Wiener spaces (see [9]), is a triple (𝐵,𝐻,𝜈), where (i)𝐵=𝐶₀[0,𝑇] is a Banach space consisting of real-valued continuous functions 𝑥(𝑡) with 𝑥(0)=0 defined on the compact interval [0,𝑇] endowed with uniform norm ‖𝑥‖=sup_{𝑡∈[0,𝑇]}|𝑥(𝑡)|, (ii)𝐻=𝐶^′₀[0,𝑇] is a real separable infinite dimensional Hilbert space consisting of absolutely continuous functions ℎ(𝑡) with ℎ(0)=0, such that 𝐷ℎ≡𝑑ℎ/𝑑𝑡∈𝐿₂[0,𝑇] endowed with the inner product ℎ₁,ℎ₂=^𝑇₀𝐷ℎ₁(𝑠)𝐷ℎ₂(𝑠)𝑑𝑠, (13)(iii)𝜈=𝑚_𝑤 is the Wiener measure on the Borel 𝜎-algebra ℬ(𝐶₀[0,𝑇]) of 𝐶₀[0,𝑇] with 𝑚_𝑤({𝑥∶𝑥(𝑡)≤𝑎})=^𝑎_−∞−𝑢exp²2𝑡𝑑𝑢. (14)

Let 𝐼 be the unitary operator from 𝐿₂[0,𝑇], onto 𝐶^′₀[0,𝑇] given by 𝐼𝑣(𝑡)=∫^𝑡₀𝑣(𝑠)𝑑𝑠 for 𝑣∈𝐿₂[0,𝑇] and let 𝐶^∗₀[]=[.0,𝑇𝐼𝑣∶𝑣iscontinuousexceptforafinitenumberoffinitejumpdiscontinuitiesandisofboundedvariationon0,𝑇]} (15) For any ℎ∈𝐶^′₀[0,𝑇] and 𝑔∈𝐶^∗₀[0,𝑇], let the operation ⊙ between 𝐶^′₀[0,𝑇] and 𝐶^∗₀[0,𝑇] be defined by ℎ⊙𝑔=𝐼𝐷ℎ𝐷𝑔, (16) where 𝐷ℎ𝐷𝑔 denotes the pointwise multiplication of the functions 𝐷ℎ and 𝐷𝑔.

It is readily seen that {𝑥(𝑡)∶(𝑥,𝑡)∈𝐶₀[0,𝑇]×[0,𝑇]} is a standard Wiener process on the probability space (𝐶₀[0,𝑇],ℬ(𝐶₀[0,𝑇]),𝑚_𝑤). In this case, we know that, for each ℎ∈𝐶^′₀[0,𝑇] and 𝑠-a.e. 𝑥∈𝐶₀[0,𝑇], (ℎ,𝑥)^∼=^𝑇₀𝐷ℎ(𝑡)𝑑𝑥(𝑡), (17) where ∫^𝑇₀𝐷ℎ(𝑡)𝑑𝑥(𝑡) denotes the Paley-Wiener-Zygmund stochastic integral [10–12].

Remark 3. Let ℳ be the set of all 𝑚_𝑤-measurable subsets of 𝐶₀[0,𝑇]. Then, (𝐶₀[0,𝑇],ℳ,𝑚_𝑤) is a complete measure space. It is well known that ℳ coincides with 𝜎(ℬ(𝐶₀[0,𝑇])), the completion of ℬ(𝐶₀[0,𝑇]).

Let 𝑔∈𝐶^′₀[0,𝑇] with |𝑔|=⟨𝑔,𝑔⟩>0. Then, the stochastic integral 𝒵_𝑔(𝑥,𝑡)=^𝑡₀𝐷𝑔(𝑠)𝑑𝑥(𝑠),𝑡∈[0,𝑇], (18) which was introduced by Park and Skoug in [13], is a Gaussian process with mean zero and covariance function _{𝐶0[0,𝑇]}𝒵_𝑔(𝑥,𝑠)𝒵_𝑔(𝑥,𝑡)𝑑𝑚_𝑤(𝑥)=₀^{min{𝑠,𝑡}}𝐷𝑔(𝑢)²𝑑𝑢. (19) In addition, 𝒵_𝑔(𝑥,𝑡) is stochastically continuous in 𝑡 on [0,𝑇]. For more detailed studies of this process, see [14–17]. Furthermore, if 𝑔 is an element of 𝐶^∗₀[0,𝑇], then, for all 𝑥∈𝐶₀[0,𝑇], 𝒵_𝑔(𝑥,𝑡) is continuous in 𝑡, and so is 𝒵_𝑔(𝑥,⋅) in 𝐶₀[0,𝑇].

From [14, Lemma 1], we note that, for each ℎ∈𝐶^′₀[0,𝑇] and each 𝑔∈𝐶^′₀[0,𝑇] with 𝐷𝑔∈𝐿_∞[0,𝑇], 𝐷ℎ(^𝑇₀𝑡)𝑑𝒵_𝑔(𝑥,𝑡)=^𝑇₀𝐷ℎ(𝑡)𝐷𝑔(𝑡)𝑑𝑥(𝑡) (20) for 𝑠-a.e. 𝑥∈𝐶₀[0,𝑇].

Given 𝑔∈𝐶^∗₀[0,𝑇], define an operator 𝐴_𝑔∶𝐶₀[0,𝑇]→𝐶₀[0,𝑇] by 𝐴_𝑔𝑥=𝒵_𝑔(𝑥,⋅). (21) Then, for all ℎ∈𝐻, ℎ,𝐴_𝑔𝑥^∼=^𝑇₀𝐷ℎ(𝑡)𝑑𝒵_𝑔()=𝑥,𝑡_𝑇0𝐷ℎ(𝑡)𝐷𝑔(𝑡)𝑑𝑥(𝑡)=ℎ⊙𝑔,𝑥^∼. (22) Thus, 𝐴_𝑔 is 𝑔^⊙-admixable in view of Definition 1.

3. A Rotation of Admixable Operators

In this section, we establish a rotation property for the abstract Wiener integral involving admixable operators. We first introduce an integration formula which plays a key role.

Lemma 4. Let 𝒜={𝛼₁,…,𝛼_𝑛} be an orthogonal set in 𝐻 and let 𝑋_𝒜 be given by (7). Let 𝑓∶ℝ^𝑛→ℂ be a Lebesgue measurable function. Then _𝐵𝑓𝑋_𝒜(𝑥)(𝑥)=𝑑𝜈_𝐵𝑓𝛼₁,𝑥^∼,…,𝛼_𝑛,𝑥^∼𝑑𝜈_∗(𝑥)=^𝑛_𝑗=12𝜋𝛼_𝑗^2−1/2×_ℝ𝑛𝑓𝑢₁,…,𝑢_𝑛×exp−^𝑛_𝑗=1𝑢²_𝑗2𝛼_𝑗²𝑑𝑢₁⋯𝑑𝑢_𝑛, (23) where by =^∗ one means that if either side exists, both sides exist and equality holds.

The following integration formula is also used several times in this paper: _ℝexp−𝑎𝑢²+𝑏𝑢𝑑𝑢=𝜋𝑎𝑏exp²4𝑎 (24) for complex numbers 𝑎 and 𝑏 with Re(𝑎)>0.

Remark 5. Let 𝛼¹ and 𝛼² be elements in 𝐻 with |𝛼¹|=|𝛼²|≡𝜎²>0. Then, the random variables 𝑋₁(𝑥)=(𝛼¹,𝑥)^∼ and 𝑋₂(𝑥)=(𝛼²,𝑥)^∼ will have the same distribution 𝑁(0,𝜎²).

Let 𝒜₁={𝛼¹₁,…,𝛼¹_𝑛} and 𝒜₂={𝛼²₁,…,𝛼²_𝑛} be orthogonal sets in 𝐻 with |𝛼¹_𝑗|=|𝛼²_𝑗| for each 𝑗∈{1,…,𝑛}. Using the aforementioned facts and applying (23), we see that, for any Lebesgue measurable function 𝑓 on ℝ^𝑛, _𝐵𝑓𝑋_𝒜1(𝑥)𝑑𝜈(𝑥)=^∗_𝐵𝑓𝑋_𝒜2(𝑥)𝑑𝜈(𝑥). (25)

To simplify the expressions in our results, we use the following notations: 𝑓(𝑢)≡𝑓𝑢₁,…,𝑢_𝑛,𝑓𝑢+𝑋_𝒜(𝑥)≡𝑓𝑢₁+𝛼₁,𝑥^∼,…,𝑢_𝑛+𝛼_𝑛,𝑥^∼ (26) for 𝑢=(𝑢₁,…,𝑢_𝑛)∈ℝ^𝑛 and 𝑋_𝒜(𝑥) given by (7).

Proposition 6. Let 𝒜₁={𝛼¹₁,…,𝛼¹_𝑛} and 𝒜₂={𝛼²₁,…,𝛼²_𝜈} be orthogonal sets in 𝐻. Then, for any Lebesgue measurable function 𝑓 on ℝ^𝑛, _𝐵2𝑓𝑋_𝒜1𝑥₁+𝑋_𝒜2𝑥₂𝑑(𝜈×𝜈)𝑥₁,𝑥_2∗=_𝐵𝑓𝑋_𝒲(2)(𝑥)𝑑𝜈(𝑥), (27) where 𝒲₍₂₎={𝑤₁,…,𝑤_𝑛} is an orthogonal set in 𝐻 which satisfies the condition 𝑤_𝑗²=𝛼¹_𝑗²+𝛼²_𝑗² (28) for each 𝑗∈{1,…,𝑛}. Also, both of the expressions in (27) are given by the last expression in (29).

Proof. First, using (23), the Fubini theorem, and (24), we have that _𝐵2𝑓𝑋_𝒜1𝑥₁+𝑋_𝒜2𝑥₂𝑑(𝜈×𝜈)𝑥₁,𝑥₂=_𝐵𝐵𝑓𝑋_𝒜1𝑥₁+𝑋_𝒜2𝑥₂𝑑𝜈𝑥₁𝑑𝜈𝑥₂=^𝑛_𝑗=12𝜋𝛼¹_𝑗^2−1/2×_ℝ𝑛𝐵𝑓𝑢+𝑋_𝒜2𝑥₂𝑑𝜈𝑥₂×exp−^𝑛_𝑗=1𝑢²_𝑗2𝛼¹_𝑗²𝑑𝑢=^𝑛_𝑗=12𝜋𝛼¹_𝑗^2−1/2𝑛_𝑗=12𝜋𝛼²_𝑗^2−1/2×_{ℝ𝑛ℝ𝑛}𝑓(𝑣)−𝑢+×exp^𝑛_𝑗=1𝑣²_𝑗2𝛼²_𝑗²𝑑𝑣×exp−^𝑛_𝑗=1𝑢²_𝑗2𝛼¹_𝑗²𝑑𝑢=^𝑛_𝑗=12𝜋𝛼¹_𝑗^2−1/2𝑛_𝑗=12𝜋𝛼²_𝑗^2−1/2×_ℝ𝑛𝑓(𝑟)⃗exp−^𝑛_𝑗=1𝑟_𝑗−𝑢_𝑗²2𝛼²_𝑗²×exp−^𝑛_𝑗=1𝑢²_𝑗2𝛼¹_𝑗²𝑑𝑟⃗=𝑢𝑑_𝑛𝑗=12𝜋𝛼¹_𝑗^2−1/2𝑛_𝑗=12𝜋𝛼²_𝑗^2−1/2×_ℝ𝑛𝑓(𝑟)⃗_ℝ𝑛exp−^𝑛_𝑗=1𝑢²_𝑗2𝛼¹_𝑗²−^𝑛_𝑗=1𝑟_𝑗−𝑢_𝑗²2𝛼²_𝑗²𝑑𝑢𝑑𝑟⃗=^𝑛_𝑗=12𝜋𝛼¹_𝑗²+𝛼²_𝑗^2−1/2×_ℝ𝑛𝑓(𝑟)⃗exp−^𝑛_𝑗=1𝑟²_𝑗2𝛼¹_𝑗²+𝛼²_𝑗²𝑑𝑟.⃗ (29)
Next, let {𝑒₁,…,𝑒_𝑛} be any orthonormal set in 𝐻. For each 𝑗∈{1,…,𝑛}, let 𝑤_𝑗=(|𝛼¹_𝑗|²+|𝛼²_𝑗|²)^1/2𝑒_𝑗. Then 𝒲₍₂₎={𝑤₁,…,𝑤_𝑛} is an orthogonal set in 𝐻 and satisfies (28) above. In this case, using (23), the Fubini theorem, and (24), we see that ∫_𝐵𝑓(𝑋_𝒲(2)(𝑥))𝑑𝜈(𝑥) is given by the last expression of (29). In view of Remark 5, we obtain the desired result.

The following corollary follows by the use of mathematical induction.

Corollary 7. For each 𝑘∈{1,…,𝑚}, let 𝒜_𝑘={𝛼^𝑘₁,…,𝛼^𝑘_𝑛} be an orthogonal set in 𝐻. Then for any Lebesgue measurable function 𝑓 on ℝ^𝑛, _𝐵𝑚𝑓𝑋_𝒜1𝑥₁+⋯+𝑋_𝒜𝑚𝑥_𝑚𝑑𝜈^𝑚𝑥₁,…,𝑥_𝑚∗=_𝐵𝑓𝑋_𝒲(𝑚)(𝑥)𝑑𝜈(𝑥), (30) where 𝒲_(𝑚)={𝑤₁,…,𝑤_𝑛} is an orthogonal set in 𝐻 which satisfies the condition 𝑤_𝑗²=^𝑚_𝑘=1𝛼^𝑘_𝑗² (31) for each 𝑗∈{1,…,𝑛}. Also, both of the expressions in (30) are given by the expression ^𝑛_𝑗=12𝜋^𝑚_𝑘=1𝛼^𝑘_𝑗^2−1/2×_ℝ𝑛𝑓(𝑟)⃗exp−^𝑛_𝑗=1𝑟²_𝑗2∑^𝑚_𝑘=1𝛼^𝑘_𝑗²𝑑𝑟.⃗ (32)

The next corollary follows directly from Proposition 6.

Corollary 8. Let 𝒜₁ and 𝒜₂ be as in Proposition 6. Assume that 𝒜₁∪𝒜₂ is an orthogonal set in 𝐻. Then, for any Borel measurable function 𝑓 on ℝ^𝑛, (27) is valid. In this case, 𝒲₍₂₎ is given by the set {𝛼¹₁+𝛼²₁,…,𝛼¹_𝑛+𝛼²_𝑛}.

Throughout the rest of this section, for convenience, we use the following notation: for a finite sequence 𝒢={𝑔₁,…,𝑔_𝑚} in 𝐵^∗, let 𝑠(𝒢)≡𝑠𝑔₁,…,𝑔_𝑚=^⊙𝑚_𝑘=1𝑔_𝑘^⊙2. (33)

For our rotation property presented, namely, Theorem 11, we will consider the pair of finite subsets ℋ={ℎ₁,…,ℎ_𝑛} of 𝐻−{0} and 𝒢={𝑔₁,…,𝑔_𝑚} of 𝐵^∗−{0} (this allows |𝒢|≤𝑚) such that (c1){ℎ₁,…,ℎ_𝑛} is orthogonal in 𝐻, (c2){ℎ₁⊙𝑔_𝑘,…,ℎ_𝑛⊙𝑔_𝑘} is orthogonal in 𝐻 for all 𝑘∈{1,…,𝑚}.

Let us return to the classical Wiener space (𝐶₀[0,𝑇],𝐶^′₀[0,𝑇],𝑚_𝑤). See Section 2. For each 𝑙∈{1,2,…}, let 𝛽_𝑙(𝑡)=sin(((𝑙−1/2)𝜋/𝑇)𝑡) on [0,𝑇]. Then, Θ≡{𝛽_𝑙}^∞_𝑙=1 is an orthogonal sequence in 𝐶^′₀[0,𝑇]. Additionally, Θ={(√2𝑇/(𝑙−1/2)𝜋)𝛽_𝑙}^∞_𝑙=1 is a complete orthonormal set in 𝐶^′₀[0,𝑇]. In fact, 𝛽_𝑙∈𝐶^∗₀[0,𝑇] for all 𝑙∈{1,2,…}. We can take different pair (ℋ,𝒢) from Θ, which satisfies conditions (c1) and (c2) above. For instance, let ℋ={𝛽_𝑙1,…,𝛽_𝑙𝑛} and let 𝒢={𝛽_𝑘1,…,𝛽_𝑘𝑚}, where 𝑙₁<𝑙₂<⋯<𝑙_𝑛<𝑘₁≤𝑘₂≤⋯≤𝑘_𝑚.

Remark 9. Let ℋ={ℎ₁,…,ℎ_𝑛} and 𝒢={𝑔₁,…,𝑔_𝑚} satisfy the conditions (c1) and (c2). Then, the stochastic inner products ℎ_𝑗,𝐴_𝑠(𝒢)𝑥^∼=ℎ_𝑗⊙𝑠(𝒢),𝑥^∼,𝑗=1,…,𝑛, (34) form a set of independent Gaussian random variables on 𝐵 with mean 0 and variance ℎ_𝑗⊙𝑠(𝒢)²=ℎ_𝑗⊙𝑠(𝒢),ℎ_𝑗(𝒢=ℎ⊙𝑠)_𝑗⊙𝑠(𝒢)⊙𝑠(𝒢),ℎ_𝑗=ℎ_𝑗⊙^𝑚_𝑘=1𝑔_𝑘^⊙2,ℎ_𝑗=^𝑚_𝑘=1ℎ_𝑗⊙𝑔_𝑘^⊙2,ℎ_𝑗=^𝑚_𝑘=1ℎ_𝑗⊙𝑔_𝑘^⊙2,ℎ_𝑗=^𝑚_𝑘=1ℎ_𝑗⊙𝑔_𝑘,ℎ_𝑗⊙𝑔_𝑘=^𝑚_𝑘=1ℎ_𝑗⊙𝑔_𝑘², (35) where 𝑠(𝒢)≡𝑠(𝑔₁,…,𝑔_𝑚) is given by (33).

Remark 10. Given an orthogonal subset ℋ, let 𝐵^∗_ℋ be the space of every element 𝑔 in 𝐵^∗−{0} such that {ℎ⊙𝑔∶ℎ∈ℋ} is orthogonal in 𝐻. Then, for any finite subset 𝒢 of 𝐵^∗_ℋ, the pair (ℋ,𝒢) satisfies conditions (c1) and (c2) above.

Given an orthogonal set ℋ in 𝐻, let 𝔊_ℋ⁽¹⁾ be the class of all cylinder-type functionals, 𝐹, given by (9) for 𝜈-a.e. 𝑥∈𝐵, where the corresponding function 𝑓 of 𝐹 satisfies the condition _ℝ𝑛𝑓(𝑣)exp−^𝑛_𝑗=1𝑣²_𝑗2ℎ_𝑗⊙𝑔²𝑑𝑣<+∞ (36) for all 𝑔∈𝐵^∗_ℋ.

We are now ready to present our rotation property of abstract Wiener measure associated with admixable operators.

Theorem 11. Let ℋ={ℎ₁,…,ℎ_𝑛} be an orthogonal set in 𝐻 and let 𝐹∈𝔊_ℋ⁽¹⁾ be given by (9). Let 𝒢={𝑔₁,…,𝑔_𝑚} be a finite subset of 𝐵^∗_ℋ, and, for each 𝑘∈{1,…,𝑚}, let 𝐴_𝑔𝑘 be the 𝑔^⊙_𝑘-admixable operator on 𝐵. Then, _𝐵𝑚𝐹^𝑚_𝑘=1𝐴_𝑔𝑘𝑥_𝑘𝑑𝜈^𝑚𝑥₁,…,𝑥_𝑚=_𝐵𝐹𝐴_𝑠(𝒢)𝑥𝑑𝜈(𝑥), (37) where 𝑠(𝒢) is given by (33).

Proof. For each 𝑗∈{1,…,𝑛} and each 𝑘∈{1,…,𝑚}, let 𝛼^𝑘_𝑗=ℎ_𝑗⊙𝑔_𝑘. For each 𝑘∈{1,…,𝑚}, 𝒜_𝑘={𝛼^𝑘₁,…,𝛼^𝑘_𝑛} is orthogonal in 𝐻 by condition (c2). Hence, for each 𝑘∈{1,…,𝑚}, the stochastic inner products ℎ_𝑗,𝐴_𝑔𝑘𝑥^∼=𝛼^𝑘_𝑗,𝑥^∼,𝑗=1,…,𝑛, (38) form a set of independent Gaussian random variables with mean 0 and variance |𝛼^𝑘_𝑗|².
We observe that 𝑋_ℋ^𝑚_𝑘=1𝐴_𝑔𝑘𝑥_𝑘=ℎ₁,^𝑚_𝑘=1𝐴_𝑔𝑘𝑥_𝑘^∼,…,ℎ_𝑛,^𝑚_𝑘=1𝐴_𝑔𝑘𝑥_𝑘^∼=^𝑚_𝑘=1ℎ₁,𝐴_𝑔𝑘𝑥_𝑘^∼,…,^𝑚_𝑘=1ℎ_𝑛,𝐴_𝑔𝑘𝑥_𝑘^∼=^𝑚_𝑘=1ℎ₁,𝐴_𝑔𝑘𝑥_𝑘^∼,…,ℎ_𝑛,𝐴_𝑔𝑘𝑥_𝑘^∼=^𝑚_𝑘=1𝛼^𝑘₁,𝑥_𝑘^∼,…,𝛼^𝑘_𝑛,𝑥_𝑘^∼=^𝑚_𝑘=1𝑋_𝒜𝑘𝑥_𝑘. (39) Thus, using (9), (39), and (30), we obtain _𝐵𝑚𝐹^𝑚_𝑘=1𝐴_𝑔𝑘𝑥_𝑘𝑑𝜈^𝑚𝑥₁,…,𝑥_𝑚=_𝐵𝑚𝑓𝑋_ℋ^𝑚_𝑘=1𝐴_𝑔𝑘𝑥_𝑘𝑑𝜈^𝑚𝑥₁,…,𝑥_𝑚=_𝐵𝑚𝑓^𝑚_𝑘=1𝑋_𝒜𝑘𝑥_𝑘𝑑𝜈^𝑚𝑥₁,…,𝑥_𝑚=_𝐵𝑓𝑋_𝒲(𝑚)(𝑥)𝑑𝜈(𝑥), (40) where 𝒲_(𝑚)={𝑤₁,…,𝑤_𝑛} is an orthogonal set in 𝐻, such that 𝑤_𝑗²=^𝑚_𝑘=1𝛼^𝑘_𝑗²=^𝑚_𝑘=1ℎ_𝑗⊙𝑔_𝑘² (41) for each 𝑗∈{1,…,𝑛}.
Next, we note that, for each 𝑗,𝑙∈{1,…,𝑛} with 𝑗≠𝑙, ℎ_𝑗⊙𝑠(𝒢),ℎ_𝑙⊙𝑠(𝒢)=ℎ_𝑗⊙𝑠(𝒢)⊙𝑠(𝒢),ℎ_𝑙=ℎ_𝑗⊙^𝑚_𝑘=1𝑔_𝑘^⊙2,ℎ_𝑙=^𝑚_𝑘=1ℎ_𝑗⊙𝑔_𝑘^⊙2,ℎ_𝑙=^𝑚_𝑘=1ℎ_𝑗⊙𝑔_𝑘^⊙2,ℎ_𝑙=^𝑚_𝑘=1ℎ_𝑗⊙𝑔_𝑘,ℎ_𝑙⊙𝑔_𝑘=0. (42) Hence, from (35) and (42), we see that {ℎ₁⊙𝑠(𝒢),…,ℎ_𝑛⊙𝑠(𝒢)} is an orthogonal set in 𝐻 with |ℎ_𝑗⊙𝑠(𝒢)|²=∑^𝑚_𝑘=1|ℎ_𝑗⊙𝑔_𝑘|² for all 𝑗∈{1,…,𝑛}, and so we can choose 𝒲_(𝑚) to be the orthogonal set {ℎ₁⊙𝑠(𝒢),…,ℎ_𝑛⊙𝑠(𝒢)}. In this case, we see that _𝐵𝑓𝑋_𝒲(𝑚)(𝑥)(𝑥)=𝑑𝜈_𝐵𝑓ℎ₁⊙𝑠(𝒢),𝑥^∼,…,ℎ_𝑛⊙𝑠(𝒢),𝑥^∼𝑑𝜈(𝑥)=_𝐵𝑓ℎ₁,𝐴_𝑠(𝒢)𝑥^∼,…,ℎ_𝑛,𝐴_𝑠(𝒢)𝑥^∼(𝑥)=𝑑𝜈_𝐵𝐹𝐴_𝑠(𝒢)𝑥𝑑𝜈(𝑥). (43) Equation (37) follows from (40) and (43).

Example 12. Let us return to the classical Wiener space (𝐶₀[0,𝑇],𝐶^′₀[0,𝑇],𝑚_𝑤) again. We introduce the family of functions Γ≡{𝛾_𝜏∶𝜏∈[0,𝑇]} from 𝐶^′₀[0,𝑇]: 𝛾_𝜏(𝑠)=[][].𝑠,𝑠∈0,𝜏𝜏,𝑠∈𝜏,𝑇 (44) These functions have the reproducing property ℎ,𝛾_𝜏=^𝑇₀𝐷ℎ(𝑠)𝐷𝛾_𝜏(𝑠)=𝑑𝑠_𝑇0𝐷ℎ(𝑠)𝜒_[0,𝜏](𝑠)=𝑑𝑠_𝜏0𝐷ℎ(𝑠)𝑑𝑠=ℎ(𝜏) (45) for all ℎ∈𝐶^′₀[0,𝑇]. In fact, Γ⊂𝐶^∗₀[0,𝑇].
For 𝛾_𝜏∈Γ, we consider the admixable operator 𝐴_𝛾𝜏 given by (21). We note that, for each 𝑔∈𝐶^∗₀[0,𝑇], 𝑔,𝐴_𝛾𝜏𝑥^∼=𝑔⊙𝛾_𝜏,𝑥^∼=^𝑇₀𝐷𝑔(𝑠)𝐷𝛾_𝜏(𝑠)(𝑠)=𝑑𝑥_𝜏0𝐷𝑔(𝑠)𝑑𝑥(𝑠)=𝒵_𝑔(𝑥,𝜏)=𝐴_𝑔𝑥(𝜏). (46) From this, we see that, for 𝑔_𝐼(𝑡)=𝑡=∫^𝑡₀𝑑𝑠∈𝐶^∗₀[0,𝑇] and 𝑎∈ℝ, 𝑔_𝐼,𝐴_𝑎𝛾𝜏𝑥^∼=𝑎𝑥(𝜏), (47) and so, using (37), we find that, for any 𝑓∈𝐿₁(ℝ) and every 𝑎,𝑏∈ℝ, _{(𝐶0[0,𝑇])2}𝑓𝑎𝑥₁(𝜏)+𝑏𝑥₂(𝜏)𝑑𝑚_𝑤×𝑚_𝑤𝑥₁,𝑥₂=_{(𝐶0[0,𝑇])2}𝑓𝑔_𝐼,𝐴_𝑎𝛾𝜏𝑥₁^∼+𝑔_𝐼,𝐴_𝑏𝛾𝜏𝑥₂^∼𝑑𝑚_𝑤×𝑚_𝑤𝑥₁,𝑥₂=_{𝐶0[0,𝑇]}𝑓𝑔_𝐼,𝐴_{𝑠(𝑎𝛾𝜏,𝑏𝛾𝜏)}𝑥^∼𝑑𝑚_𝑤(𝑥). (48) Using (16), we now have 𝑠𝑎𝛾_𝜏,𝑏𝛾_𝜏(𝑡)=^⊙𝑎𝛾_𝜏^⊙2+𝑏𝛾_𝜏^⊙2(𝑡)=𝑎𝛾_𝜏^⊙2+𝑏𝛾_𝜏^⊙2(𝑡)=𝑎²+𝑏²𝛾_𝜏(𝑡) (49) on [0,𝑇]. Thus, by (48), (49), and (47), we have _{(𝐶0[0,𝑇])2}𝑓𝑎𝑥₁(𝜏)+𝑏𝑥₂(𝜏)𝑑𝑚_𝑤×𝑚_𝑤𝑥₁,𝑥₂=_{𝐶0[0,𝑇]}𝑓𝑎²+𝑏²𝑥(𝜏)𝑑𝑚_𝑤(𝑥). (50)

Proposition 13 (Cameron and Storvick, [2]). Let 𝐹(𝑎²+𝑏²𝑥) be Wiener integrable on 𝐶₀[0,𝑇] for 𝑎,𝑏∈ℝ. As a result, 𝐹(𝑎𝑥₁+𝑏𝑥₂) is integrable on 𝐶²₀[0,𝑇], and _{𝐶20[0,𝑇]}𝐹𝑎𝑥₁+𝑏𝑥₂𝑑𝑚_𝑤×𝑚_𝑤𝑥₁,𝑥₂=_{𝐶0[0,𝑇]}𝐹𝑎²+𝑏²𝑥𝑑𝑚_𝑤(𝑥). (51)

Proof. Let 𝑛 be any positive integer, and let 0=𝜏₀<𝜏₁<⋯<𝜏_𝑛≤𝑇 be any partition of [0,𝑇]. It suffices to show that (51) holds for any tame function 𝐹(𝑥)=𝑓𝑥𝜏₁,𝑥𝜏₂−𝑥𝜏₁,…,𝑥𝜏_𝑛−𝑥𝜏_𝑛−1 (52) with 𝑓∈𝐿₁(ℝ^𝑛).
Let ⊙ be defined by (16) between 𝐶^′₀[0,𝑇] and 𝐶^∗₀[0,𝑇]. For 𝑔_𝐼(𝑡)=𝑡=∫^𝑡₀𝑑𝑠∈𝐶^∗₀[0,𝑇], let the 𝑔^⊙_𝐼-admixable operator 𝐴_𝑔𝐼 be given by (21), and, for each 𝑗∈{1,…,𝑛}, let ℎ_𝑗(𝑡)=^𝑡₀𝜒_{[𝜏𝑗−1,𝜏𝑗]}(𝑠)𝑑𝑠 (53) on [0,𝑇]. For any 𝑎,𝑏∈ℝ−{0}, (c1){ℎ₁,…,ℎ_𝑛} is an orthogonal set in 𝐶^′₀[0,𝑇], and (c2){ℎ₁⊙𝑎𝑔_𝐼,…,ℎ_𝑛⊙𝑎𝑔_𝐼} and {ℎ₁⊙𝑏𝑔_𝐼,…,ℎ_𝑛⊙𝑏𝑔_𝐼} are orthogonal sets in 𝐶^′₀[0,𝑇].
Also, for any 𝑎,𝑏∈ℝ−{0}, 𝐴_{𝑠(𝑎𝑔𝐼,𝑏𝑔𝐼)}=𝐴_{𝑎2+𝑏2𝑔𝐼}, and, for each 𝑗∈{1,…,𝑛}, ℎ_𝑗,𝐴_𝑎𝑔𝐼𝑥^∼=𝑎^𝑇₀𝜒_{[𝜏𝑗−1,𝜏𝑗]}(𝑠)(𝑠)𝑥𝜏𝑑𝑥=𝑎_𝑗−𝑥𝜏_𝑗−1. (54) Thus, the left side of (51) with 𝐹 given by (52) is rewritten by _{(𝐶0[0,𝑇])2}𝐹𝑎𝑥₁+𝑏𝑥₂𝑑𝑚_𝑤×𝑚_𝑤𝑥₁,𝑥₂=_{(𝐶0[0,𝑇])2}𝑓ℎ₁,𝐴_𝑎𝑔𝐼𝑥₁^∼+ℎ₁,𝐴_𝑏𝑔𝐼𝑥₂^∼,…,ℎ_𝑛,𝐴_𝑎𝑔𝐼𝑥₁^∼+ℎ_𝑛,𝐴_𝑏𝑔𝐼𝑥₂^∼𝑑𝑚_𝑤×𝑚_𝑤𝑥₁,𝑥₂. (55) Thus, from Theorem 11, we obtain _{(𝐶0[0,𝑇])2}𝐹𝑎𝑥₁+𝑏𝑥₂𝑑𝑚_𝑤×𝑚_𝑤𝑥₁,𝑥₂=_{𝐶0[0,𝑇]}𝑓ℎ₁,𝐴_{𝑎2+𝑏2𝑔𝐼}𝑥^∼,…,ℎ_𝑛,𝐴_{𝑎2+𝑏2𝑔𝐼}𝑥^∼𝑑𝑚_𝑤(𝑥)=_{𝐶0[0,𝑇]}𝐹𝑎²+𝑏²𝑥𝑑𝑚_𝑤(𝑥). (56)
Using standard methods, similar to those in [1], we can obtain the result for general functionals 𝐹 on 𝐶₀[0,𝑇].

4. Fourier-Feynman Transforms and Convolutions Associated with Admixable Operators

In this section, to apply our results from the previous section, we first define an 𝐿_𝑝 analytic Fourier-Feynman transform associated with admixable operators on 𝐵. Then, we establish the existence theorem and the inverse transform theorem of this transform for some classes of cylinder-type functionals on 𝐵 having the form (9) for 𝑠-a.e. 𝑥∈𝐵. Moreover, we present various relationships involving the convolution and the transforms.

Throughout the rest of this paper, let ℂ, ℂ₊, and ℂ₊ denote, respectively, the complex numbers, the complex numbers with positive real part, and the nonzero complex numbers with nonnegative real part.

Let 𝑔∈𝐵^∗, and let 𝐴_𝑔 be the corresponding admixable operator on 𝐵. Let 𝐹∶𝐵→ℂ be a scale-invariant measurable functional such that 𝐽_𝐹𝑔;𝜆=_𝐵𝐹𝜆^−1/2𝐴_𝑔𝑥𝑑𝜈(𝑥) (57) exists as a finite number for all 𝜆>0. If there exists a function 𝐽^∗_𝐹(𝑔;𝜆) analytic on ℂ₊ such that 𝐽^∗_𝐹(𝑔;𝜆)=𝐽_𝐹(𝑔;𝜆) for all 𝜆>0, then 𝐽^∗_𝐹(𝑔;𝜆) is defined to be the analytic Wiener integral (associated with the 𝑔^⊙-admixable operator 𝐴_𝑔) of 𝐹 over 𝐵 with parameter 𝜆. For 𝜆∈ℂ₊, we write 𝐼^anw𝜆_𝑔[𝐹]=𝐽^∗_𝐹𝑔;𝜆. (58) Let 𝑞≠0 be a real number, and let 𝐹 be a functional such that ∫^anw𝜆_𝐵𝐹(𝐴_𝑔𝑥)𝑑𝜈(𝑥)=𝐽^∗_𝐹(𝑔;𝜆) exists for all 𝜆∈ℂ₊. If the following limit exists, we call it the analytic Feynman integral of 𝐹 with parameter 𝑞, and we write 𝐼^anf𝑞_𝑔[𝐹]=lim_{𝜆→−𝑖𝑞}𝐼^anw𝜆_𝑔[𝐹], (59) where 𝜆 approaches −𝑖𝑞 through values in ℂ₊.

Note that if 𝐴_𝑔 is the identity operator on 𝐵, then these definitions agree with the previous definitions of the analytic Wiener integral and the analytic Feynman integral [18–20].

We are now ready to state the definition of the analytic Fourier-Feynman transform associated with admixable operator (admix-FFT).

Definition 14. Let (𝐵,𝐻,𝜈) be an abstract Wiener space. For 𝑔∈𝐵^∗, 𝜆∈ℂ₊, and 𝑦∈𝐵, let 𝑇_𝜆,𝑔(𝐹)𝑦=^anw𝜆_𝐵𝐹𝑦+𝐴_𝑔𝑥𝑑𝜈(𝑥), (60) where 𝐴_𝑔 is the 𝑔^⊙-admixable operator on 𝐵. Let 𝑞 be a nonzero real number. For 𝑝∈(1,2], we define the 𝐿_𝑝 analytic 𝑔^⊙-admix-FFT, 𝑇^(𝑝)_𝑞,𝑔(𝐹) of 𝐹, by the formula (𝜆∈ℂ₊), 𝑇^(𝑝)_𝑞,𝑔(𝐹)𝑦=l.i.m._{𝜆→−𝑖𝑞}𝑇_𝜆,𝑔(𝐹)𝑦, (61) if it exists; that is, for each 𝜌>0, lim_{𝜆→−𝑖𝑞𝐵}𝑇_𝜆,𝑔(𝐹)𝜌𝑦−𝑇^(𝑝)_𝑞,𝑔(𝐹)𝜌𝑦^𝑝′𝑑𝜈𝑦=0, (62) where 1/𝑝+1/𝑝^′=1. We define the 𝐿₁ analytic 𝑔^⊙-admix-FFT, 𝑇⁽¹⁾_𝑞,𝑔(𝐹) of 𝐹, by the formula (𝜆∈ℂ₊), 𝑇⁽¹⁾_𝑞,𝑔(𝐹)𝑦=lim_{𝜆→−𝑖𝑞}𝑇_𝜆,𝑔(𝐹)𝑦, (63) if it exists.

We note that, for 𝑝∈[1,2], 𝑇^(𝑝)_𝑞,𝑔(𝐹) is defined only 𝑠-a.e.. We also note that if 𝑇^(𝑝)_𝑞,𝑔(𝐹) exists and if 𝐹≈𝐺, then 𝑇^(𝑝)_𝑞,𝑔(𝐺) exists and 𝑇^(𝑝)_𝑞,𝑔(𝐺)≈𝑇^(𝑝)_𝑞,𝑔(𝐹).

Next, we give the definition of the convolution product (CP).

Definition 15. Let 𝐹 and 𝐺 be scale-invariant measurable functionals on 𝐵. For 𝜆∈ℂ₊ and 𝑔∈𝐵^∗, we define their CP with respect to 𝐴_𝑔 (if it exists) by (𝐹∗𝐺)_𝜆,𝑔𝑦=∫^anw𝜆_𝐵𝐹𝑦+𝐴_𝑔𝑥√2𝐺𝑦−𝐴_𝑔𝑥√2𝑑𝜈(𝑥),𝜆∈ℂ₊,∫^anf𝑞_𝐵𝐹𝑦+𝐴_𝑔𝑥√2𝐺𝑦−𝐴_𝑔𝑥√2𝑑𝜈(𝑥),𝜆=−𝑖𝑞,𝑞∈ℝ,𝑞≠0. (64) When 𝜆=−𝑖𝑞, we denote (𝐹∗𝐺)_𝜆,𝑔 by (𝐹∗𝐺)_𝑞,𝑔.

For any scale-invariant measurable functional 𝐹, we see that, for 𝜆>0, ^anw𝜆_𝐵𝐹𝑦+𝐴_𝑔𝑥𝑑𝜈(𝑥)=_𝐵𝐹𝑦+𝜆^−1/2𝐴_𝑔𝑥𝑑𝜈(𝑥) (65) if it exists.

Let ℳ(ℝ^𝑛) denote the space of complex-valued, countably additive (and hence finite) Borel measures on ℬ(ℝ^𝑛), the Borel 𝜎-algebra of ℝ^𝑛. It is well known that a complex-valued Borel measure 𝜏 necessarily has a finite total variation ‖𝜏‖, and ℳ(ℝ^𝑛) is a Banach algebra under the norm ‖⋅‖ and with convolution as multiplication.

For 𝜏∈ℳ(ℝ^𝑛), the Fourier transform 𝜏 of 𝜏 is a complex-valued function defined on ℝ^𝑛 by the formula 𝜏(𝑢)=_ℝ𝑛exp𝑖^𝑛_𝑗=1𝑢_𝑗𝑣_𝑗𝑑𝜏(𝑣), (66) where 𝑢=(𝑢₁,…,𝑢_𝑛) and 𝑣=(𝑣₁,…,𝑣_𝑛) are in ℝ^𝑛.

Let ℋ={ℎ₁,…,ℎ_𝑛} be an orthonormal set in 𝐻. Define the functional 𝐹∶𝐵→ℂ by 𝐹(𝑥)=𝜏𝑋_ℋ(𝑥)=𝜏ℎ₁,𝑥^∼,…,ℎ_𝑛,𝑥^∼ (67) for 𝑠-a.e. 𝑥∈𝐵, where 𝜏 is the Fourier transform of 𝜏 in ℳ(ℝ^𝑛). Then 𝐹 is a bounded cylinder-type functional because |𝜏(𝑢)|≤‖𝜏‖<+∞.

Let 𝔗_ℋ be the set of all functionals 𝐹 on 𝐵 having the form (67). Note that 𝐹∈𝔗_ℋ implies that 𝐹 is scale-invariant measurable on 𝐵. Throughout this section, we fix the orthogonal set ℋ.

We now state the existence theorem for the analytic Feynman integral of the functionals in 𝔗_ℋ.

Theorem 16. Let 𝐹∈𝔗_ℋ be given by (67). Then, for all 𝑔∈𝐵^∗_ℋ and all nonzero real numbers 𝑞, the analytic Feynman integral 𝐼^{𝑎𝑛𝑓𝑞}_𝑔[𝐹] of 𝐹 exists and is given by the formula 𝐼^{𝑎𝑛𝑓𝑞}_𝑔[𝐹]=_ℝ𝑛−𝑖exp2𝑞^𝑛_𝑗=1ℎ_𝑗⊙𝑔²𝑣²_𝑗𝑑𝜏(𝑣). (68)

Proof. By (67), (66), the Fubini theorem, (23), and (24), we see that, for all 𝜆>0, 𝐽_𝐹=𝑔;𝜆_𝐵𝐹𝜆^−1/2𝐴_𝑔𝑥(𝑥)=𝑑𝜈_ℝ𝑛𝐵exp𝑖𝜆^𝑛−1/2_𝑗=1ℎ_𝑗⊙𝑔,𝑥^∼𝑣_𝑗(𝑥)(𝑣)=𝑑𝜈𝑑𝜏_𝑛𝑗=1𝜆2𝜋ℎ_𝑗⊙𝑔^21/2×_{ℝ𝑛ℝ𝑛}exp𝑖^𝑛_𝑗=1𝑢_𝑗𝑣_𝑗−𝜆2^𝑛_𝑗=1𝑢²_𝑗ℎ_𝑗⊙𝑔²𝑑𝑢(𝑣)=𝑑𝜏_𝑛𝑗=1𝜆2𝜋ℎ_𝑗⊙𝑔^21/2×_ℝ𝑛^𝑛_𝑗=1ℝexp𝑖𝑢_𝑗𝑣_𝑗−𝜆2𝑢²_𝑗ℎ_𝑗⊙𝑔²𝑑𝑢_𝑗(𝑣)=𝑑𝜏_ℝ𝑛−1exp2𝜆^𝑛_𝑗=1ℎ_𝑗⊙𝑔²𝑣²_𝑗𝑑𝜏(𝑣). (69)
Now, let 𝐽^∗_𝐹(𝑔;𝜆)=∫_ℝ𝑛exp{−(1/2𝜆)∑^𝑛_𝑗=1|ℎ_𝑗⊙𝑔|²𝑣²_𝑗}𝑑𝜏(𝑣) for 𝜆∈ℂ₊. Then, 𝐽^∗_𝐹(𝑔;𝜆)=𝐽_𝐹(𝑔;𝜆) for all 𝜆>0 and |𝐽^∗_𝐹(𝑔;𝜆)|≤∫_ℝ𝑛𝑑|𝜏|(𝑣)≤‖𝜏‖<∞ for all 𝜆∈ℂ₊. Thus, applying the dominated convergence theorem, we see that 𝐽^∗_𝐹(𝑔;𝜆) is continuous on ℂ₊. Also, because 𝑘(𝜆)≡exp{−(1/2𝜆)∑^𝑛_𝑗=1|ℎ_𝑗⊙𝑔|²𝑣²_𝑗} is analytic on ℂ₊, applying the Fubini theorem, we have _Δ𝐽^∗_𝐹𝑔;𝜆𝑑𝜆=_ℝ𝑛Δ𝑘(𝜆)𝑑𝜆𝑑𝜈(𝑣)=0 (70) for all rectifiable simple closed curve Δ lying in ℂ₊. Thus, by the Morera theorem, 𝐽^∗_𝐹(𝑔;𝜆) is analytic on ℂ₊. Therefore, the analytic Wiener integral 𝐼^anw𝜆_𝑔[𝐹]=𝐽^∗_𝐹(𝑔;𝜆) exists. Finally, applying the dominated convergence theorem, we know that 𝐼^anf𝑞_𝑔[𝐹]=lim_{𝜆→−𝑖𝑞}𝐼^anw𝜆_𝑔[𝐹] is given by the right side of (68).

Next, we establish the existence of the admix-FFT for functionals in 𝔗_ℋ.

Theorem 17. Let 𝐹∈𝔗_ℋ be given by (67). Then, for all 𝑝∈[1,2] and all 𝑔∈𝐵^∗, the analytic 𝐿_𝑝𝑔^⊙-admix-FFT, 𝑇^(𝑝)_𝑞,𝑔(𝐹), exists for all nonzero real numbers 𝑞, belongs to 𝔗_ℋ, and is given by the formula 𝑇^(𝑝)_𝑞,𝑔(𝐹)𝑦=_ℝ𝑛exp𝑖^𝑛_𝑗=1ℎ_𝑗,𝑦^∼𝑣_𝑗𝑑𝜏^𝑞_𝑔(𝑣) (71) for 𝑠-a.e. 𝑦∈𝐵, where 𝜏^𝑞_𝑔 is the complex measure on ℝ^𝑛 given by 𝜏^𝑞_𝑔(𝐸)=_𝐸−𝑖exp2𝑞^𝑛_𝑗=1ℎ_𝑗⊙𝑔²𝑣²_𝑗𝑑𝜏(𝑣) (72) for 𝐸∈ℬ(ℝ^𝑛).

Proof. Proceeding as in the proof of Theorem 16, we see that, for all 𝜆∈ℂ₊, all 𝑔∈𝐵^∗_ℋ, and for 𝑠-a.e. 𝑦∈𝐵, 𝑇_𝜆,𝑔(𝐹)𝑦=_𝐵𝐹𝑦+𝜆^−1/2𝐴_𝑔𝑥(𝑥)=𝑑𝜈_ℝ𝑛exp𝑖^𝑛_𝑗=1ℎ_𝑗,𝑦^∼𝑣_𝑗−12𝜆^𝑛_𝑗=1ℎ_𝑗⊙𝑔²𝑣²_𝑗𝑑𝜏(𝑣), (73) is an analytic function of 𝜆 on ℂ₊, and that for any 𝑞∈ℝ−{0} and 𝑠-a.e. 𝑦∈𝐵, 𝑇⁽¹⁾_𝑞,𝑔(𝐹)𝑦=lim_{𝜆→−𝑖𝑞}𝑇_𝜆,𝑔(𝐹𝑦=_ℝ𝑛)exp𝑖^𝑛_𝑗=1ℎ_𝑗,𝑦^∼𝑣_𝑗−𝑖2𝑞^𝑛_𝑗=1ℎ_𝑗⊙𝑔²𝑣²_𝑗𝑑𝜏(𝑣). (74) Clearly, the set function 𝜏^𝑞_𝑔 given by (72) is a complex measure on ℬ(ℝ^𝑛), and so the right side of (74) can be rewritten as the right side of (71).
Next, we note that 𝑇_𝜆,𝑔(𝐹)𝑦≤_ℝ𝑛𝑑|𝜏|(𝑣)=‖𝜏‖ (75) for all 𝜆∈ℂ₊ and 𝑇⁽¹⁾_𝑞,𝑔(𝐹)𝑦≤_ℝ𝑛𝑑|𝜏|(𝑣)=‖𝜏‖. (76) Using these, we see that, for all 𝑝∈(1,2], all 𝜌>0, and all 𝜆∈ℂ₊, 𝑇_𝜆,𝑔(𝐹)𝜌𝑦−𝑇⁽¹⁾_𝑞,𝑔(𝐹)𝜌𝑦^𝑝′≤(2‖𝜏‖)^𝑝′, (77) and so, by the dominated convergence theorem, we see that, for any nonzero real 𝑞, lim_{𝜆→−𝑖𝑞𝐵}𝑇_𝜆,𝑔(𝐹)𝜌𝑦−𝑇⁽¹⁾_𝑞,𝑔(𝐹)𝜌𝑦^𝑝′𝑑𝜈𝑦=0. (78) Hence, 𝑇^(𝑝)_𝑞,𝑔(𝐹)(𝑦) exists and is given by the right side of (74) for all desired values of 𝑝 and 𝑞 and all 𝑔∈𝐵^∗_ℋ. Thus, the theorem is proved.

Theorem 18. Let 𝐹∈𝔗_ℋ be given by (67). Then, for all 𝑝∈[1,2], all 𝑔∈𝐵^∗_ℋ, and all nonzero real 𝑞, 𝑇^(𝑝)_−𝑞,𝑔𝑇^(𝑝)_𝑞,𝑔(𝐹)≈𝐹. (79) As such, the admix-FFT, 𝑇^(𝑝)_𝑞,𝑔, has the inverse transform {𝑇^(𝑝)_𝑞,𝑔}⁻¹=𝑇^(𝑝)_−𝑞,𝑔.

Theorem 19. Let 𝐹 and 𝐺 be elements of 𝔗_ℋ with corresponding finite Borel measures 𝜏 and 𝜇 in ℳ(𝐻). Then, for all 𝑘∈𝐵^∗_ℋ, the CP, (𝐹∗𝐺)_𝑞,𝑘, exists for all nonzero real numbers 𝑞, belongs to 𝔗_ℋ, and is given by the formula (𝐹∗𝐺)_𝑞,𝑘𝑦=_ℝ𝑛exp𝑖^𝑛_𝑗=1ℎ_𝑗,𝑦^∼𝑟_𝑗𝑑𝜛^𝑞_𝑘∘𝜑⁻¹(ℎ) (80) for 𝑠-a.e. 𝑦∈𝐵, where 𝜑∶ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 is a continuous function defined by 𝜑(𝑤)=(𝑤)𝑣,𝑣+√2, (81) and 𝜛^𝑞_𝑘 is a complex measure on ℬ(ℝ^𝑛) given by 𝜛^𝑞_𝑘(𝐸)=_𝐸−𝑖exp4𝑞^𝑛_𝑗=1ℎ_𝑗⊙𝑘²𝑣_𝑗−𝑤_𝑗²𝑑𝜏(𝑣)𝑑𝜇(𝑤) (82) for 𝐸∈ℬ(ℝ^𝑛).

Proof. Using (64), the Fubini theorem, (23), and (24), we have that, for all 𝜆>0 and 𝑠-a.e. 𝑦∈𝐵, (𝐹∗𝐺)_𝜆,𝑘𝑦=_𝐵𝐹𝑦+𝐴_𝑘𝑥√2𝐺𝑦−𝐴_𝑘𝑥√2(𝑥)=𝑑𝜈_ℝ𝑛𝑖exp√2^𝑛_𝑗=1ℎ_𝑗,𝑦^∼𝑣_𝑗+𝑤_𝑗×_𝐵𝑖exp√2𝜆^𝑛_𝑗=1ℎ_𝑗⊙𝑘,𝑥^∼×𝑣_𝑗−𝑤_𝑗(𝑥)(𝑣)(𝑤)=𝑑𝜈𝑑𝜏𝑑𝜇_ℝ𝑛𝑖exp√2^𝑛_𝑗=1ℎ_𝑗,𝑦^∼𝑣_𝑗+𝑤_𝑗−14𝜆^𝑛_𝑗=1ℎ_𝑗⊙𝑘²𝑣_𝑗−𝑤_𝑗²𝑑𝜏(𝑣)𝑑𝜇(𝑤). (83) Using the same argument as in the proof of Theorem 17, we can show that the last expression in the previous equation is an analytic function of 𝜆 on ℂ₊ and is a bounded continuous function of 𝜆 on ℂ₊ because 𝜏 and 𝜇 are finite Borel measures. Hence, (𝐹∗𝐺)_𝑞,𝑘 exists and is given by (𝐹∗𝐺)_𝑞,𝑘𝑦=_ℝn𝑖exp√2^𝑛_𝑗=1ℎ_𝑗,𝑦^∼𝑣_𝑗+𝑤_𝑗−𝑖4𝑞^𝑛_𝑗=1ℎ_𝑗⊙𝑘²𝑣_𝑗−𝑤_𝑗²𝑑𝜏(𝑣)𝑑𝜇(𝑤) (84) for all 𝑞∈ℝ−{0} and 𝑠-a.e. 𝑦∈𝐵.
Consider the set function 𝜛^𝑞_𝑘 and the continuous function 𝜑 given by (82) and (81), respectively. Clearly, the set function 𝜛^𝑞_𝑘 is a complex measure on ℬ(ℝ^𝑛). Hence, 𝜛^𝑞_𝑘∘𝜑 is an element of ℳ(ℝ^𝑛), and so the right side of (84) can be rewritten as the right side of (80). Thus, the theorem is proved.

Lemma 20. Let ℋ={ℎ₁,…,ℎ_𝑛} be any orthogonal set in 𝐻. For every 𝑔∈𝐵^∗_ℋ, every 𝑣=(𝑣₁,…,𝑣_𝑛) and 𝑤=(𝑤₁,…,𝑤_𝑛) in ℝ^𝑛, let 𝑌^ℋ_𝑣𝑔,,𝑍^ℋ_𝑤𝑔,∶𝐵²→ℝ be given by 𝑌^ℋ_𝑣𝑔,𝑥₁,𝑥₂=^𝑛_𝑗=1ℎ_𝑗,𝐴_𝑔𝑥₁+𝐴_𝑔𝑥₂^∼√2𝑣_𝑗,𝑍^ℋ_𝑤𝑔,𝑥₁,𝑥₂=^𝑛_𝑗=1ℎ_𝑗,𝐴_𝑔𝑥₁−𝐴_𝑔𝑥₂^∼√2𝑤_𝑗, (85) respectively. As a result, 𝑌^ℋ_𝑣𝑔, and 𝑍^ℋ_𝑤𝑔, are independent random variables.

Proof. Since the random variables 𝑌^ℋ_𝑣𝑔, and 𝑍^ℋ_𝑤𝑔, are Gaussian with mean zero, it suffices to show that _𝐵2𝑌^ℋ_𝑣𝑔,𝑥₁,𝑥₂𝑍^ℋ_𝑤𝑔,𝑥₁,𝑥₂𝑑(𝜈×𝜈)𝑥₁,𝑥₂=0. (86)
We know that {ℎ_𝑗⊙𝑔,…,ℎ_𝑛⊙𝑔} is an orthogonal set in 𝐻; thus, {(ℎ_𝑗⊙𝑔,𝑥)^∼,…,(ℎ_𝑛⊙𝑔,𝑥)^∼} is a set of independent Gaussian random variables with mean zero on 𝐵. However, using the Fubini theorem, we obtain _𝐵2𝑌^ℋ_𝑣𝑔,𝑥₁,𝑥₂𝑍^ℋ_𝑤𝑔,𝑥₁,𝑥₂𝑑(𝜈×𝜈)𝑥₁,𝑥₂=12^𝑛𝑛_𝑗=1𝑘=1𝑣_𝑗𝑤_𝑘𝐵ℎ_𝑗⊙𝑔,𝑥₁^∼×ℎ_𝑘⊙𝑔,𝑥₁^∼𝑑𝜈𝑥₁𝑑𝜈𝑥₂+𝑣_𝑗𝑤_𝑘𝐵ℎ_𝑗⊙𝑔,𝑥₁^∼𝑑𝜈𝑥₁×_𝐵ℎ_𝑘⊙𝑔,𝑥₂^∼𝑑𝜈𝑥₂+𝑣_𝑗𝑤_𝑘𝐵ℎ_𝑗⊙𝑔,𝑥₂^∼𝑑𝜈𝑥₂×_𝐵ℎ_𝑘⊙𝑔,𝑥₁^∼𝑑𝜈𝑥₁−𝑣_𝑗𝑤_𝑘𝐵ℎ_𝑗⊙𝑔,𝑥₂^∼×ℎ⊙𝑔,𝑥₂^∼𝑑𝜈𝑥₁𝑑𝜈𝑥₂=12^𝑛𝑛_𝑗=1𝑘=1𝑣_𝑗𝑤_𝑘𝐵ℎ_𝑗⊙𝑔,𝑥₁^∼ℎ_𝑘⊙𝑔,𝑥₁^∼𝑑𝜈𝑥₁−𝑣_𝑗𝑤_𝑘𝐵ℎ_𝑗⊙𝑔,𝑥₂^∼×ℎ_𝑘⊙𝑔,𝑥₂^∼𝑑𝜈𝑥₂=12^𝑛_𝑗=1𝑣_𝑗𝑤_𝑗𝐵ℎ_𝑗⊙𝑔,𝑥₁^∼2𝑑𝜈𝑥₁−𝑣_𝑗𝑤_𝑗𝐵ℎ_𝑗⊙𝑔,𝑥₂^∼𝑑𝜈𝑥₂+12^𝑛_𝑗≠𝑘𝑣_𝑗𝑤_𝑘𝐵ℎ_𝑗⊙𝑔,𝑥₁^∼ℎ_𝑘⊙𝑔,𝑥₁^∼𝑑𝜈𝑥₁−𝑣_𝑗𝑤_𝑘𝐵ℎ_𝑗⊙𝑔,𝑥₂^∼ℎ⊙𝑔,𝑥₂^∼𝑑𝜈𝑥₂=0, (87) which concludes the proof of Lemma 20.

Remark 21. For each 𝑣∈ℝ^𝑛, let 𝜓(𝑣;𝑥)=exp𝑖^𝑛_𝑗=1ℎ_𝑗,𝑥^∼𝑣_𝑗,𝑥∈𝐵. (88) As such, (67) is rewritten as 𝐹(𝑥)=_ℝ𝑛𝜓(𝑣;𝑥)𝑑𝜏(𝑣), (89) and the functional 𝜓(𝑣;⋅) is an element of 𝔊_ℋ⁽¹⁾ for each 𝑣.

Applying the same method as used in the proof of Theorem 3.5 in [4], we have the following theorem. For the proof of Theorem 22, we can apply Lemma 20 and Theorem 11 to the functional 𝜓 given by (88).

Theorem 22. Let 𝐹, 𝐺, 𝜏, and 𝜇 be as in Theorem 19, and let 𝑔 be an element of 𝐵^∗_ℋ. Then, for all 𝑝∈[1,2] and all nonzero real 𝑞, 𝑇^(𝑝)_𝑞,𝑔(𝐹∗𝐺)_𝑞,𝑔𝑦=𝑇^(𝑝)_𝑞,𝑔(𝐹)𝑦√2𝑇^(𝑝)_𝑞,𝑔(𝐺)𝑦√2 (90) for 𝑠-a.e. 𝑦∈𝐵.

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 Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0014552).