Change of Scale Formulas for Wiener Integrals Related to Fourier-Feynman Transform and Convolution

Abstract

Cameron and Storvick discovered change of scale formulas for Wiener integrals of functionals in Banach algebra 𝒮 on classical Wiener space. Yoo and Skoug extended these results for functionals in the Fresnel class ℱ(𝐵) and in a generalized Fresnel class ℱ_𝐴1,𝐴2 on abstract Wiener space. We express Fourier-Feynman transform and convolution product of functionals in 𝒮 as limits of Wiener integrals. Moreover we obtain change of scale formulas for Wiener integrals related to Fourier-Feynman transform and convolution product of these functionals.

1. Introduction

It has long been known that Wiener measure and Wiener measurability behave badly under the change of scale transformation [1] and under translations [2]. Cameron and Storvick [3] expressed the analytic Feynman integral on classical Wiener space as a limit of Wiener integrals. In doing so, they discovered nice change of scale formulas for Wiener integrals on classical Wiener space (𝐶₀[0,1],𝑚) [4]. In [5, 6], Yoo and Skoug extended these results to an abstract Wiener space (𝐻,𝐵,𝜈). Moreover Yoo et al. [7, 8] established a change of scale formula for Wiener integrals of some unbounded functionals on (a product) abstract Wiener space. Recently Yoo et al. [9] obtained a change of scale formula for a function space integral on a generalized Wiener space 𝐶_𝑎,𝑏[0,𝑇].

On the other hand, in [10], Cameron and Storvick introduced an 𝐿₂ analytic Fourier-Feynman transform. In [11], Johnson and Skoug developed an 𝐿_𝑝 analytic Fourier-Feynman transform for 1≤𝑝≤2 that extended the results in [10]. In [12], Huffman et al. defined a convolution product for functionals on Wiener space and, for a cylinder type functional, showed that the Fourier-Feynman transform of the convolution product is a product of Fourier-Feynman transforms. For a detailed survey of the previous work on the Fourier-Feynman transform and related topics, see [13].

In this paper, we express the Fourier-Feynman transform and convolution product of functionals in Banach algebra 𝒮 as limits of Wiener integrals on 𝐶₀[0,𝑇]. Moreover we obtain change of scale formulas for Wiener integrals related to Fourier-Feynman transform and convolution product of these functionals. Some preliminary results of this paper were introduced as an oral presentation in 2013 Annual Meeting of the Korean Mathematical Society [14].

Let 𝐶₀[0,𝑇] denote the Wiener space, that is, the space of real valued continuous functions 𝑥 on [0,𝑇] with 𝑥(0)=0. Let ℳ denote the class of all Wiener measurable subsets of 𝐶₀[0,𝑇] and let 𝑚 denote Wiener measure. Then (𝐶₀[0,𝑇],ℳ,𝑚) is a complete measure space and we denote the Wiener integral of a functional 𝐹 by _{𝐶0[0,𝑇]}𝐹(𝑥)𝑑𝑚(𝑥). (1)

A subset 𝐸 of 𝐶₀[0,𝑇] is said to be scale-invariant measurable [15] provided 𝜌𝐸 is measurable for each 𝜌>0, and a scale-invariant measurable set 𝑁 is said to be scale-invariant null provided 𝑚(𝜌𝑁)=0 for each 𝜌>0. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (𝑠-a.e.).

Let ℂ₊ and ℂ^∼₊ denote the sets of complex numbers with positive real part and the complex numbers with nonnegative real part, respectively. Let 𝐹 be a complex valued measurable functional on 𝐶₀[0,𝑇] such that the Wiener integral 𝐽_𝐹(𝜆)=_{𝐶0[0,𝑇]}𝐹𝜆^−1/2𝑥𝑑𝑚(𝑥) (2) exists as a finite number for all 𝜆>0. If there exists a function 𝐽^∗_𝐹(𝜆) analytic in ℂ₊ such that 𝐽^∗_𝐹(𝜆)=𝐽_𝐹(𝜆) for all 𝜆>0, then 𝐽^∗_𝐹(𝜆) is defined to be the analytic Wiener integral of 𝐹 over 𝐶₀[0,𝑇] with parameter 𝜆, and for 𝜆∈ℂ₊ we write ^anw𝜆_{𝐶0[0,𝑇]}𝐹(𝑥)𝑑𝑚(𝑥)=𝐽^∗_𝐹(𝜆). (3) If the following limit exists for nonzero real 𝑞, then we call it the analytic Feynman integral of 𝐹 over 𝐶₀[0,𝑇] with parameter 𝑞 and we write ^anf𝑞_{𝐶0[]0,𝑇}𝐹(𝑥)𝑑𝑚(𝑥)=lim_{𝜆→−𝑖𝑞}^anw𝜆_{𝐶0[]0,𝑇}𝐹(𝑥)𝑑𝑚(𝑥), (4) where 𝜆 approaches −𝑖𝑞 through ℂ₊.

Now we will introduce the class of functionals that we work on in this paper. The Banach algebra 𝒮 was introduced in [16] by Cameron and Storvick. It consists of functionals expressible in the form 𝐹(𝑥)=_{𝐿2[0,𝑇]}exp{𝑖⟨𝑣,𝑥⟩}𝑑𝑓(𝑣) (5) for 𝑠-a.e. 𝑥 in 𝐶₀[0,𝑇], where the associated measure 𝑓 is a complex Borel measure on 𝐿₂[0,𝑇] and ⟨𝑣,𝑥⟩ denote the Paley-Wiener-Zygmund stochastic integral ∫^𝑇₀𝑣(𝑡)𝑑𝑥(𝑡).

2. Fourier Feynman Transform and a Change of Scale Formula

In this section we give a relationship between the Wiener integral and the Fourier-Feynman transform on 𝐶₀[0,𝑇] for functionals in the Banach algebra 𝒮; that is, we express the Fourier-Feynman transform of functionals in 𝒮 as a limit of Wiener integrals on 𝐶₀[0,𝑇]. We begin this section by introducing the definition of analytic Fourier-Feynman transform for functionals defined on 𝐶₀[0,𝑇]. Let 1≤𝑝<∞ and let 𝑞 be a nonzero real number.

Definition 1. Let 𝐹 be a functional on 𝐶₀[0,𝑇]. For 𝜆∈ℂ₊ and 𝑦∈𝐶₀[0,𝑇], let 𝑇_𝜆(𝐹)𝑦=^anw𝜆_{𝐶0[0,𝑇]}𝐹𝑥+𝑦𝑑𝑚(𝑥). (6) For 1<𝑝<∞, we define the 𝐿_𝑝 analytic Fourier-Feynman transform 𝑇_𝑞^(𝑝)(𝐹) of 𝐹 on 𝐶₀[0,𝑇] by the formula (𝜆∈ℂ₊)𝑇_𝑞^(𝑝)(𝐹)𝑦=l.i.m._{𝜆→−𝑖𝑞}𝑇_𝜆(𝐹)𝑦, (7) whenever this limit exists; that is, for each 𝜌>0, lim_{𝜆→−𝑖𝑞𝐶0[0,𝑇]}𝑇_𝜆(𝐹)𝜌𝑥−𝑇_𝑞^(𝑝)(𝐹)𝜌𝑥^𝑝′𝑑𝑚(𝑥)=0, (8) where 1/𝑝+1/𝑝^′=1. We define the 𝐿₁ analytic Fourier-Feynman transform 𝑇_𝑞⁽¹⁾(𝐹) of 𝐹 by (𝜆∈ℂ₊)𝑇_𝑞⁽¹⁾(𝐹)𝑦=lim_{𝜆→−𝑖𝑞}𝑇_𝜆(𝐹)𝑦, (9) for 𝑠-a.e. 𝑦∈𝐶₀[0,𝑇], whenever this limit exists [10–12, 17].

By the definition (4) of the analytic Feynman integral and the 𝐿₁ analytic Fourier-Feynman transform (9), it is easy to see that for a nonzero real number 𝑞, 𝑇_𝑞⁽¹⁾(𝐹)𝑦=^anf𝑞_{𝐶0[]0,𝑇}𝐹𝑥+𝑦𝑑𝑚(𝑥). (10) In particular, if 𝐹∈𝒮, then 𝐹 is analytic Feynman integrable and 𝑇_𝑞⁽¹⁾(𝐹)(0)=^anf𝑞_{𝐶0[0,𝑇]}𝐹(𝑥)𝑑𝑚(𝑥). (11)

Huffman et al. established the existence of the Fourier-Feynman transform on 𝐶₀[0,𝑇] for functionals in 𝒮.

Theorem 2 (Theorem 3.1 of [17]). Let 𝐹∈𝒮 be given by (5). Then for all 𝜆∈ℂ₊𝑇_𝜆(𝐹)𝑦=_{𝐿2[0,𝑇]}𝑖−1exp𝑣,𝑦‖𝑣‖2𝜆²²𝑑𝑓(𝑣) (12) for 𝑠-a.e. 𝑦∈𝐶₀[0,𝑇]. Moreover for all 1≤𝑝<∞ and for all nonzero real number 𝑞, the Fourier-Feynman transform 𝑇_𝑞^(𝑝)(𝐹) exists and is given by 𝑇_𝑞^(𝑝)(𝐹)𝑦=_{𝐿2[0,𝑇]}𝑖−𝑖exp𝑣,𝑦‖𝑣‖2𝑞²²𝑑𝑓(𝑣) (13) for 𝑠-a.e. 𝑦∈𝐶₀[0,𝑇].

We next introduce an integration formula which is useful in this paper. The proof of this lemma is essentially the same as Lemma 3 of [3] and hence we will state it without proof.

Lemma 3. Let 𝜆∈ℂ₊, let {𝜙₁,…,𝜙_𝑛} be an orthonormal set in 𝐿₂[0,𝑇], and let 𝑣∈𝐿₂[0,𝑇]. Then _{𝐶0[0,𝑇]}exp1−𝜆2^𝑛_𝑘=1𝜙_𝑘,𝑥²+𝑖𝑣,𝑥+𝑦𝑑𝑚(𝑥)=𝜆^−𝑛/2exp𝜆−12𝜆^𝑛_𝑘=1𝜙_𝑘,𝑣²−12‖𝑣‖²₂+𝑖𝑣,𝑦. (14)

Now we give a relationship between the analytic Fourier-Feynman transform and the Wiener integral on 𝐶₀[0,𝑇] for functionals in 𝒮. In this theorem we express the Fourier-Feynman transform of functionals in 𝒮 as a limit of Wiener integrals.

Theorem 4. Let 𝐹∈𝒮 be given by (5). Let {𝜙_𝑛} be a complete orthonormal set of functionals in 𝐿₂[0,𝑇]. Let 𝑞 be a nonzero real number and let {𝜆_𝑛} be a sequence of complex numbers in ℂ₊ such that 𝜆_𝑛→−𝑖𝑞. Then we have 𝑇_𝑞^(𝑝)(𝐹)𝑦=lim_𝑛→∞𝜆_𝑛^𝑛/2_{𝐶0[0,𝑇]}exp1−𝜆_𝑛2^𝑛_𝑘=1𝜙_𝑘,𝑥²×𝐹𝑥+𝑦𝑑𝑚(𝑥) (15) for 𝑠-a.e. 𝑦∈𝐶₀[0,𝑇].

Proof. Let Γ(𝜆_𝑛) be the Wiener integral on the right hand side of (15). Then by (5) and the Fubini theorem, Γ𝜆_𝑛=_{𝐿2[0,𝑇]}𝐾𝑣,𝜆_𝑛𝑑𝑓(𝑣), (16) where 𝐾𝑣,𝜆_𝑛=_{𝐶0[0,𝑇]}exp1−𝜆_𝑛2^𝑛_𝑘=1𝜙_𝑘,𝑥²+𝑖⟨𝑣,𝑥+𝑦⟩𝑑𝑚(𝑥). (17) By Lemma 3, we evaluate the above Wiener integral to obtain 𝐾𝑣,𝜆_𝑛=𝜆_𝑛^−𝑛/2𝜆exp^𝑛−12𝜆_𝑛^𝑛_𝑘=1𝜙_𝑘,𝑣²−12‖𝑣‖²₂+𝑖𝑣,𝑦. (18) Now by Parseval’s theorem lim_𝑛→∞𝜆_𝑛^𝑛/2𝐾𝑣,𝜆_𝑛−𝑖=exp‖𝑣‖2𝑞²²+𝑖⟨𝑣,𝑦⟩ (19) and so by the bounded convergence theorem lim_𝑛→∞𝜆_𝑛^𝑛/2Γ𝜆_𝑛=lim_𝑛→∞𝜆_𝑛^𝑛/2_{𝐿2[0,𝑇]}𝐾𝑣,𝜆_𝑛(𝑣)=𝑑𝑓_{𝐿2[0,𝑇]}−𝑖exp‖𝑣‖2𝑞²²+𝑖𝑣,𝑦𝑑𝑓(𝑣). (20) Finally by (13) in Theorem 2, the proof is completed.

As we have seen in (10) and (11) above, if 𝑝=1, then the Fourier-Feynman transform 𝑇_𝑞⁽¹⁾(𝐹)(0) is equal to the analytic Feynman integral of 𝐹. Hence we have the following corollary.

Corollary 5 (Theorem 2 of [3]). Let 𝐹∈𝒮 be given by (5). Let {𝜙_𝑛} be a complete orthonormal set of functionals in 𝐿₂[0,𝑇]. Let 𝑞 be a nonzero real number and let {𝜆_𝑛} be a sequence of complex numbers in ℂ₊ such that 𝜆_𝑛→−𝑖𝑞. Then we have ^anf𝑞_{𝐶0[]0,𝑇}𝐹(𝑥)𝑑𝑚(𝑥)=lim_𝑛→∞𝜆_𝑛^𝑛/2_{𝐶0[0,𝑇]}exp1−𝜆_𝑛2^𝑛_𝑘=1𝜙_𝑘,𝑥²×𝐹(𝑥)𝑑𝑚(𝑥). (21)

The following is a relationship between 𝑇_𝜆(𝐹) and the Wiener integral for functionals in 𝒮.

Theorem 6. Let 𝐹∈𝒮 be given by (5). Let {𝜙_𝑛} be a complete orthonormal set of functionals in 𝐿₂[0,𝑇]. Then for each 𝜆∈ℂ₊ we have 𝑇_𝜆(𝐹)𝑦=lim_𝑛→∞𝜆^𝑛/2_{𝐶0[0,𝑇]}exp1−𝜆2^𝑛_𝑘=1𝜙_𝑘,𝑥²×𝐹𝑥+𝑦𝑑𝑚(𝑥) (22) for 𝑠-a.e. 𝑦∈𝐶₀[0,𝑇].

Proof. To prove this theorem, we modify the proof of Theorem 4 by replacing 𝜆_𝑛 by 𝜆 whenever it occurs. Then we have lim_𝑛→∞𝜆^𝑛/2𝐾()−1𝑣,𝜆=exp‖𝑣‖2𝜆²²+𝑖𝑣,𝑦. (23) We apply the dominated convergence theorem to obtain lim_𝑛→∞𝜆^𝑛/2Γ(𝜆)=_{𝐿2[0,𝑇]}−1exp‖𝑣‖2𝜆²²+𝑖𝑣,𝑦𝑑𝑓(𝑣). (24) By (12) in Theorem 2, the proof is completed.

Our main result in this section, namely, a change of scale formula for Wiener integral related to Fourier-Feynman transform of functionals in 𝒮, now follows from Theorem 6.

Theorem 7. Let 𝐹∈𝒮 be given by (5). Let {𝜙_𝑛} be a complete orthonormal set of functionals in 𝐿₂[0,𝑇]. Then for each 𝜌>0_{𝐶0[0,𝑇]}𝐹𝜌𝑥+𝑦𝑑𝑚(𝑥)=lim_𝑛→∞𝜌^−𝑛_{𝐶0[0,𝑇]}𝜌exp²−12𝜌^2𝑛_𝑘=1𝜙_𝑘,𝑥²×𝐹𝑥+𝑦𝑑𝑚(𝑥) (25) for 𝑠-a.e. 𝑦∈𝐶₀[0,𝑇].

Proof. First note that for 𝜆>0𝑇_𝜆(𝐹)𝑦=_{𝐶0[0,𝑇]}𝐹𝜆^−1/2𝑥+𝑦𝑑𝑚(𝑥). (26) Letting 𝜆=𝜌⁻² in (22), we have (25) and this completes the proof.

Letting 𝑦=0 in (25), we have the following change of scale formula for Wiener integrals on classical Wiener space.

Corollary 8 (Theorem 2 of [4]). Let 𝐹∈𝒮 be given by (5). Let {𝜙_𝑛} be a complete orthonormal set of functionals in 𝐿₂[0,𝑇]. Then we have _{𝐶0[0,𝑇]}𝐹𝜌𝑥𝑑𝑚(𝑥)=lim_𝑛→∞𝜌^−𝑛_{𝐶0[0,𝑇]}𝜌exp²−12𝜌^2𝑛_𝑘=1𝜙_𝑘,𝑥²𝐹(𝑥)𝑑𝑚(𝑥) (27) for each 𝜌>0.

In our next example we will explicitly compute a Wiener integral of a functional under a change of scale transformation.

Example 9. Let {𝜙_𝑛} be a complete orthonormal set of functionals in 𝐿₂[0,𝑇]. Define 𝐹(𝑥)=exp𝛼⟨𝜙₁,𝑥⟩ (28) for 𝑥∈𝐶₀[0,𝑇] and 𝛼 is a real or complex number. We evaluate the Wiener integrals on each side of (25). The left hand side of (25) can be evaluated as follows: 𝐿≡_{𝐶0[0,𝑇]}𝐹(𝑥)=𝜌𝑥+𝑦𝑑𝑚_{𝐶0[0,𝑇]}exp𝛼𝜌⟨𝜙₁,𝑥⟩+𝛼⟨𝜙₁,𝑦⟩𝑑𝑚(𝑥). (29) By the Paley-Wiener-Zygmund theorem (see [18]), we have 𝐿=(2𝜋)^−1/2exp𝛼𝜙₁,𝑦_ℝ1exp𝛼𝜌𝑢−2𝑢²𝛼𝑑𝑢=exp²𝜌²2+𝛼𝜙₁,𝑦. (30) Next we evaluate the Wiener integral on the right hand side of (25). Consider 𝑅≡_{𝐶0[0,𝑇]}𝜌exp²−12𝜌^2𝑛_𝑘=1𝜙_𝑘,𝑥²𝐹𝑥+𝑦𝑑𝑚(𝑥)=_{𝐶0[0,𝑇]}𝜌exp²−12𝜌^2𝑛_𝑘=1𝜙_𝑘,𝑥²+𝛼𝜙₁,𝑥+𝛼𝜙₁,𝑦𝑑𝑚(𝑥). (31) By the Paley-Wiener-Zygmund theorem again, we have 𝑅=(2𝜋)^−𝑛/2exp𝛼𝜙₁×,𝑦_ℝ𝑛𝜌exp²−12𝜌^2𝑛_𝑘=1𝑢²_𝑘+𝛼𝑢₁−12^𝑛_𝑘=1𝑢²_𝑘𝑑𝑢=𝜌^𝑛𝛼exp²𝜌²2+𝛼𝜙₁,𝑦. (32) Thus we have established that (25) is valid for 𝐹(𝑥)=exp{𝛼⟨𝜙₁,𝑥⟩}.

Note that in Example 9 above, 𝛼 is a real or complex number. If 𝛼 is pure imaginary, 𝐹∈𝒮 and 𝐹 is an example of a functional to which Theorem 7 applies. On the other hand, if the real part of 𝛼 is not equal to 0, then 𝐹 can be unbounded. Thus this example shows that the class of functionals for which (25) holds is more extensive than 𝒮.

3. Convolution and a Change of Scale Formula

In this section we give a relationship between the Wiener integral and the convolution product on 𝐶₀[0,𝑇] for functionals in the Banach algebra 𝒮; that is, we express the convolution product of functionals in 𝒮 as a limit of Wiener integrals on 𝐶₀[0,𝑇]. We start this section by introducing the definition of convolution product for functionals on 𝐶₀[0,𝑇].

Definition 10. Let 𝐹 and 𝐺 be functionals on 𝐶₀[0,𝑇]. For 𝜆∈ℂ₊ and 𝑦∈𝐶₀[0,𝑇], we define their convolution product (𝐹∗𝐺)_𝜆 by (𝐹∗𝐺)_𝜆𝑦=^anw𝜆_{𝐶0[]0,𝑇}𝐹𝑦+𝑥√2𝐺𝑦−𝑥√2𝑑𝑚(𝑥) (33) if it exists. Moreover for nonzero real number 𝑞, the convolution product (𝐹∗𝐺)_𝑞 is defined by (𝐹∗𝐺)_𝑞𝑦=^anf𝑞_{𝐶0[]0,𝑇}𝐹𝑦+𝑥√2𝐺𝑦−𝑥√2𝑑𝑚(𝑥) (34) if it exists [12, 17, 19, 20].

The following is the existence theorem for the convolution product of functionals in 𝒮 on 𝐶₀[0,𝑇].

Theorem 11 (Theorem 3.2 of [17]). Let 𝐹 and 𝐺 be elements of 𝒮 with associated complex Borel measures 𝑓 and 𝑔, respectively. Then for all 𝜆∈ℂ₊(𝐹∗𝐺)_𝜆𝑦=_{𝐿22[0,𝑇]}𝑖exp√2−1𝑣+𝑤,𝑦‖‖4𝜆𝑣−𝑤²₂𝑑𝑓(𝑣)𝑑𝑔(𝑤) (35) for 𝑠-a.e. 𝑦∈𝐶₀[0,𝑇]. Moreover for all nonzero real number 𝑞, the convolution product (𝐹∗𝐺)_𝑞 exists and is given by (𝐹∗𝐺)_𝑞𝑦=_{𝐿22[0,𝑇]}𝑖exp√2−𝑖𝑣+𝑤,𝑦‖‖4𝑞𝑣−𝑤²₂𝑑𝑓(𝑣)𝑑𝑔(𝑤) (36) for 𝑠-a.e. 𝑦∈𝐶₀[0,𝑇].

Now we give a relationship between the convolution product and the Wiener integral on 𝐶₀[0,𝑇] for functionals in 𝒮. In this theorem we express the convolution product of functionals in 𝒮 as a limit of Wiener integrals.

Theorem 12. Let 𝐹 and 𝐺 be elements of 𝒮 with associated complex Borel measures 𝑓 and 𝑔, respectively. Let {𝜙_𝑛} be a complete orthonormal set of functionals in 𝐿₂[0,𝑇]. Let 𝑞 be a nonzero real number and let {𝜆_𝑛} be a sequence of complex numbers in ℂ₊ such that 𝜆_𝑛→−𝑖𝑞. Then we have (𝐹∗𝐺)_𝑞𝑦=lim_𝑛→∞𝜆_𝑛^𝑛/2_{𝐶0[0,𝑇]}exp1−𝜆_𝑛2^𝑛_𝑘=1𝜙_𝑘,𝑥²×𝐹𝑦+𝑥√2𝐺𝑦−𝑥√2𝑑𝑚(𝑥) (37) for 𝑠-a.e. 𝑦∈𝐶₀[0,𝑇].

Proof. Let Γ^∗(𝜆_𝑛) be the Wiener integral on the right hand side of (37). Then by (5) and the Fubini theorem, Γ^∗𝜆_𝑛=_{𝐿22[0,𝑇]}𝐾^∗𝑣,𝑤,𝜆_𝑛𝑑𝑓(𝑣)𝑑𝑔(𝑤), (38) where 𝐾^∗𝑣,𝑤,𝜆_𝑛=_{𝐶0[0,𝑇]}exp1−𝜆_𝑛2^𝑛_𝑘=1𝜙_𝑘,𝑥²+𝑖√2⟨⟩+𝑖𝑣−𝑤,𝑥√2𝑣+𝑤,𝑦𝑑𝑚(𝑥). (39) By Lemma 3, we evaluate the above Wiener integral to obtain 𝐾^∗𝑣,𝑤,𝜆_𝑛=𝜆_𝑛^−𝑛/2𝜆exp^𝑛−14𝜆_𝑛^𝑛_𝑘=1𝜙_𝑘,𝑣−𝑤²−14‖𝑣−𝑤‖²₂+𝑖√2⟨𝑣+𝑤,𝑦⟩. (40) Now by Parseval’s theorem lim_𝑛→∞𝜆_𝑛^𝑛/2𝐾^∗𝑣,𝑤,𝜆_𝑛−𝑖=exp‖‖4𝑞𝑣−𝑤²₂+𝑖√2𝑣+𝑤,𝑦 (41) and so by the bounded convergence theorem lim_𝑛→∞𝜆_𝑛^𝑛/2Γ^∗𝜆_𝑛=lim_𝑛→∞𝜆_𝑛^𝑛/2_{𝐿22[0,𝑇]}𝐾^∗𝑣,𝑤,𝜆_𝑛(𝑣)(𝑤)=𝑑𝑓𝑑𝑔_{𝐿22[0,𝑇]}−𝑖exp‖‖4𝑞𝑣−𝑤²₂+𝑖√2𝑣+𝑤,𝑦𝑑𝑓(𝑣)𝑑𝑔(𝑤). (42) Finally by (36) in Theorem 11, the proof is completed.

The following is a relationship between the convolution product (𝐹∗𝐺)_𝜆 and the Wiener integral for functionals in 𝒮.

Theorem 13. Let 𝐹 and 𝐺 be elements of 𝒮 with associated complex Borel measures 𝑓 and 𝑔, respectively. Let {𝜙_𝑛} be a complete orthonormal set of functionals in 𝐿₂[0,𝑇]. Then for each 𝜆∈ℂ₊ we have (𝐹∗𝐺)_𝜆𝑦=lim_𝑛→∞𝜆^𝑛/2_{𝐶0[0,𝑇]}exp1−𝜆2^𝑛_𝑘=1𝜙_𝑘,𝑥²×𝐹𝑦+𝑥√2𝐺𝑦−𝑥√2𝑑𝑚(𝑥) (43) for 𝑠-a.e. 𝑦∈𝐶₀[0,𝑇].

Proof. To prove this theorem, we modify the proof of Theorem 12 by replacing 𝜆_𝑛 by 𝜆 whenever it occurs. Then we have lim_𝑛→∞𝜆^𝑛/2𝐾^∗()−1𝑣,𝜆=exp‖‖4𝜆𝑣−𝑤²₂+𝑖√2𝑣+𝑤,𝑦. (44) We apply the dominated convergence theorem to obtain lim_𝑛→∞𝜆^𝑛/2Γ^∗(𝜆)=_{𝐿22[0,𝑇]}−1exp‖‖4𝜆𝑣−𝑤²₂+𝑖√2⟨𝑣+𝑤,𝑦⟩𝑑𝑓(𝑣)𝑑𝑔(𝑤). (45) By (35) in Theorem 11, the proof is completed.

Our main result in this section, namely, a change of scale formula for Wiener integral related to convolution product of functionals in 𝒮, now follows from Theorem 13.

Theorem 14. Let 𝐹 and 𝐺 be elements of 𝒮 with associated complex Borel measures 𝑓 and 𝑔, respectively. Let {𝜙_𝑛} be a complete orthonormal set of functionals in 𝐿₂[0,𝑇]. Then for each 𝜌>0_{𝐶0[0,𝑇]}𝐹𝑦+𝜌𝑥√2𝐺𝑦−𝜌𝑥√2𝑑𝑚(𝑥)=lim_𝑛→∞𝜌^−𝑛_{𝐶0[0,𝑇]}𝜌exp²−12𝜌^2𝑛_𝑘=1𝜙_𝑘,𝑥²×𝐹𝑦+𝑥√2𝐺𝑦−𝑥√2𝑑𝑚(𝑥) (46) for 𝑠-a.e. 𝑦∈𝐶₀[0,𝑇].

Proof. First note that for 𝜆>0(𝐹∗𝐺)_𝜆𝑦=_{𝐶0[0,𝑇]}𝐹𝑦+𝜆^−1/2𝑥√2𝐺𝑦−𝜆^−1/2𝑥√2𝑑𝑚(𝑥). (47) Letting 𝜆=𝜌⁻² in (43), we have (46) and this completes the proof.

In our final example we will explicitly compute a Wiener integral related to convolution product under a change of scale transformation.

Example 15. Let {𝜙_𝑛} be a complete orthonormal set of functionals in 𝐿₂[0,𝑇]. Define 𝐹(𝑥)=exp𝛼𝜙₁,𝑥,𝐺(𝑥)=exp𝛼⟨𝜙₂,𝑥⟩ (48) for 𝑥∈𝐶₀[0,𝑇] and 𝛼 is a real or complex number. We evaluate the Wiener integrals on each side of (46). The left hand side of (46) can be evaluated as follows: 𝐿≡_{𝐶0[0,𝑇]}𝐹𝑦+𝜌𝑥√2𝐺𝑦−𝜌𝑥√2(𝑥)=𝑑𝑚_{𝐶0[0,𝑇]}𝛼exp√2𝜙₁+𝜙₂+,𝑦𝛼𝜌√2𝜙₁−𝜙₂,𝑥𝑑𝑚(𝑥). (49) By the Paley-Wiener-Zygmund theorem, we have 𝐿=(2𝜋)⁻¹𝛼exp√2𝜙₁+𝜙₂×,𝑦_ℝ2exp𝛼𝜌√2𝑢₁−𝑢₂−12𝑢²₁+𝑢²₂𝑑𝑢₁𝑑𝑢₂𝛼=exp²𝜌²2+𝛼√2𝜙₁+𝜙₂,𝑦. (50) Next we evaluate the Wiener integral on the right hand side of (46): 𝑅≡_{𝐶0[0,𝑇]}𝜌exp²−12𝜌^2𝑛_𝑘=1𝜙_𝑘,𝑥²×𝐹𝑦+𝑥√2𝐺𝑦−𝑥√2(𝑥)=𝑑𝑚_{𝐶0[0,𝑇]}𝜌exp²−12𝜌^2𝑛_𝑘=1𝜙_𝑘,𝑥²+𝛼√2𝜙₁+𝜙₂+𝛼,𝑦√2𝜙₁−𝜙₂,𝑥𝑑𝑚(𝑥). (51) By the Paley-Wiener-Zygmund theorem, we have 𝑅=(2𝜋)^−𝑛/2𝛼exp√2𝜙₁+𝜙₂,𝑦_ℝ𝑛𝜌exp²−12𝜌^2𝑛_𝑘=1𝑢²_𝑘+𝛼√2𝑢₁−𝑢₂−12^𝑛_𝑘=1𝑢²_𝑘𝑑𝑢=𝜌^𝑛𝛼exp²𝜌²2+𝛼√2𝜙₁+𝜙₂,𝑦. (52) Thus we have established that (46) is valid for 𝐹(𝑥)=exp{𝛼⟨𝜙₁,𝑥⟩} and 𝐺(𝑥)=exp{𝛼⟨𝜙₂,𝑥⟩}.

Note that in Example 15 above, 𝛼 was a real or complex number. If 𝛼 is pure imaginary, 𝐹 and 𝐺 belong to 𝒮, so 𝐹 and 𝐺 are examples of functionals to which Theorem 14 applies. On the other hand, if the real part of 𝛼 is not equal to 0, then 𝐹 and 𝐺 can be unbounded. Thus this example shows that the class of functionals for which (46) holds is more extensive than 𝒮.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

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 (2010-0022563).