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

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 ofWiener integrals. In doing so, they discovered nice change of scale formulas for Wiener integrals on classical Wiener space (C 0 [0, 1], m) [4]. In [5, 6], Yoo and Skoug extended these results to an abstract Wiener space (H, B, ]). 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 L 2 analytic Fourier-Feynman transform. In [11], Johnson and Skoug developed an L p analytic FourierFeynman 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 C 0 [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 AnnualMeeting of the Korean Mathematical Society [14]. Let C 0 [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 C 0 [0, T] and let m denote Wiener measure. Then


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 ( 0 [0, 1], ) [4].In [5,6], Yoo and Skoug extended these results to an abstract Wiener space (, , ]).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  , [0, ].
On the other hand, in [10], Cameron and Storvick introduced an  2 analytic Fourier-Feynman transform.In [11], Johnson and Skoug developed an   analytic Fourier-Feynman transform for 1 ≤  ≤ 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  0 [0, ].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  0 [0, ] denote the Wiener space, that is, the space of real valued continuous functions  on [0, ] with (0) = 0. Let M denote the class of all Wiener measurable subsets of  0 [0, ] and let  denote Wiener measure.Then ( 0 [0, ], M, ) is a complete measure space and we denote the Wiener integral of a functional  by A subset  of  0 [0, ] is said to be scale-invariant measurable [15] provided  is measurable for each  > 0, and a scale-invariant measurable set  is said to be scaleinvariant null provided () = 0 for each  > 0. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (-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  be a complex valued measurable functional on  0 [0, ] such that the Wiener integral exists as a finite number for all  > 0. If there exists a function  *  () analytic in C + such that  *  () =   () for all  > 0, then  *  () is defined to be the analytic Wiener integral of  over  0 [0, ] with parameter , and for  ∈ C + we write If the following limit exists for nonzero real , then we call it the analytic Feynman integral of  over  0 [0, ] with parameter  and we write where  approaches − 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 for -a.e. in  0 [0, ], where the associated measure  is a complex Borel measure on  2 [0, ] and ⟨V, ⟩ denote the Paley-Wiener-Zygmund stochastic integral ∫  0 V()().

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  0 [0, ] 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  0 [0, ].We begin this section by introducing the definition of analytic Fourier-Feynman transform for functionals defined on  0 [0, ].Let 1 ≤  < ∞ and let  be a nonzero real number.
By the definition (4) of the analytic Feynman integral and the  1 analytic Fourier-Feynman transform (9), it is easy to see that for a nonzero real number ,  (1)   ( In particular, if  ∈ S, then  is analytic Feynman integrable and Huffman et al. established the existence of the Fourier-Feynman transform on  0 [0, ] for functionals in S. Theorem 2 (Theorem 3.1 of [17]).Let  ∈ S be given by (5).
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.
Journal of Function Spaces 3 Now we give a relationship between the analytic Fourier-Feynman transform and the Wiener integral on  0 [0, ] 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  ∈ S be given by (5).Let {  } be a complete orthonormal set of functionals in  2 [0, ].Let  be a nonzero real number and let {  } be a sequence of complex numbers in C + such that   → −.Then we have for -a.e. ∈  0 [0, ].
Proof.Let Γ(  ) be the Wiener integral on the right hand side of (15).Then by ( 5) and the Fubini theorem, where By Lemma 3, we evaluate the above Wiener integral to obtain Now by Parseval's theorem lim and so by the bounded convergence theorem lim Finally by (13) in Theorem 2, the proof is completed.
As we have seen in ( 10) and ( 11) above, if  = 1, then the Fourier-Feynman transform  (1)   ()(0) is equal to the analytic Feynman integral of .Hence we have the following corollary.
Corollary 5 (Theorem 2 of [3]).Let  ∈ S be given by (5).Let {  } be a complete orthonormal set of functionals in  2 [0, ].Let  be a nonzero real number and let {  } be a sequence of complex numbers in C + such that   → −.Then we have The following is a relationship between   () and the Wiener integral for functionals in S. Theorem 6.Let  ∈ S be given by (5).Let {  } be a complete orthonormal set of functionals in  2 [0, ].Then for each  ∈ C + we have for -a.e. ∈  0 [0, ].
Proof.To prove this theorem, we modify the proof of Theorem 4 by replacing   by  whenever it occurs.Then we have lim We apply the dominated convergence theorem to obtain lim 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.
Proof.First note that for  > 0 Letting  =  −2 in (22), we have (25) and this completes the proof.
Letting  = 0 in (25), we have the following change of scale formula for Wiener integrals on classical Wiener space.
In our next example we will explicitly compute a Wiener integral of a functional under a change of scale transformation.
for  ∈  0 [0, ] and  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: By the Paley-Wiener-Zygmund theorem (see [18]), we have Next we evaluate the Wiener integral on the right hand side of (25).Consider By the Paley-Wiener-Zygmund theorem again, we have Thus we have established that ( 25) is valid for () = exp{⟨ 1 , ⟩}.
Note that in Example 9 above,  is a real or complex number.If  is pure imaginary,  ∈ S and  is an example of a functional to which Theorem 7 applies.On the other hand, if the real part of  is not equal to 0, then  can be unbounded.Thus this example shows that the class of functionals for which (25) holds is more extensive than S.

Convolution and a Change of Scale Formula
In this section we give a relationship between the Wiener integral and the convolution product on  0 [0, ] 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  0 [0, ].We start this section by introducing the definition of convolution product for functionals on  0 [0, ].if it exists.Moreover for nonzero real number , the convolution product ( * )  is defined by if it exists [12,17,19,20].
The following is the existence theorem for the convolution product of functionals in S on  0 [0, ].
Now we give a relationship between the convolution product and the Wiener integral on  0 [0, ] 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  and  be elements of S with associated complex Borel measures  and , respectively.Let {  } be a complete orthonormal set of functionals in  2 [0, ].Let  be a nonzero real number and let {  } be a sequence of complex numbers in C + such that   → −.Then we have for -a.e. ∈  0 [0, ].
Proof.Let Γ * (  ) be the Wiener integral on the right hand side of (37).Then by (5) and the Fubini theorem, where By Lemma 3, we evaluate the above Wiener integral to obtain Now by Parseval's theorem lim and so by the bounded convergence theorem lim Finally by (36) in Theorem 11, the proof is completed.
The following is a relationship between the convolution product ( * )  and the Wiener integral for functionals in S.
Theorem 13.Let  and  be elements of S with associated complex Borel measures  and , respectively.Let {  } be a complete orthonormal set of functionals in  2 [0, ].Then for each  ∈ C + we have for -a.e. ∈  0 [0, ].
Proof.To prove this theorem, we modify the proof of Theorem 12 by replacing   by  whenever it occurs.Then we have lim Journal of Function Spaces We apply the dominated convergence theorem to obtain lim 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  and  be elements of S with associated complex Borel measures  and , respectively.Let {  } be a complete orthonormal set of functionals in  2 [0, ].Then for each  > 0 for -a.e. ∈  0 [0, ].
Proof.First note that for  > 0 Letting  =  −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.
Note that in Example 15 above,  was a real or complex number.If  is pure imaginary,  and  belong to S, so  and  are examples of functionals to which Theorem 14 applies.On the other hand, if the real part of  is not equal to 0, then  and  can be unbounded.Thus this example shows that the class of functionals for which (46) holds is more extensive than S.