Shifting and Variational Properties for Fourier-Feynman Transform and Convolution

L p analytic Fourier-Feynman transform for 1 ≤ p ≤ 2 that extended the results in [2]. In [4, 5], Huffman et al. defined a convolution product for functionals on Wiener space and showed that the Fourier-Feynman transform of a convolution product is a product of Fourier-Feynman transforms. Recently Kim et al. [6] obtained change of scale formulas for Wiener integrals related to Fourier-Feynman transform and convolution. For a detailed survey of the previous work on Fourier-Feynman transform and related topics, see [7]. Let M denote the class of all Wiener measurable subsets of C 0 [0, T] and let m denote Wiener measure. Then

In [4,5], Huffman et al. defined a convolution product for functionals on Wiener space and showed that the Fourier-Feynman transform of a convolution product is a product of Fourier-Feynman transforms.Recently Kim et al. [6] obtained change of scale formulas for Wiener integrals related to Fourier-Feynman transform and convolution.For a detailed survey of the previous work on Fourier-Feynman transform and related topics, see [7].
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

𝐹 (𝑥) 𝑑𝑚 (𝑥) .
( A subset  of  0 [0, ] is said to be scale-invariant measurable [8] provided  is measurable for each  > 0, and a scale-invariant measurable set  is said to be scale-invariant null provided () = 0 for each  > 0. A property that holds except on a scale-invariant null set is said to hold scaleinvariant 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  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 with in this paper.The Banach algebra S, which exp { ⟨V, ⟩}  (V) (5) for s-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()().In this paper, we study shifting, scaling, modulation, and variational properties for Fourier-Feynman transform of functionals in S. Shifting properties are some of the important properties of Fourier transform.In Section 2, we develop shifting properties for Fourier-Feynman transform.For example, time shifting, frequency shifting, scaling, and modulation properties for Fourier-Feynman transform are given.
In Section 3, we study variational properties for Fourier-Feynman transform of functionals in S and in the last section we develop shifting, scaling, and modulation properties for convolution product of functionals in S.
The Banach algebra S is a very rich class of functionals.It is known that important functionals in quantum mechanics and Feynman integration theory belong to S. For example, functionals of the form were discussed in the book by Feynman and Hibbs [10] on path integrals and in Feynman's original paper [11].For appropriate  : [0, ] 2 × R 2 → C, functionals of form (6) are known to belong to S [12].Hence the results in this paper can be immediately applied to many functionals of form (6).
Since   is linear, obviously  ()  is linear; that is, for all constants ,  and functionals ,  on  0 [0, ], whenever each transform exists.By Definition (4) of the analytic Feynman integral and  1 analytic Fourier-Feynman transform (10), it is easy to see that  (1)   ( In particular, if  ∈ S, then  is analytic Feynman integrable and ()  () .
Huffman et al. established the existence of Fourier-Feynman transform on  0 [0, ] for functionals in S.
The Fourier transform F turns a function  into a new function F[].Because the transform is used in signal analysis, we usually use  for time as the variable with  and  as the variable of the transform F[]; that is, Engineers refer to the variable  in the transformed function as the frequency of the signal  [13].
We will use the same convention in this paper; that is, for a Fourier-Feynman transform  ()   []() of (), we call the variable  a time and the variable  a frequency.
Our first result in this section shows that the time shifting of the Fourier-Feynman transform is equal to the frequency shifting of the Fourier-Feynman transform.Theorem 3. Let  be a functional on  0 [0, ] and let  0 ∈  0 [0, ].Then one has if all sides exist.
Proof.For all  > 0 and for s-a.e. ∈  0 [0, ], if the Wiener integral exists.Hence we have the result.
The following theorem is reminiscent of the time shifting theorem for the Fourier transform.Hence we call the following theorem the time shifting formula for Fourier-Feynman transform on Wiener space.It says that if we shift back  0 and replace () by ( −  0 ), then the Fourier-Feynman transform of this shifted function is equal to the Fourier-Feynman transform of () exp{⟨ 0 , ⟩} multiplied by an exponential factor.
Cameron and Storvick [14] presented a new translation theorem for the analytic Feynman integral on Wiener space.Moreover Ahn et al. [15] gave a simple proof of an abstract Wiener space version of the translation theorem.Taking  = 1 and  = 0 in (19) and considering (13) we obtain Cameron and Storvick's translation theorem as follows.Hence Theorem 4 can be viewed as Cameron and Storvick's translation theorem for the Fourier-Feynman transform.

Then one has
Next theorem is reminiscent of the frequency shifting theorem for the Fourier transform.Using Theorem 3 we have the following property for the frequency shifting of the Fourier-Feynman transform.
The following theorem is called a scaling theorem because we want the transform not of (), but of (), in which  can be thought of as a scaling factor.Theorem 7 (scaling).Let  ∈ S be given by ( 5) and let  be a nonzero real number.Then one has for s-a.e. ∈  0 [0, ].
Our next corollary follows immediately from the scaling theorem above by putting  = −1.This result is called time reversal because we replace  by − in () to get (−).The transform of this new functional is obtained by simply replacing  by − in the transform of ().
Our next theorem is useful in obtaining the Fourier-Feynman transforms of new functionals from the Fourier-Feynman transforms of old functionals for which we know their Fourier-Feynman transform.
Since the Dirac measure concentrated at V = 0 in  2 [0, ] is a complex Borel measure, the constant function  ≡ 1 belongs to S. Hence we have the following corollary.

Variational Properties for Fourier-Feynman Transform
In using the Fourier transform to solve differential equations, we need an expression relating the transform of   to that of .In this section we develop similar relationships for Fourier-Feynman transform on Wiener space; that is, we provide variational properties for Fourier-Feynman transform of functionals in the Banach algebra S.
where   is a complex Borel measure on  2 [0, ] defined by for each Borel subset  of  0 [0, ].
Proof.We will prove that Now the result follows if we can pass the differentiation under the integral sign.But this is done because by the Fubini theorem which is finite and so ∫  2 [0,] |⟨V,  1 ⟩⟨V,  2 ⟩|||(V) < ∞ for s-a.e. 1 ,  2 in  0 [0, ].Now by mathematical induction we obtain general result (38).
In our next theorem, for functionals in S we establish a relationship between the Fourier-Feynman transform of the variation and the variation of the Fourier-Feynman transform.Also see Corollary 4.3 of [15] for a similar result.Theorem 14.Let  ∈ S be given by ( 5) with for s-a.e. and  1 , . . .,   in  0 [0, ].Also, both of the expressions in (45) are given by the expression for s-a.e. and  1 , . . .,   in  0 [0, ].
Proof.By (38) and Theorem 2, we have for s-a.e. and  1 , . . .,   in  0 [0, ].Now by (39) we know that the last expression can be rewritten as (46).On the other hand, by the same method as in the proof of Theorem 12, we see that the right hand side of (45) is also expressed as (46).

Shifting Properties for Convolution Product
We developed in Section 2 some properties relevant to shifting and computational rules for the Fourier-Feynman transform of functionals in the Banach algebra S. In this section we study similar properties for the convolution product of functionals in S. Let us begin with the definition of the convolution product of functionals on Wiener space.Theorem 18 (Theorem 3.2 of [5]).Let  and  be elements of S with corresponding finite Borel measures  and , respectively.Then their convolution product ( * )  exists and is given by the formula for s-a.e. ∈  0 [0, ].
Our first result in this section is a relationship between time shifting and frequency shifting of the convolution product on Wiener space.Theorem 19.Let  and  be functionals on  0 [0, ] and let if each side exists.
Proof.For all  > 0 and for s-a.e. ∈  0 [0, ], we have if the Wiener integrals exist.Hence we have the result.
The following theorem is reminiscent of the time shifting theorem for Fourier-Feynman transform (Theorem 4) in Section 2. But in this theorem we have to shift back  0 for  and shift front  0 for  to obtain a concrete form of a time shifting formula for the convolution product.
Considering the second part of the proof of Theorem 20 above, we see that, for  and  given as in Theorem 18,