RELATIONSHIPS AMONG TRANSFORMS, CONVOLUTIONS, AND FIRST VARIATIONS

. In this paper, we establish several interesting relationships involving the Fourier-Feynman transform, the convolution product, and the ﬁrst variation for functionals F on Wiener space of the form α j ,x denotes the Paley-Wiener-Zygmund stochastic integral (cid:1) .


Introduction.
Let C 0 [0,T ] denote one-parameter Wiener space; that is the space of R-valued continuous functions x on [0,T ] with x(0) = 0. The concept of an L 1 analytic Fourier-Feynman transform was introduced by Brue in [1]. In [3], Cameron and Storvick introduced an L 2 analytic Fourier-Feynman transform. In [11], Johnson and Skoug developed an L p analytic Fourier-Feynman transform theory for 1 ≤ p ≤ 2 which extended the results in [1,3] and gave various relationships between the L 1 and the L 2 theories. In [7], Huffman, Park, and Skoug defined a convolution product for functionals on Wiener space and in [9,7,8], they established various results involving transforms and convolutions. In [5], Cameron and Storvick evaluated the Feynman integral of the first variation of certain functionals on Wiener space and in [13], Park, Skoug, and Storvick examined various relationships existing among the first variation, the Fourier-Feynman transform, and the convolution product for functionals on Wiener space which belong to Banach algebra in [4]. Section 3 of this paper includes all the relationships involving exactly two of the three concepts of "transform," "convolution product," and "first variation" of functionals of the type mentioned in the abstract. In Section 4, we examine all the relationships involving all three of these concepts, but where each concept is used exactly once.
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.). If two functionals F and G are equal s-a.e., we write F ≈ G.
Let C + = {λ ∈ C : Re λ > 0} and C ∼ + = {λ ∈ C : λ = 0 and Re λ ≥ 0}. Let F be a C-valued scale-invariant measurable functional on C 0 [0,T ] such that exists for all λ > 0. If there exists a function J * (λ) analytic in C + such that J * (λ) = J(λ) for all λ > 0, then J * (λ) is defined to be the analytic Wiener integral of F over C 0 [0,T ] with parameter λ and, for λ ∈ C + , we write Let q = 0 be a real number and let F be a functional such that 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

Notation.
(i) For λ ∈ C + and y ∈ C 0 [0,T ], let and we call H the scale invariant limit in the mean of order p . A similar definition is understood when n is replaced by the continuously varying parameter λ.
We are finally ready to state the definition of the L p analytic Fourier-Feynman transform [11], the definition of the convolution product [7], and the definition of the first variation of a function [2,5].
Definition. Let q = 0 be a real number. For 1 < p ≤ 2, we define the L p analytic Fourier-Feynman transform T q (p; F) of F by the formula (λ ∈ C + ) whenever this limit exists. Also, the L 1 analytic Fourier-Feynman transform 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; F) ≈ T q (p; G).
Definition. Let F and G be functionals on C 0 [0,T ]. For λ ∈ C ∼ + , we define their convolution product (if it exists) by

Remarks.
(i) When λ = −iq, we denote (F * G) λ by (F * G) q . (ii) Our definition of the convolution product is different from the definition given by Yeh in [14] and used by Yoo in [15]. In [14,15], Yeh and Yoo studied the relationship between their convolution product and Fourier-Wiener transforms.
Next, we give the definition of the first variation δF of a functional F . We finish this section by describing the class of functionals that we work with in this paper. Let n be a positive integer (fixed throughout this paper). Let 3. Relationships involving two concepts. In [7], several relationships involving the Fourier-Feynman transform and the convolution product were established for functionals in B(p;0). In this section, we also study relationships involving the first variation. In our first lemma, which follows easily from the definitions of δF (x | w) and B(p; m), we obtain a formula for the first variation of functionals in B(p; m).

Definition. Let
, w ∈ A and m ∈ N be given. Let F ∈ B(p; m) be given by (2.11). Then .

Corollary 3.2. Let p, m and F be as in Lemma 3.1 and assume that
Our next corollary to Lemma 3.1 gives us a formula for δ l F .
Notation. For u = (u 1 ,...,u n ) ∈ R n , we write: and In [7,Sec. 2], it was shown that T q (p; F) exists for each p ∈ [1,2], each F ∈ B(p;0), and each nonzero q ∈ R. In addition, (3.7) Furthermore, T q (p; F) is an element of B(p ;0), where (1/p) + (1/p ) = 1. Next, let m ∈ N and F ∈ B(p; m) be given. Since B(p; m) ⊆ B(p;0), we know that T q (p; F) exists and is given by (3.7). The proof that T q (p; F) belongs to B(p ; m) for m > 0 is similar to the proof given in [7] for the case m = 0.
In our first theorem, we show that the transform with respect to the first argument of the variation equals the variation of the transform. (2.11), and let w ∈ A. Then, for all real q = 0 and s-a.e. y ∈ C 0 [0,T ], Also, both of the expressions in (3.8) are given by the expression which, as a function of y, is a element of B(p ; m − 1).

Proof.
First, using the definition of the first variation and equation (3.7), we see that (3.10) Next, using equation (3.1), we see that (3.11) Then, evaluating the above analytic Feynman integral, we obtain (3.9) as desired.
Next, taking further variations of the expression given in (3.9), we obtain the following corollary.

12)
which, as a function of y, is an element of B(p ; m − l).
In our next theorem, we show that the transform with respect to the second argument of the variation equals the variation of the functional.
which, as a function of y, is an element of B(p; m − 1).
Our next lemma involves the convolution product of functionals from various B(p; m) classes. Proof. First note that, for s-a.e. y ∈ C 0 [0,T ], where n/2 f p g p .
Now, a standard argument shows that l belongs to C 0 (R n ). Hence, (F * G) q is an element of B(∞; m).
In our next theorem, we obtain a formula for the first variation of the convolution product. Theorem 3.3. Let p, m, q, F , and G be as in Lemma 3.2, and let w ∈ A. Then for s-a.e. y ∈ C 0 [0,T ], δ(F * G) q (y | w) exists and is given by the last expression in equation (3.19) below. Furthermore, as a function of y, δ(F * G) q (y | w) is an element of B(∞; m − 1).
Proof. By Lemma 3.2, (F * G) q is an element of B(∞; m) and so, by Lemma 3.1, Next, we obtain formulas for the convolution product of the first variation of functionals. In Theorem 3.4, we take the convolution with respect to the first argument of the variations while in Theorem 3.5, we take the convolution with respect to the second argument of the variations. F ∈ B(p; m) and G ∈ B(p ; m), it follows, from Lemma 3.1, that δF (y | w) ∈ B(p; m − 1) and δG(y | w) ∈ B(p ; m − 1). Hence, by Lemma 3.2, (δF (· | w) * δG(· | w)) q (y) is an element of B(∞; m − 1). Also, by equations (2.9) and (3.1),

3, that T q (2; F) and T q (2; G) both exist and are elements of B(2; m). Hence, (T q (2; F) * T q (2; G)) q is an element of B(∞; m). Equation (3.24) then follows upon the evaluation of the analytic Feynman integral
Proof. First, we note that r ∈ [1,2]. For the case m = 0, it was shown, in [10, p. 29], that (F * G) q ∈ B(r ;0) and that equation ( By choosing specific values for p 1 and p 2 in Theorem 3.7, we obtain the following corollary.

Relationships involving three concepts.
In this section, we look at all the relationships involving the "transform," the "convolution," and "variation" where each operation is used exactly once. There are more than six possibilities since one can take both the transform and the convolution with respect to the first or the second argument of the variation. However, there are some repetitions, for example, we observed, in Theorem 3.2, that the transform with respect to the second argument of the variation equals the variation of the functional. It turns out that there are nine distinct possibilities. We state formally five of these results as theorems (namely, Theorem 4.1 through Theorem 4.5 ) and the other four results as formulas (namely, equation (4.11) through equation (4.14) ). However, all nine of these results hold for s-a.e. y ∈ C 0 [0,T ] and all real q = 0.
In our first theorem, we obtain a formula for the transform with respect to the first argument of the variation of the convolution product.  Let m be a positive integer, let w ∈ A, and let p 1 ,p 2 ,r ,F, and G be as in Theorem 3.7. Then T q r ; δ(F * G) q (· | w) (y) = δT q r ; (F * G) q (y | w) = T q (p 1 ; F) y/ 2 δT q (p 2 ; G) y/ 2 w/ 2 + δT q (p 1 ; F) y/ 2 w/ 2 T q (p 2 ; G) y/ 2 , as a function of y, is an element of B(r ; m − 1).
Proof. The first equality in (4.1) follows from (3.8). But, by Theorem 3.7, (F * G) q ∈ B(r ; m) and so, using (3.26), we see that which equals the last expression on the right-hand side of equation (4.1).

4)
which, as a function of y, is an element of B(r ; m − 1).
Next, we seek formulas for the transforms of the convolution product with respect to the first argument of the variations. Here, there are two cases, namely, we can take the transform of the expression δF (· | w) * δG(· | w) q (y) (4.5) either with respect to y (Theorem 4.3 below), or else with respect to w (Theorem 4.4 below). T q r ; (δF (· | w) * δ(· | w)) q (y) = δT q (p 1 ; F) y/ 2 w δT q (p 2 ; G) y/ 2 w , (4.6) which, as a function of y, is an element of B(r ; m − 1).
Remark. Again, choosing specific values for p 1 and p 2 in Theorem 4.3 (as we did in Corollary 3.5), one gets various versions of equation (4.6).
which, as a function of y, is an element of B(∞; m − 1).

Proof.
To obtain (4.7), we simply substitute the last expression in equation (3.20), with w replaced with w + x, into the analytic Feynman integral (4.8) and then evaluate this integral using (3.23).
Our next goal is to obtain formulas for the transforms of the convolution product with respect to the second argument of the variations. Again, there are two cases since we can take the transform of the expressions in equation (3.21) either with respect to w (Theorem 4.5 below) or else with respect to y (equation (4.11) below). Theorem 4.5. Let p ∈ [1,2], let m be a positive integer, let F ∈ B(p; m), and let G ∈ B(p ; m). Then T q p; δF (y | ·) * δG(y | ·) q (w) = δF y | w/ 2 δG y | w/ 2 , (4.9) which, as a function of y, is an element of B(1; m − 1).
Under the hypotheses of Theorem 4.5, the transform of the expressions in equation (3.21) with respect to y yields n/2 n j=1 R n f j α, y + u g j α, y + u exp iq 2 u 2 d u. (4.11) Next, we want to take the variation of the expressions in equation (3.24). So, let m be a positive integer, let F and G be elements of B(2; m), and let w ∈ A. Then δ T q (2; F) * T q (2; G) q (y | w) which, as a function of y, is an element of B(∞; m − 1).
We finish up this section by finding formulas for the convolution product of the transform of the variation. Again, there are two cases, namely, we can take the convolution with respect to the first argument (equation (4.13) below) or the second argument (equation (4.14) below) of the variation. However, in both cases, the transform is taken with respect to the first argument of the variation. So, let m be a positive integer, let F and G be elements of B(2; m), and let w ∈ A. Then δT q (2; F)(· | w) * δT q (2; G)(· | w) q (y) which, as a function of y, is an element of B(∞; m − 1). Again, let m be a positive integer, let F , and G be elements of B(2; m), and let w ∈ A. Then, using (3.9), (3.23), and then (3.9) again, we obtain δT q (2; F )(y | ·) * δT q (2; G)(y | ·) q (w)  which, as a function of y, is an element of B(∞; m − 1).