Conditional Fourier-Feynman Transforms and Convolutions over Continuous Paths

Abstract In the present paper, we evaluate the analytic conditional FourierFeynman transforms and convolution products of bounded functions which are important in Feynman integration theories and quantum mechanics. We then investigate the inverse transforms of the functions with their relationships and finally that the conditional analytic FourierFeynman transforms of the conditional convolution products for the functions, can be expressed in terms of the products of the conditional Fourier-Feynman transform of each function.


Introduction and preliminaries
Let C 0 [0, t] denote the Wiener space, that is, the space of real-valued continuous paths x on the closed interval [0, t] with x(0) = 0. On the space C 0 [0, t], Chang and Skoug [2] introduced the concepts of conditional Fourier-Feynman transform and conditional convolution product and then, examined the effects that drift has on the conditional Fourier-Feynman transform, the conditional convolution product, and various relationships that occur between them.Moreover, on C[0, t], the space of real-valued continuous paths on [0, t], Kim [12] extended the relationships between the conditional convolution product and the L p (1 ≤ p ≤ ∞)-analytic conditional Fourier-Feynman transform of the functions in a Banach algebra which corresponds to the Cameron-Storvick's Banach algebra S [1].The second author [3,4,5,6,7] also established the relationships between them for various functions on C[0, t].In particular, he [6] derived an evaluation formula for the L p -analytic conditional Fourier-Feynman transforms and convolution products of bounded functions with the conditioning functions X n and X n+1 on C[0, t] given by X n (x) = (x(t 0 ), x(t 1 ), • • • , x(t n )) and X n+1 (x) = (x(t 0 ), x(t 1 ), • • • , x(t n ), x(t n+1 )), where n is a positive integer and 0 = t 0 < t 1 < • • • < t n < t n+1 = t is a partition of [0, t], and then, derived their relationships.Note that X n is independent of the present positions of paths in C[0, t] while X n+1 wholly depends on the present positions.
In this paper, when n = 0, we further develop the relationships on the space (C[0, t], w ϕ ), an analogue of Wiener space associated with a probability measure ϕ on the Borel class of R [11,13,14].In fact, using simple formulas for the conditional expectations given X 0 and X 1 , we proceed to evaluate the L p -analytic conditional Fourier-Feynman transforms and convolution products for the functions of the form for w ϕ -a.e.
σ is a complex Borel measure of bounded variation on L 2 [0, t] and ρ is a bounded complex Borel measure on R r .We then investigate the inverse transforms and the relationships between them of the functions given by (1).Finally, we show that the L p -analytic conditional Fourier-Feynman transform of the conditional convolution product for the functions Ψ 1 and Ψ 2 given by (1), can be expressed by the formula for a nonzero real q, w ϕ -a.e.y ∈ C[0, t] and P X 0 -a.e.ξ 0 , ζ 0 ∈ R. Thus the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions, can be interpreted as the product of the analytic conditional Fourier-Feynman transform of each function.Note that they are independent of the initial positions of paths in C[0, t] while they depend only on the present time t.Let C, C + and C ∼ + denote the sets of complex numbers, complex numbers with positive real parts and nonzero complex numbers with nonnegative real parts, respectively.
Let C = C[0, t] be the space of all real-valued continuous functions on the closed interval [0, t] and let (C[0, t], B(C[0, t]), w ϕ ) [11,13,14] (v, x) is called the Paley-Wiener-Zygmund integral of v according to x.Note that by •, • Ê r , the dot product on the r-dimensional Euclidean space R r is also denoted.
Let F : C[0, t] → C be integrable and X be a random vector on C[0, t] assuming that the value space of X is a normed space equipped with the Borel σ-algebra.Then, we have the conditional expectation E[F |X] of F given X from a well known probability theory.Furthermore, there exists a P X -integrable complex-valued function ψ on the value space of X such that E[F |X](x) = (ψ • X)(x) for w ϕ -a.e.x ∈ C[0, t], where P X is the probability distribution of X.The function ψ is called the conditional w ϕ -integral of F given X and it is also denoted by Throughout this paper, let unless otherwise specified, where the expectation is taken over the variable x.Moreover, we have from [8], If I λ F (0, ξ 0 , ξ 1 ) has the analytic extension J * λ (F )(ξ 0 , ξ 1 ) on C + as a function of λ, then it is called the conditional analytic Wiener w ϕ -integral of F given X 1 with parameter λ and denoted by Moreover, if for a nonzero real q, E anw λ [F |X 1 ](ξ 0 , ξ 1 ) has the limit as λ approaches to −iq through C + , then it is called the conditional analytic Feynman w ϕ -integral of F given X 1 with parameter q and denoted by For a given extended real number p with 1 < p ≤ ∞, suppose that p and p are related by 1  p + 1 p = 1(possibly p = 1 if p = ∞).Let F n and F be measurable functions such that lim n→∞ C |F n (x) − F (x)| p dw ϕ (x) = 0. Then we write l.i.m. n→∞ (w p )(F n ) = F and call F the limit in the mean of order p .A similar definition is understood when n is replaced by a continuously varying parameter.
Let F and G be defined on For a nonzero real q and w ϕ -a.e.y ∈ C[0, t], define the L 1 -analytic conditional Fourier-Feynman transform T (1) , where λ approaches to −iq through C + .We also define the conditional convolution product [(F * G) λ |X 1 ] of F and G given X 1 by the formula, for w ϕ -a.e.y ∈ C[0, t] . Similar definitions and notations are understood with ξ 0 ∈ R if X 1 is replaced by X 0 .

The position-dependent transforms
Throughout this paper, let β lj e j be the linear combinations of the e j s and let A = [β lj ] r×r be the coefficient matrix of the combinations.Define the linear transformation T A : R r → R r given by T A z = zA, where z is any row-vector in R r .For and for λ ∈ C ∼ + let where ).Note that by the Bessel's inequality, Using the same method as used in the proof of Theorem 2.6 in [10], we can prove the following lemma.
Let M(R r ) be the space of all functions φ on R r defined by where ρ is in the space M(R r ) of complex Borel measure of bounded variation over R r .Let M(L 2 [0, t]) be the class of all C-valued Borel measures of bounded variation over L 2 [0, t] and let S wϕ be the space of all functions F which for σ ∈ M(L 2 [0, t]) have the form for w ϕ -a.e.x ∈ C[0, t].Note that S wϕ is a Banach algebra which is equivalent to M(L 2 [0, t]) with the norm F = σ , the total variation of σ [11].

The position-independent transforms
In this section, we evaluate the present position-independent conditional Fourier-Feynman transform and conditional convolution product of the functions as given in the previous section.For these purpose, we need the following lemma by the well-known integration formula Ê exp{−au 2 +ibu}du = ( π a ) and z ∈ R r , let H 1 (y, ξ 0 , ξ 1 , v, z) be given by (2) and for λ ∈ C ∼ + , let Then for λ > 0, λ 2πt Using the same process as used in the proof of Theorem 2.2, we can easily prove the following theorem by (4), Theorem 2.2, Lemma 3.1, the Morera's theorem and the dominated convergence theorem.Theorem 3.2 Under the assumptions as given in Theorem 2.2, we have for a nonzero real q, w ϕ -a.e.y ∈ C[0, t] and P X 0 -a.e.ξ 0 ∈ R where H 1 , H 2 and H 4 are given by ( 2), ( 3) and (9), respectively.

Relationships between the transforms and convolutions
In this section, we investigate the inverse conditional transform of the conditional Fourier-Feynman transforms of the functions as given in the previous sections.We also show that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions, can be expressed as the products of the analytic conditional Fourier-Feynman transform of each function.
Theorem 4.1 Let the assumptions and notations be as given in Theorem 2.2.For (ξ 0 , ξ 1 ), (ζ 0 , ζ 1 ) ∈ R 2 and y ∈ C[0, t], define where H 1 is given by (2).Then, for by Lemma 2.1 and Theorem 2.2, where H 1 , H 2 and H 3 are given by ( 2), ( 3) and ( 5), respectively.By (4) and the Morera's Theorem, we have the existence of which converges to 0 as λ approaches to −iq through C + by the dominated convergence theorem.which converges to 0 as λ approaches to −iq through C + by the dominated convergence theorem.