A Rotation of Admixable Operators on Abstract Wiener Space with Applications

Abstract

We investigate certain rotation properties of the abstract Wiener measure. To determine our rotation property for the Wiener measure, we introduce the concept of an admixable operator via an algebraic structure on abstract Wiener space. As for applications, we define the analytic Fourier-Feynman transform and the convolution product associated with the admixable operators and proceed to establish the relationships between this transform and the corresponding convolution product.

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Recommended by Henryk Hudzik

Department of Mathematics, Dankook University, Cheonan 330-714, Republic of Korea

Received 17 December 2012; Accepted 24 June 2013

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and Preliminaries

Let ( _{C 0} [ 0 , T ] , [physics M-matrix] , _{m w} ) denote one-parameter Wiener space. Bearman's rotation theorem [1] for Wiener measure has played an important role in various research areas in mathematics and physics involving Wiener integration theory. Bearman's theorem was further developed by Cameron and Storvick [2] and by Johnson and Skoug [3] in their studies of Wiener integral equations. Recently, in [4], using results in [5], Chang et al. obtained results involving a very general multiple Fourier-Feynman transform on Wiener space.

Let H be a real separable infinite-dimensional Hilbert space with inner product Y9; · , · YA; and norm | · | = Y9; · , · YA; . Let _{|| · || 0} be a measurable norm on H with respect to the Gaussian cylinder set measure _{ν 0} on H . Let B denote the completion of H with respect to _{|| · || 0} . Let i denote the natural injection from H to B . The adjoint operator ^{i *} of i is one to one and maps ^{B *} continuously onto a dense subset of ^{H *} , where ^{B *} and ^{H *} are topological duals of B and H , respectively. By identifying ^{H *} with H and ^{B *} with ^{i * B *} , we have a triple ^{B *} ⊂ ^{H *} [approximate] H ⊂ B and Y9; x , y YA; = ( x , y ) for all x in H and y in ^{B *} , where ( · , · ) denotes the natural dual pairing between B and ^{B *} . By the well-known result of Gross [6], _{ν 0} [composite function] ^{i - 1} provides a unique countably additive extension, ν , to the Borel σ -algebra [Bernoulli] ( B ) of B . ν is a probability measure on the Borel σ -algebra [Bernoulli] ( B ) of B which satisfies [figure omitted; refer to PDF] The triple ( B , H , ν ) is called an abstract Wiener space. For more details, see [6-9].

Let { _{e j} ^{} j = 1 ∞} be a complete orthonormal set in H such that _{e j} 's are in ^{B *} . For each h ∈ H and x ∈ B , we define the stochastic inner product ( h , x ^{) ~} by [figure omitted; refer to PDF] For every h (...0; 0) in H , ( h , x ^{) ~} exists for ν -a.e. x ∈ B , and it is a Gaussian random variable on B with mean zero and variance | h ^{| 2} ; that is, (1) holds with y ∈ ^{B *} replaced with h ∈ H . In fact, the stochastic inner product ( h , x ^{) ~} is essentially independent of the choice of the complete orthonormal set used in its definition. Also, if both h and x are in H , then Parseval's identity gives ( h , x ^{) ~} = Y9; h , x YA; . Furthermore, ( h , λ x ^{) ~} = ( λ h , x ^{) ~} = λ ( h , x ^{) ~} for any λ ∈ ... , h ∈ H , and x ∈ B . We also see that, if { _{h 1} , ... , _{h n} } is an orthonormal set in H , then the random variables ( _{h j} , x ^{) ~} 's are independent.

Let ...B2; ( B ) be the class of ν -measurable subsets of B . A subset E of B is said to be scale-invariant measurable [3, 7] provided ρ E is ...B2; ( B ) -measurable for every ρ > 0 , and a scale-invariant measurable subset N of B is said to be scale-invariant null provided ν ( ρ N ) = 0 for every ρ > 0 . A property that holds except on a scale-invariant null set is said to hold scale invariant almost everywhere ( s -a.e.). A functional F on B is said to be scale-invariant measurable provided F is defined on a scale-invariant measurable set and F ( ρ · ) is ...B2; ( B ) -measurable for every ρ > 0 . If two functionals F and G on B are equal s -a.e., that is, for each ρ > 0 , ν ( { x ∈ B : F ( ρ x ) ...0; G ( ρ x ) } ) = 0 , then we write F [approximate] G .

Next, we introduce the concept of an admixable operator on B .

Definition 1.

Let [ecedil]9; be an operation between H and ^{B *} which satisfies the conditions:

(1) ^{B *} × ^{B *} ∋ ( _{g 1} , _{g 2} ) ... _{g 1} [ecedil]9; _{g 2} = _{g 2} [ecedil]9; _{g 1} ∈ ^{B *} .

(2) H × ^{B *} ∋ ( h , g ) ... h [ecedil]9; g = g [ecedil]9; h ∈ H .

(3) If h [ecedil]9; g = 0 for h ∈ H and g ∈ ^{B *} , then h = 0 or g = 0 .

(4) For every h ∈ H and every _{g 1} , _{g 2} ∈ ^{B *} , [figure omitted; refer to PDF]

(5) For every _{h 1} , _{h 2} ∈ H and every g ∈ ^{B *} , [figure omitted; refer to PDF]

(6) For every _{g 1} , _{g 2} ∈ ^{B *} , there exists _{g 3} ∈ ^{B *} such that [figure omitted; refer to PDF]

: where ^{g [ecedil]9; 2} = g [ecedil]9; g . In this case, we write _{g 3} = ^{g 1 [ecedil]9; 2} + ^{g 2 [ecedil]9; 2} [ecedil]9; .

(7) For every _{h 1} , _{h 2} ∈ H and every g ∈ ^{B *} , [figure omitted; refer to PDF]

Given g ∈ ^{B *} , let _{A g} : B [arrow right] B be a linear operator associated with g . The operator _{A g} is said to be ^{g [ecedil]9;} -admixable provided ( h , _{A g} x ^{) ~} = ( h [ecedil]9; g , x ^{) ~} for all h ∈ H .

For a finite subset ...B1; = { _{v 1} , ... , _{v m} } of H , let _{X ...B1;} : B [arrow right] ^{... m} be the random vector given by [figure omitted; refer to PDF] A functional F is called a cylinder-type functional on B if there exists a linearly independent subset ...B1; = { _{v 1} , ... , _{v m} } of H such that [figure omitted; refer to PDF] where ψ is a complex-valued Lebesgue measurable function on ^{... m} . It is easy to show that, for the given cylinder-type functional F of the form (8), there exists an orthogonal subset [Hamiltonian (script capital H)] = { _{h 1} , ... , _{h n} } of H such that F is expressed as [figure omitted; refer to PDF] where f is a complex-valued Lebesgue measurable function on ^{... n} . Thus, we lose no generality in assuming that every cylinder-type functional on B is of the form (9).

Lemma 2 (Chung, [7]).

Let ( B , H , ν ) be an abstract Wiener space, and let [Hamiltonian (script capital H)] = { _{h 1} , ... , _{h n} } be an orthogonal set in H . Let f be a complex-valued function defined on ^{... n} . Then, for the cylinder-type functional F given by (9) on B ,

(i) F is [Bernoulli] ( B ) -measurable if and only if f is Borel measurable on ^{... n} ,

(ii) F is ...B2; ( B ) -measurable if and only if f is Lebesgue measurable on ^{... n} .

For g ∈ ^{B *} , let _{A g} be the ^{g [ecedil]9;} -admixable operator on B . In this case, for any orthogonal subset [Hamiltonian (script capital H)] of H , [figure omitted; refer to PDF]

The seminal results by Bearman in [1] are summarized as follows (see [2]): if F ( ^{a 2} + ^{b 2} x ) is Wiener integrable on _{C 0} [ 0 , T ] for a , b ∈ ... , then F ( a _{x 1} + b _{x 2} ) is integrable on ( _{C 0} [ 0 , T ] ^{) 2} and [figure omitted; refer to PDF]

The main purpose of this paper is to establish a rotation property for the abstract Wiener integral, [figure omitted; refer to PDF] where F is given by (9) and _{A g} is determined by _{A g 1} and _{A g 2} .

2. A Typical Example of an Abstract Wiener Space

The classical Wiener space _{C 0} [ 0 , T ] , which is one of the most important examples of abstract Wiener spaces (see [9]), is a triple ( B , H , ν ) , where

(i) B = _{C 0} [ 0 , T ] is a Banach space consisting of real-valued continuous functions x ( t ) with x ( 0 ) = 0 defined on the compact interval [ 0 , T ] endowed with uniform norm || x || = _{sup t ∈ [ 0 , T ]} | x ( t ) | ,

(ii) H = ^{C 0 [variant prime]} [ 0 , T ] is a real separable infinite dimensional Hilbert space consisting of absolutely continuous functions h ( t ) with h ( 0 ) = 0 , such that D h ...1; d h / d t ∈ _{L 2} [ 0 , T ] endowed with the inner product [figure omitted; refer to PDF]

(iii): ν = _{m w} is the Wiener measure on the Borel σ -algebra [Bernoulli] ( _{C 0} [ 0 , T ] ) of _{C 0} [ 0 , T ] with [figure omitted; refer to PDF]

Let I be the unitary operator from _{L 2} [ 0 , T ] , onto ^{C 0 [variant prime]} [ 0 , T ] given by I v ( t ) = ^{∫ 0 t} ... v ( s ) d s for v ∈ _{L 2} [ 0 , T ] and let [figure omitted; refer to PDF] For any h ∈ ^{C 0 [variant prime]} [ 0 , T ] and g ∈ ^{C 0 *} [ 0 , T ] , let the operation [ecedil]9; between ^{C 0 [variant prime]} [ 0 , T ] and ^{C 0 *} [ 0 , T ] be defined by [figure omitted; refer to PDF] where D h D g denotes the pointwise multiplication of the functions D h and D g .

It is readily seen that { x ( t ) : ( x , t ) ∈ _{C 0} [ 0 , T ] × [ 0 , T ] } is a standard Wiener process on the probability space ( _{C 0} [ 0 , T ] , [Bernoulli] ( _{C 0} [ 0 , T ] ) , _{m w} ) . In this case, we know that, for each h ∈ ^{C 0 [variant prime]} [ 0 , T ] and s -a.e. x ∈ _{C 0} [ 0 , T ] , [figure omitted; refer to PDF] where ^{∫ 0 T} ... D h ( t ) d ~ x ( t ) denotes the Paley-Wiener-Zygmund stochastic integral [10-12].

Remark 3.

Let [physics M-matrix] be the set of all _{m w} -measurable subsets of _{C 0} [ 0 , T ] . Then, ( _{C 0} [ 0 , T ] , [physics M-matrix] , _{m w} ) is a complete measure space. It is well known that [physics M-matrix] coincides with σ ( [Bernoulli] ( _{C 0} [ 0 , T ] ) ) , the completion of [Bernoulli] ( _{C 0} [ 0 , T ] ) .

Let g ∈ ^{C 0 [variant prime]} [ 0 , T ] with | g | = Y9; g , g YA; > 0 . Then, the stochastic integral [figure omitted; refer to PDF] which was introduced by Park and Skoug in [13], is a Gaussian process with mean zero and covariance function [figure omitted; refer to PDF] In addition, _{...B5; g} ( x , t ) is stochastically continuous in t on [ 0 , T ] . For more detailed studies of this process, see [14-17]. Furthermore, if g is an element of ^{C 0 *} [ 0 , T ] , then, for all x ∈ _{C 0} [ 0 , T ] , _{...B5; g} ( x , t ) is continuous in t , and so is _{...B5; g} ( x , · ) in _{C 0} [ 0 , T ] .

From [14, Lemma 1], we note that, for each h ∈ ^{C 0 [variant prime]} [ 0 , T ] and each g ∈ ^{C 0 [variant prime]} [ 0 , T ] with D g ∈ _{L ∞} [ 0 , T ] , [figure omitted; refer to PDF] for s -a.e. x ∈ _{C 0} [ 0 , T ] .

Given g ∈ ^{C 0 *} [ 0 , T ] , define an operator _{A g} : _{C 0} [ 0 , T ] [arrow right] _{C 0} [ 0 , T ] by [figure omitted; refer to PDF] Then, for all h ∈ H , [figure omitted; refer to PDF] Thus, _{A g} is ^{g [ecedil]9;} -admixable in view of Definition 1.

3. A Rotation of Admixable Operators

In this section, we establish a rotation property for the abstract Wiener integral involving admixable operators. We first introduce an integration formula which plays a key role.

Lemma 4.

Let ...9C; = { _{α 1} , ... , _{α n} } be an orthogonal set in H and let _{X ...9C;} be given by (7). Let f : ^{... n} [arrow right] ... be a Lebesgue measurable function. Then [figure omitted; refer to PDF] where by = * one means that if either side exists, both sides exist and equality holds.

The following integration formula is also used several times in this paper: [figure omitted; refer to PDF] for complex numbers a and b with Re ( a ) > 0 .

Remark 5.

Let ^{α 1} and ^{α 2} be elements in H with | ^{α 1} | = | ^{α 2} | ...1; ^{σ 2} > 0 . Then, the random variables _{X 1} ( x ) = ( ^{α 1} , x ^{) ~} and _{X 2} ( x ) = ( ^{α 2} , x ^{) ~} will have the same distribution N ( 0 , ^{σ 2} ) .

Let _{...9C; 1} = { ^{α 1 1} , ... , ^{α n 1} } and _{...9C; 2} = { ^{α 1 2} , ... , ^{α n 2} } be orthogonal sets in H with | ^{α j 1} | = | ^{α j 2} | for each j ∈ { 1 , ... , n } . Using the aforementioned facts and applying (23), we see that, for any Lebesgue measurable function f on ^{... n} , [figure omitted; refer to PDF]

To simplify the expressions in our results, we use the following notations: [figure omitted; refer to PDF] for u [arrow right] = ( _{u 1} , ... , _{u n} ) ∈ ^{... n} and _{X ...9C;} ( x ) given by (7).

Proposition 6.

Let _{...9C; 1} = { ^{α 1 1} , ... , ^{α n 1} } and _{...9C; 2} = { ^{α 1 2} , ... , ^{α ν 2} } be orthogonal sets in H . Then, for any Lebesgue measurable function f on ^{... n} , [figure omitted; refer to PDF] where _{...B2; ( 2 )} = { _{w 1} , ... , _{w n} } is an orthogonal set in H which satisfies the condition [figure omitted; refer to PDF] for each j ∈ { 1 , ... , n } . Also, both of the expressions in (27) are given by the last expression in (29).

Proof.

First, using (23), the Fubini theorem, and (24), we have that [figure omitted; refer to PDF]

Next, let { _{e 1} , ... , _{e n} } be any orthonormal set in H . For each j ∈ { 1 , ... , n } , let _{w j} = ( | ^{α j 1 | 2} + | ^{α j 2 | 2 ) 1 / 2} _{e j} . Then _{...B2; ( 2 )} = { _{w 1} , ... , _{w n} } is an orthogonal set in H and satisfies (28) above. In this case, using (23), the Fubini theorem, and (24), we see that _{∫ B} ... f ( _{X ...B2; ( 2 )} ( x ) ) d ν ( x ) is given by the last expression of (29). In view of Remark 5, we obtain the desired result.

The following corollary follows by the use of mathematical induction.

Corollary 7.

For each k ∈ { 1 , ... , m } , let _{...9C; k} = { ^{α 1 k} , ... , ^{α n k} } be an orthogonal set in H . Then for any Lebesgue measurable function f on ^{... n} , [figure omitted; refer to PDF] where _{...B2; ( m )} = { _{w 1} , ... , _{w n} } is an orthogonal set in H which satisfies the condition [figure omitted; refer to PDF] for each j ∈ { 1 , ... , n } . Also, both of the expressions in (30) are given by the expression [figure omitted; refer to PDF]

The next corollary follows directly from Proposition 6.

Corollary 8.

Let _{...9C; 1} and _{...9C; 2} be as in Proposition 6. Assume that _{...9C; 1} ∪ _{...9C; 2} is an orthogonal set in H . Then, for any Borel measurable function f on ^{... n} , (27) is valid. In this case, _{...B2; ( 2 )} is given by the set { ^{α 1 1} + ^{α 1 2} , ... , ^{α n 1} + ^{α n 2} } .

Throughout the rest of this section, for convenience, we use the following notation: for a finite sequence ...A2; = { _{g 1} , ... , _{g m} } in ^{B *} , let [figure omitted; refer to PDF]

For our rotation property presented, namely, Theorem 11, we will consider the pair of finite subsets [Hamiltonian (script capital H)] = { _{h 1} , ... , _{h n} } of H - { 0 } and ...A2; = { _{g 1} , ... , _{g m} } of ^{B *} - { 0 } (this allows | ...A2; | ...4; m ) such that

(c1) { _{h 1} , ... , _{h n} } is orthogonal in H ,

(c2) { _{h 1} [ecedil]9; _{g k} , ... , _{h n} [ecedil]9; _{g k} } is orthogonal in H for all k ∈ { 1 , ... , m } .

Let us return to the classical Wiener space ( _{C 0} [ 0 , T ] , ^{C 0 [variant prime]} [ 0 , T ] , _{m w} ) . See Section 2. For each l ∈ { 1,2 , ... } , let _{β l} ( t ) = sin ( ( ( l - 1 / 2 ) π / T ) t ) on [ 0 , T ] . Then, Θ ...1; { _{β l} ^{} l = 1 ∞} is an orthogonal sequence in ^{C 0 [variant prime]} [ 0 , T ] . Additionally, Θ ~ = { ( 2 T / ( l - 1 / 2 ) π ) _{β l} ^{} l = 1 ∞} is a complete orthonormal set in ^{C 0 [variant prime]} [ 0 , T ] . In fact, _{β l} ∈ ^{C 0 *} [ 0 , T ] for all l ∈ { 1,2 , ... } . We can take different pair ( [Hamiltonian (script capital H)] , ...A2; ) from Θ , which satisfies conditions (c1) and (c2) above. For instance, let [Hamiltonian (script capital H)] = { _{β l 1} , ... , _{β l n} } and let ...A2; = { _{β k 1} , ... , _{β k m} } , where _{l 1} < _{l 2} < ... < _{l n} < _{k 1} ...4; _{k 2} ...4; ... ...4; _{k m} .

Remark 9.

Let [Hamiltonian (script capital H)] = { _{h 1} , ... , _{h n} } and ...A2; = { _{g 1} , ... , _{g m} } satisfy the conditions (c1) and (c2). Then, the stochastic inner products [figure omitted; refer to PDF] form a set of independent Gaussian random variables on B with mean 0 and variance [figure omitted; refer to PDF] where s ( ...A2; ) ...1; s ( _{g 1} , ... , _{g m} ) is given by (33).

Remark 10.

Given an orthogonal subset [Hamiltonian (script capital H)] , let ^{B [Hamiltonian (script capital H)] *} be the space of every element g in ^{B *} - { 0 } such that { h [ecedil]9; g : h ∈ [Hamiltonian (script capital H)] } is orthogonal in H . Then, for any finite subset ...A2; of ^{B [Hamiltonian (script capital H)] *} , the pair ( [Hamiltonian (script capital H)] , ...A2; ) satisfies conditions (c1) and (c2) above.

Given an orthogonal set [Hamiltonian (script capital H)] in H , let ^{... [Hamiltonian (script capital H)] ( 1 )} be the class of all cylinder-type functionals, F , given by (9) for ν -a.e. x ∈ B , where the corresponding function f of F satisfies the condition [figure omitted; refer to PDF] for all g ∈ ^{B [Hamiltonian (script capital H)] *} .

We are now ready to present our rotation property of abstract Wiener measure associated with admixable operators.

Theorem 11.

Let [Hamiltonian (script capital H)] = { _{h 1} , ... , _{h n} } be an orthogonal set in H and let F ∈ ^{... [Hamiltonian (script capital H)] ( 1 )} be given by (9). Let ...A2; = { _{g 1} , ... , _{g m} } be a finite subset of ^{B [Hamiltonian (script capital H)] *} , and, for each k ∈ { 1 , ... , m } , let _{A g k} be the ^{g k [ecedil]9;} -admixable operator on B . Then, [figure omitted; refer to PDF] where s ( ...A2; ) is given by (33).

Proof.

For each j ∈ { 1 , ... , n } and each k ∈ { 1 , ... , m } , let ^{α j k} = _{h j} [ecedil]9; _{g k} . For each k ∈ { 1 , ... , m } , _{...9C; k} = { ^{α 1 k} , ... , ^{α n k} } is orthogonal in H by condition (c2). Hence, for each k ∈ { 1 , ... , m } , the stochastic inner products [figure omitted; refer to PDF] form a set of independent Gaussian random variables with mean 0 and variance | ^{α j k | 2} .

We observe that [figure omitted; refer to PDF] Thus, using (9), (39), and (30), we obtain [figure omitted; refer to PDF] where _{...B2; ( m )} = { _{w 1} , ... , _{w n} } is an orthogonal set in H , such that [figure omitted; refer to PDF] for each j ∈ { 1 , ... , n } .

Next, we note that, for each j , l ∈ { 1 , ... , n } with j ...0; l , [figure omitted; refer to PDF] Hence, from (35) and (42), we see that { _{h 1} [ecedil]9; s ( ...A2; ) , ... , _{h n} [ecedil]9; s ( ...A2; ) } is an orthogonal set in H with | _{h j} [ecedil]9; s ( ...A2; ) ^{| 2} = ^{∑ k = 1 m} ... | _{h j} [ecedil]9; _{g k} ^{| 2} for all j ∈ { 1 , ... , n } , and so we can choose _{...B2; ( m )} to be the orthogonal set { _{h 1} [ecedil]9; s ( ...A2; ) , ... , _{h n} [ecedil]9; s ( ...A2; ) } . In this case, we see that [figure omitted; refer to PDF] Equation (37) follows from (40) and (43).

Example 12.

Let us return to the classical Wiener space ( _{C 0} [ 0 , T ] , ^{C 0 [variant prime]} [ 0 , T ] , _{m w} ) again. We introduce the family of functions Γ ...1; { _{γ τ} : τ ∈ [ 0 , T ] } from ^{C 0 [variant prime]} [ 0 , T ] : [figure omitted; refer to PDF] These functions have the reproducing property [figure omitted; refer to PDF] for all h ∈ ^{C 0 [variant prime]} [ 0 , T ] . In fact, Γ ⊂ ^{C 0 *} [ 0 , T ] .

For _{γ τ} ∈ Γ , we consider the admixable operator _{A γ τ} given by (21). We note that, for each g ∈ ^{C 0 *} [ 0 , T ] , [figure omitted; refer to PDF] From this, we see that, for _{g I} ( t ) = t = ^{∫ 0 t} ... d s ∈ ^{C 0 *} [ 0 , T ] and a ∈ ... , [figure omitted; refer to PDF] and so, using (37), we find that, for any f ∈ _{L 1} ( ... ) and every a , b ∈ ... , [figure omitted; refer to PDF] Using (16), we now have [figure omitted; refer to PDF] on [ 0 , T ] . Thus, by (48), (49), and (47), we have [figure omitted; refer to PDF]

Proposition 13 (Cameron and Storvick, [2]).

Let F ( ^{a 2} + ^{b 2} x ) be Wiener integrable on _{C 0} [ 0 , T ] for a , b ∈ ... . As a result, F ( a _{x 1} + b _{x 2} ) is integrable on ^{C 0 2} [ 0 , T ] , and [figure omitted; refer to PDF]

Proof.

Let n be any positive integer, and let 0 = _{τ 0} < _{τ 1} < ... < _{τ n} ...4; T be any partition of [ 0 , T ] . It suffices to show that (51) holds for any tame function [figure omitted; refer to PDF] with f ∈ _{L 1} ( ^{... n} ) .

Let [ecedil]9; be defined by (16) between ^{C 0 [variant prime]} [ 0 , T ] and ^{C 0 *} [ 0 , T ] . For _{g I} ( t ) = t = ^{∫ 0 t} ... d s ∈ ^{C 0 *} [ 0 , T ] , let the ^{g I [ecedil]9;} -admixable operator _{A g I} be given by (21), and, for each j ∈ { 1 , ... , n } , let [figure omitted; refer to PDF] on [ 0 , T ] . For any a , b ∈ ... - { 0 } ,

(c1) { _{h 1} , ... , _{h n} } is an orthogonal set in ^{C 0 [variant prime]} [ 0 , T ] , and

(c2) { _{h 1} [ecedil]9; a _{g I} , ... , _{h n} [ecedil]9; a _{g I} } and { _{h 1} [ecedil]9; b _{g I} , ... , _{h n} [ecedil]9; b _{g I} } are orthogonal sets in ^{C 0 [variant prime]} [ 0 , T ] .

Also, for any a , b ∈ ... - { 0 } , _{A s ( a g I , b g I )} = _{A ^{a 2} + ^{b 2} g I} , and, for each j ∈ { 1 , ... , n } , [figure omitted; refer to PDF] Thus, the left side of (51) with F given by (52) is rewritten by [figure omitted; refer to PDF] Thus, from Theorem 11, we obtain [figure omitted; refer to PDF]

Using standard methods, similar to those in [1], we can obtain the result for general functionals F on _{C 0} [ 0 , T ] .

4. Fourier-Feynman Transforms and Convolutions Associated with Admixable Operators

In this section, to apply our results from the previous section, we first define an _{L p} analytic Fourier-Feynman transform associated with admixable operators on B . Then, we establish the existence theorem and the inverse transform theorem of this transform for some classes of cylinder-type functionals on B having the form (9) for s -a.e. x ∈ B . Moreover, we present various relationships involving the convolution and the transforms.

Throughout the rest of this paper, let ... , _{... +} , and _{... ~ +} denote, respectively, the complex numbers, the complex numbers with positive real part, and the nonzero complex numbers with nonnegative real part.

Let g ∈ ^{B *} , and let _{A g} be the corresponding admixable operator on B . Let F : B [arrow right] ... be a scale-invariant measurable functional such that [figure omitted; refer to PDF] exists as a finite number for all λ > 0 . If there exists a function ^{J F *} ( g ; λ ) analytic on _{... +} such that ^{J F *} ( g ; λ ) = _{J F} ( g ; λ ) for all λ > 0 , then ^{J F *} ( g ; λ ) is defined to be the analytic Wiener integral (associated with the ^{g [ecedil]9;} -admixable operator _{A g} ) of F over B with parameter λ . For λ ∈ _{... +} , we write [figure omitted; refer to PDF] Let q ...0; 0 be a real number, and let F be a functional such that ^{∫ B an _{w λ}} ... F ( _{A g} x ) d ν ( x ) = ^{J F *} ( g ; λ ) exists for all λ ∈ _{... +} . If the following limit exists, we call it the analytic Feynman integral of F with parameter q , and we write [figure omitted; refer to PDF] where λ approaches - i q through values in _{... +} .

Note that if _{A g} is the identity operator on B , then these definitions agree with the previous definitions of the analytic Wiener integral and the analytic Feynman integral [18-20].

We are now ready to state the definition of the analytic Fourier-Feynman transform associated with admixable operator (admix-FFT).

Definition 14.

Let ( B , H , ν ) be an abstract Wiener space. For g ∈ ^{B *} , λ ∈ _{... +} , and y ∈ B , let [figure omitted; refer to PDF] where _{A g} is the ^{g [ecedil]9;} -admixable operator on B . Let q be a nonzero real number. For p ∈ ( 1,2 ] , we define the _{L p} analytic ^{g [ecedil]9;} -admix-FFT, ^{T q , g ( p )} ( F ) of F , by the formula ( λ ∈ _{... +} ) , [figure omitted; refer to PDF] if it exists; that is, for each ρ > 0 , [figure omitted; refer to PDF] where 1 / p + 1 / ^{p [variant prime]} = 1 . We define the _{L 1} analytic ^{g [ecedil]9;} -admix-FFT, ^{T q , g ( 1 )} ( F ) of F , by the formula ( λ ∈ _{... +} ) , [figure omitted; refer to PDF] if it exists.

We note that, for p ∈ [ 1,2 ] , ^{T q , g ( p )} ( F ) is defined only s -a.e.. We also note that if ^{T q , g ( p )} ( F ) exists and if F [approximate] G , then ^{T q , g ( p )} ( G ) exists and ^{T q , g ( p )} ( G ) [approximate] ^{T q , g ( p )} ( F ) .

Next, we give the definition of the convolution product (CP).

Definition 15.

Let F and G be scale-invariant measurable functionals on B . For λ ∈ _{... ~ +} and g ∈ ^{B *} , we define their CP with respect to _{A g} (if it exists) by [figure omitted; refer to PDF] When λ = - i q , we denote ( F * G _{) λ , g} by ( F * G _{) q , g} .

For any scale-invariant measurable functional F , we see that, for λ > 0 , [figure omitted; refer to PDF] if it exists.

Let [physics M-matrix] ( ^{... n} ) denote the space of complex-valued, countably additive (and hence finite) Borel measures on [Bernoulli] ( ^{... n} ) , the Borel σ -algebra of ^{... n} . It is well known that a complex-valued Borel measure τ necessarily has a finite total variation || τ || , and [physics M-matrix] ( ^{... n} ) is a Banach algebra under the norm || · || and with convolution as multiplication.

For τ ∈ [physics M-matrix] ( ^{... n} ) , the Fourier transform τ ^ of τ is a complex-valued function defined on ^{... n} by the formula [figure omitted; refer to PDF] where u [arrow right] = ( _{u 1} , ... , _{u n} ) and v [arrow right] = ( _{v 1} , ... , _{v n} ) are in ^{... n} .

Let [Hamiltonian (script capital H)] = { _{h 1} , ... , _{h n} } be an orthonormal set in H . Define the functional F : B [arrow right] ... by [figure omitted; refer to PDF] for s -a.e. x ∈ B , where τ ^ is the Fourier transform of τ in [physics M-matrix] ( ^{... n} ) . Then F is a bounded cylinder-type functional because | τ ^ ( u [arrow right] ) | ...4; || τ || < + ∞ .

Let _{... [Hamiltonian (script capital H)]} be the set of all functionals F on B having the form (67). Note that F ∈ _{... [Hamiltonian (script capital H)]} implies that F is scale-invariant measurable on B . Throughout this section, we fix the orthogonal set [Hamiltonian (script capital H)] .

We now state the existence theorem for the analytic Feynman integral of the functionals in _{... [Hamiltonian (script capital H)]} .

Theorem 16.

Let F ∈ _{... [Hamiltonian (script capital H)]} be given by (67). Then, for all g ∈ ^{B [Hamiltonian (script capital H)] *} and all nonzero real numbers q , the analytic Feynman integral ^{I g a n _{f q}} [ F ] of F exists and is given by the formula [figure omitted; refer to PDF]

Proof.

By (67), (66), the Fubini theorem, (23), and (24), we see that, for all λ > 0 , [figure omitted; refer to PDF]

Now, let ^{J F *} ( g ; λ ) = _{∫ ^{... n}} ... exp { - ( 1 / 2 λ ) ^{∑ j = 1 n} ... | _{h j} [ecedil]9; g ^{| 2 v j 2} } d τ ( v [arrow right] ) for λ ∈ _{... +} . Then, ^{J F *} ( g ; λ ) = _{J F} ( g ; λ ) for all λ > 0 and | ^{J F *} ( g ; λ ) | ...4; _{∫ ^{... n}} ... d | τ | ( v [arrow right] ) ...4; || τ || < ∞ for all λ ∈ _{... +} . Thus, applying the dominated convergence theorem, we see that ^{J F *} ( g ; λ ) is continuous on _{... +} . Also, because k ( λ ) ...1; exp { - ( 1 / 2 λ ) ^{∑ j = 1 n} ... | _{h j} [ecedil]9; g ^{| 2 v j 2} } is analytic on _{... +} , applying the Fubini theorem, we have [figure omitted; refer to PDF] for all rectifiable simple closed curve Δ lying in _{... +} . Thus, by the Morera theorem, ^{J F *} ( g ; λ ) is analytic on _{... +} . Therefore, the analytic Wiener integral ^{I g an _{w λ}} [ F ] = ^{J F *} ( g ; λ ) exists. Finally, applying the dominated convergence theorem, we know that ^{I g an _{f q}} [ F ] = _{lim λ [arrow right] - i q} ^{I g an _{w λ}} [ F ] is given by the right side of (68).

Next, we establish the existence of the admix-FFT for functionals in _{... [Hamiltonian (script capital H)]} .

Theorem 17.

Let F ∈ _{... [Hamiltonian (script capital H)]} be given by (67). Then, for all p ∈ [ 1,2 ] and all g ∈ ^{B *} , the analytic _{L p} ^{g [ecedil]9;} -admix-FFT, ^{T q , g ( p )} ( F ) , exists for all nonzero real numbers q , belongs to _{... [Hamiltonian (script capital H)]} , and is given by the formula [figure omitted; refer to PDF] for s -a.e. y ∈ B , where ^{τ g q} is the complex measure on ^{... n} given by [figure omitted; refer to PDF] for E ∈ [Bernoulli] ( ^{... n} ) .

Proof.

Proceeding as in the proof of Theorem 16, we see that, for all λ ∈ _{... +} , all g ∈ ^{B [Hamiltonian (script capital H)] *} , and for s -a.e. y ∈ B , [figure omitted; refer to PDF] is an analytic function of λ on _{... +} , and that for any q ∈ ... - { 0 } and s -a.e. y ∈ B , [figure omitted; refer to PDF] Clearly, the set function ^{τ g q} given by (72) is a complex measure on [Bernoulli] ( ^{... n} ) , and so the right side of (74) can be rewritten as the right side of (71).

Next, we note that [figure omitted; refer to PDF] for all λ ∈ _{... +} and [figure omitted; refer to PDF] Using these, we see that, for all p ∈ ( 1,2 ] , all ρ > 0 , and all λ ∈ _{... +} , [figure omitted; refer to PDF] and so, by the dominated convergence theorem, we see that, for any nonzero real q , [figure omitted; refer to PDF] Hence, ^{T q , g ( p )} ( F ) ( y ) exists and is given by the right side of (74) for all desired values of p and q and all g ∈ ^{B [Hamiltonian (script capital H)] *} . Thus, the theorem is proved.

Theorem 18.

Let F ∈ _{... [Hamiltonian (script capital H)]} be given by (67). Then, for all p ∈ [ 1,2 ] , all g ∈ ^{B [Hamiltonian (script capital H)] *} , and all nonzero real q , [figure omitted; refer to PDF] As such, the admix-FFT, ^{T q , g ( p )} , has the inverse transform { ^{T q , g ( p ) } - 1} = ^{T - q , g ( p )} .

Theorem 19.

Let F and G be elements of _{... [Hamiltonian (script capital H)]} with corresponding finite Borel measures τ and μ in [physics M-matrix] ( H ) . Then, for all k ∈ ^{B [Hamiltonian (script capital H)] *} , the CP, ( F * G _{) q , k} , exists for all nonzero real numbers q , belongs to _{... [Hamiltonian (script capital H)]} , and is given by the formula [figure omitted; refer to PDF] for s -a.e. y ∈ B , where [straight phi] : ^{... n} × ^{... n} [arrow right] ^{... n} is a continuous function defined by [figure omitted; refer to PDF] and ^{[varpi] k q} is a complex measure on [Bernoulli] ( ^{... n} ) given by [figure omitted; refer to PDF] for E ∈ [Bernoulli] ( ^{... n} ) .

Proof.

Using (64), the Fubini theorem, (23), and (24), we have that, for all λ > 0 and s -a.e. y ∈ B , [figure omitted; refer to PDF] Using the same argument as in the proof of Theorem 17, we can show that the last expression in the previous equation is an analytic function of λ on _{... +} and is a bounded continuous function of λ on _{... ~ +} because τ and μ are finite Borel measures. Hence, ( F * G _{) q , k} exists and is given by [figure omitted; refer to PDF] for all q ∈ ... - { 0 } and s -a.e. y ∈ B .

Consider the set function ^{[varpi] k q} and the continuous function [straight phi] given by (82) and (81), respectively. Clearly, the set function ^{[varpi] k q} is a complex measure on [Bernoulli] ( ^{... n} ) . Hence, ^{[varpi] k q} [composite function] [straight phi] is an element of [physics M-matrix] ( ^{... n} ) , and so the right side of (84) can be rewritten as the right side of (80). Thus, the theorem is proved.

Lemma 20.

Let [Hamiltonian (script capital H)] = { _{h 1} , ... , _{h n} } be any orthogonal set in H . For every g ∈ ^{B [Hamiltonian (script capital H)] *} , every v [arrow right] = ( _{v 1} , ... , _{v n} ) and w [arrow right] = ( _{w 1} , ... , _{w n} ) in ^{... n} , let ^{Y g , v [arrow right] [Hamiltonian (script capital H)]} , ^{Z g , w [arrow right] [Hamiltonian (script capital H)]} : ^{B 2} [arrow right] ... be given by [figure omitted; refer to PDF] respectively. As a result, ^{Y g , v [arrow right] [Hamiltonian (script capital H)]} and ^{Z g , w [arrow right] [Hamiltonian (script capital H)]} are independent random variables.

Proof.

Since the random variables ^{Y g , v [arrow right] [Hamiltonian (script capital H)]} and ^{Z g , w [arrow right] [Hamiltonian (script capital H)]} are Gaussian with mean zero, it suffices to show that [figure omitted; refer to PDF]

We know that { _{h j} [ecedil]9; g , ... , _{h n} [ecedil]9; g } is an orthogonal set in H ; thus, { ( _{h j} [ecedil]9; g , x ^{) ~} , ... , ( _{h n} [ecedil]9; g , x ^{) ~} } is a set of independent Gaussian random variables with mean zero on B . However, using the Fubini theorem, we obtain [figure omitted; refer to PDF] which concludes the proof of Lemma 20.

Remark 21.

For each v [arrow right] ∈ ^{... n} , let [figure omitted; refer to PDF] As such, (67) is rewritten as [figure omitted; refer to PDF] and the functional ψ ( v [arrow right] ; · ) is an element of ^{... [Hamiltonian (script capital H)] ( 1 )} for each v [arrow right] .

Applying the same method as used in the proof of Theorem 3.5 in [4], we have the following theorem. For the proof of Theorem 22, we can apply Lemma 20 and Theorem 11 to the functional ψ given by (88).

Theorem 22.

Let F , G , τ , and μ be as in Theorem 19, and let g be an element of ^{B [Hamiltonian (script capital H)] *} . Then, for all p ∈ [ 1,2 ] and all nonzero real q , [figure omitted; refer to PDF] for s -a.e. y ∈ B .

Acknowledgments

The authors would like to express their gratitude to the referees for their valuable comments and suggestions which have improved the original paper. 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 (2011-0014552).

References

[1] J. E. Bearman, "Rotations in the product of two Wiener spaces," Proceedings of the American Mathematical Society , vol. 3, pp. 129-137, 1952.

[2] R. H. Cameron, D. A. Storvick, "An operator valued Yeh-Wiener integral, and a Wiener integral equation," Indiana University Mathematics Journal , vol. 25, no. 3, pp. 235-258, 1976.

[3] G. W. Johnson, D. L. Skoug, "Scale-invariant measurability in Wiener space," Pacific Journal of Mathematics , vol. 83, no. 1, pp. 157-176, 1979.

[4] J. G. Choi, D. Skoug, S. J. Chang, "A multiple generalized Fourier-Feynman transform via a rotation on Wiener space," International Journal of Mathematics , vol. 23, no. 7, 2012.

[5] J. G. Choi, S. J. Chang, "A rotation on Wiener space with applications," ISRN Applied Mathematics , vol. 2012, 2012.

[6] L. Gross, "Abstract Wiener spaces," Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability , vol. 2, pp. 31-42, 1965.

[7] D. M. Chung, "Scale-invariant measurability in abstract Wiener spaces," Pacific Journal of Mathematics , vol. 130, no. 1, pp. 27-40, 1987.

[8] G. Kallianpur, C. Bromley, M. Pinsky, "Generalized Feynman integrals using analytic continuation in several complex variables," Stochastic Analysis and Applications , vol. 7, pp. 217-267, Marcel Dekker, New York, NY, USA, 1984.

[9] H. H. Kuo Gaussian Measures in Banach Spaces , vol. 463, of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975.

[10] G. W. Johnson, D. L. Skoug, "Notes on the Feynman integral. II," Journal of Functional Analysis , vol. 41, no. 3, pp. 277-289, 1981.

[11] R. E. A. C. Paley, N. Wiener, A. Zygmund, "Notes on random functions," Mathematische Zeitschrift , vol. 37, no. 1, pp. 647-668, 1933.

[12] C. Park, D. Skoug, "A note on Paley-Wiener-Zygmund stochastic integrals," Proceedings of the American Mathematical Society , vol. 103, no. 2, pp. 591-601, 1988.

[13] C. Park, D. Skoug, "A Kac-Feynman integral equation for conditional Wiener integrals," Journal of Integral Equations and Applications , vol. 3, no. 3, pp. 411-427, 1991.

[14] D. M. Chung, C. Park, D. Skoug, "Generalized Feynman integrals via conditional Feynman integrals," The Michigan Mathematical Journal , vol. 40, no. 2, pp. 377-391, 1993.

[15] T. Huffman, C. Park, D. Skoug, "Generalized transforms and convolutions," International Journal of Mathematics and Mathematical Sciences , vol. 20, no. 1, pp. 19-32, 1997.

[16] C. Park, D. Skoug, "Generalized Feynman integrals: the L ( _{L 2} , _{L 2} ) theory," The Rocky Mountain Journal of Mathematics , vol. 25, no. 2, pp. 739-756, 1995.

[17] C. Park, D. Skoug, "Conditional Fourier-Feynman transforms and conditional convolution products," Journal of the Korean Mathematical Society , vol. 38, no. 1, pp. 61-76, 2001.

[18] J. M. Ahn, K. S. Chang, B. S. Kim, I. Yoo, "Fourier-Feynman transform, convolution and first variation," Acta Mathematica Hungarica , vol. 100, no. 3, pp. 215-235, 2003.

[19] K. S. Chang, B. S. Kim, I. Yoo, "Fourier-Feynman transform, convolution and first variation of functionals on abstract Wiener space," Integral Transforms and Special Functions , vol. 10, no. 3-4, pp. 179-200, 2000.

[20] K. S. Chang, T. S. Song, I. Yoo, "Analytic Fourier-Feynman transform and first variation on abstract Wiener space," Journal of the Korean Mathematical Society , vol. 38, no. 2, pp. 485-501, 2001.