Integral Transforms on a Function Space with Change of Scales Using Multivariate Normal Distributions

Using simple formulas for generalized conditional Wiener integrals on a function space which is an analogue of Wiener space, we evaluate two generalized analytic conditional Wiener integrals of a generalized cylinder function which is useful in Feynman integration theories and quantum mechanics. We then establish various integral transforms over continuous paths with change of scales for the generalized analytic conditional Wiener integrals. In these evaluation formulas and integral transforms we use multivariate normal distributions so that the orthonormalization process of projection vectors which are needed to establish the conditionalWiener integrals can be removed in the existing change of scale transforms. Consequently the transforms in the present paper can be expressed in terms of the generalized cylinder function itself.


Introduction
Let  0 [0, ] denote the classical Wiener space, the space of continuous real-valued functions  on [0, ] with (0) = 0.It is well known that Wiener measure and Wiener measurability are not invariant under change of scale and under translation [1,2].As integral transforms, change of scale formulas for Wiener integrals of various functions was developed on the classical and abstract Wiener spaces [3][4][5][6][7].Further change of scale formulas for conditional Wiener integrals was introduced by the author and his coauthors [8][9][10].In fact change of scale formulas for conditional Wiener integrals was established on  0 [0, ], on the infinite dimensional Wiener space, and on [0, ], an analogue of Wiener space [11] which is the space of real-valued continuous paths on [0, ].Some difficulties in studying the transforms for the conditional Wiener integrals of cylinder functions which play important roles in Feynman integration theories are that they cannot be expressed in terms of the original cylinder functions.To avoid these difficulties, modified cylinder functions expressed by a polygonal function with projection vectors satisfying orthonormality were used to derive the change of scale transforms [8][9][10].In this paper we use multivariate normal distributions so that the orthonormalization process of projection vectors which are needed to establish the conditional Wiener integrals can be removed in the existing change of scale transforms.Consequently the transforms in the present paper can be expressed by the cylinder function itself and generalize the results in [8][9][10].
Let  ∈ [0, ] and let ℎ be of bounded variation with ℎ ̸ = 0 a.e. on ( Using simple formulas for generalized conditional Wiener integrals on [0, ] with the conditioning functions   and 2

Journal of Function Spaces
+1 [12] we evaluate generalized analytic conditional Wiener integrals of the following generalized cylinder function: where  ∈   (R  ) with 1 ≤  ≤ ∞ and {V 1 , . . ., V  } is an orthonormal subset of  2 [0, ].We then establish various change of scale transforms for the generalized analytic conditional Wiener integrals of   with   and  +1 .In these evaluation formulas and scale transforms we use multivariate normal distributions so that Gram-Schmidt orthonormalization process of {P ⊥ ℎV 1 , . . ., P ⊥ ℎV  } can be removed in the existing change of scale transforms for a suitable orthogonal projection P ⊥ on  2 [0, ].In contrast with the existing change of scale transforms in [8][9][10], the transforms in this paper are expressed in terms of the cylinder function   itself and generalize some results in those references.
For ⃗ , ⃗  ∈ R  ,  ∈ C, and any nonsingular positive × matrix   on R let where ⟨⋅, ⋅⟩ R denotes the dot product on R  .For a function . By Theorems 6 and 7 in [12], we have the following theorems.
Theorem 1.Let  be a complex valued function on [0, ] and let   be integrable over where  ,+1 is given by ( 14),   +1 is the probability distribution of  +1 on (R +2 , (R +2 )), and the expectation on the right hand side of the equation is taken over the variable .
for    +1 a.e.⃗  +1 ∈ R +2 , where  and  ,+1 are given by ( 12) and ( 13), respectively, and    +1 is the probability distribution of   +1 on (R +2 , B(R +2 )).By Theorem 2 we also have for where ⃗  +1 = ( 0 , ) has a limit as  approaches to − through C + , then it is called the conditional analytic Feynman   -integral of   given  +1 with the parameter  and is denoted by [  ] is also defined by if it exists, where the limit is taken through C + .Applying Theorem 2.3 in [13], we can easily prove the following theorem.Theorem 3. Let {ℎ 1 , ℎ 2 , . . ., ℎ  } be an orthonormal system of  2 [0, ].Then (ℎ 1 , ⋅), . . ., (ℎ  , ⋅) are independent and each (ℎ  , ⋅) has the standard normal distribution.Moreover if  : where   is the identity matrix on R  and * = means that if either side exists then both sides exist and they are equal.
Since  is a probability measure on R we have the following corollary.

Corollary 4. Under the assumptions as given in Theorem 3
The following lemmas are useful to prove the results in the next sections and their proofs are simple.
(1) The multiplication operator  ℎ in Lemma 5 is well defined because ℎ is of bounded variation which implies the boundedness of ℎ.  ℎ will denote the operator as given in the lemma unless otherwise specified. ( In this case the symbol (V,  , ( ⃗   )) does not mean the Paley-Wiener-Zygmund integral of V ∈  2 [0, ].It is only a formal expression for ∑  =1 ⟨V  ,   ⟩(  −  −1 ) which is as given in Lemma 6.

Multivariate Normal Distributions
In this section we derive a multivariate normal distribution which will be needed in the next section.Lemma 8. Let {V 1 , . . ., V  } be a set of independent vectors in  2 [0, ].Then the covariance matrix Σ = [  ] × of the random variables (V  , ⋅),  = 1, . . ., , exists and is positive definite.Moreover   is given by and the determinant |Σ| of Σ is positive so that Σ is nonsingular and the inverse matrix Σ −1 of Σ is also positive definite.

Theorem 9. Let the assumptions and notations be as given in Lemma 8. Then for every Borel measurable function 𝑓 :
where Ψ  is given by ( 15),   is the identity matrix on R, and Σ 1/2 is the positive definite matrix satisfying (Σ 1/2 ) 2 = Σ.
By the same process as used in Lemma 2.1 of [15], Theorem 9, and the change of variable theorem, we have the following corollary.

Analytic Feynman Integrals and Conditional Analytic Feynman Integrals
We begin this section with introducing the cylinder function on the analogue of Wiener space.Let {V 1 , V 2 , . . ., V  } be an orthonormal subset of  2 [0, ], let  be any positive integer, let 1 ≤  ≤ ∞, and let  be given by for   a.e. ∈ [0, ], where  ∈   (R  ).Without loss of generality we can take  to be Borel measurable.In the following theorem we evaluate the Wiener and Feynman integrals of   .
Theorem 11.Let  be given by ( 37) with 1 ≤  ≤ ∞ and suppose that where × and Ψ  is given by (15).Moreover if  = 1, then, for a nonzero real ,    [  ] is given by the right hand side of (38) with replacing  by −.
Proof.Let  > 0. Replacing ℎ by  −1/2 ℎ we have by Corollary 10 For any real  > 0 which is integrable over ⃗ , so that we have the theorem by the change of variable theorem, Morera's theorem, the uniqueness of analytic continuation, and the dominated convergence theorem.

Corollary 12. Under the assumptions as given in Theorem 11
Moreover if  = 1, then    [  ] is given by the right hand side of (42) with replacing  by −.
Proof.For  = 1, . . .,  and   a.e. ∈ [0, ] we have by Lemma 5 so that for  > 0 and ⃗  +1 ∈ R +2 we have by Corollary 10 where Ψ  is given by (15).By the Morera and the dominated convergence theorem we have the theorem.
We now have the following corollary by a similar calculation of (41).

Corollary 14. Under the assumptions as given in Theorem 13
] is given by the right hand side of ( 46) with replacing  by −.
With the above notations, we have the following theorem.