Generalized Analytic Fourier-Feynman Transform of Functionals in a Banach Algebra
We introduce the Fresnel type class . We also establish the existence of the generalized analytic Fourier-Feynman transform for functionals in the Banach algebra .
Let be a separable Hilbert space and let be the space of all complex-valued Borel measures on . The Fourier transform of in is defined by The set of all functions of the form (1) is denoted by and is called the Fresnel class of . Let be an abstract Wiener space. It is known [1, 2] that each functional of the form (1) can be extended to uniquely by where is a stochastic inner product between and . The Fresnel class of is the space of (equivalence classes of) all functionals of the form (2). There has been a tremendous amount of papers and books in the literature on the Fresnel integral theory and Fresnel classes and on abstract Wiener and Hilbert spaces. For an elementary introduction see [3, Chapter 20].
Furthermore, in , Kallianpur and Bromley introduced a larger class than the Fresnel class and showed the existence of the analytic Feynman integral of functionals in for a successful treatment of certain physical problems by means of a Feynman integral. The Fresnel class of is the space of (equivalence classes of) all functionals on of the following form: where and are bounded, nonnegative, and self-adjoint operators on and .
In this paper we study the functionals of the form (3) with in a very general function space . The function space , induced by generalized Brownian motion process, was introduced by Yeh [4, 5] and was used extensively in [6–13]. In this paper, we also construct a concrete theory of the generalized analytic Fourier-Feynman transform (GFFT) of functionals in a generalized Fresnel type class defined on . Other work involving GFFT theories on include [6, 7, 9, 12, 13].
The Wiener process used in [1, 2, 14–17] is stationary in time and is free of drift while the stochastic process used in this paper, as well as in [4, 6–13, 18], is nonstationary in time and is subject to a drift .
It turns out, as noted in Remark 7 below, that including a drift term makes establishing the existence of the GFFT of functionals on very difficult. However, when and on , the general function space reduces to the Wiener space .
2. Definitions and Preliminaries
Let be an absolutely continuous real-valued function on with , , and let be a strictly increasing, continuously differentiable real-valued function with and for each . The generalized Brownian motion process determined by and is a Gaussian process with mean function and covariance function . For more details, see [6, 10, 12]. By Theorem 14.2 in , the probability measure induced by , taking a separable version, is supported by (which is equivalent to the Banach space of continuous functions on with under the sup norm). Hence, is the function space induced by where is the Borel -algebra of . We then complete this function space to obtain , where is the set of all Wiener measurable subsets of .
A subset of is said to be scale-invariant measurable provided is -measurable for all , and a scale-invariant measurable set is said to be a scale-invariant null set provided for all . A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s-a.e.). A functional is said to be scale-invariant measurable provided is defined on a scale-invariant measurable set and is -measurable for every . If two functionals and defined on are equal s-a.e., we write .
Let be the space of Lebesgue measurable functions on given by where is the total variation function of . Then is a separable Hilbert space with inner product defined by In particular, note that if and only if a.e. on .
Let be a complete orthonormal set in , each of whose elements is of bounded variation on such that Then for each , the Paley-Wiener-Zygmund (PWZ) stochastic integral is defined by the following formula: for all for which the limit exists; one can show that for each , the PWZ stochastic integral exists for -a.e. , and if is of bounded variation on , then the PWZ stochastic integral equals the Riemann-Stieltjes integral for s-a.e. .
Remark 1. (1) For each , the PWZ stochastic integral is a Gaussian random variable on with mean and variance .
(2) For all , , Hence, we see that for all , , if and only if and are independent random variables.
The following Cameron-Martin subspace of plays an important role throughout this paper.
Let For , let be defined by the following formula: where denotes the Radon-Nikodym derivative of the signed measure induced by , with respect to the Borel-Stieltjes measure induced by . Then with inner product is a separable Hilbert space.
Throughout this paper, for , we will use the notation instead of . We also use the following notations for , , : Then with the inner product given by (13) is also a separable Hilbert space. It is easy to see that the two norms and are equivalent. Furthermore, we have the following assertions. (i) is an element of . (ii)For each , the random variable is Gaussian with mean and variance . (iii) for any real number , and .(iv)Let be a subset of such that , where is the Kronecker delta. Then the random variables ’s are independent.
The following integration formula is used several times in this paper: for complex numbers and with .
3. The GFFT of Functionals in a Banach Algebra
Let be the space of complex-valued, countably additive (and hence finite) Borel measures on . is a Banach algebra under the total variation norm and with convolution as multiplication.
We define the Fresnel type class of functionals on as the space of all stochastic Fourier transforms of elements of ; that is, if and only if there exists a measure in such that for s-a.e. . More precisely, since we will identify functionals which coincide s-a.e. on , can be regarded as the space of all -equivalence classes of functionals of the form (15).
The Fresnel type class is a Banach algebra with norm In fact, the correspondence is injective, carries convolution into pointwise multiplication and is a Banach algebra isomorphism where and are related by (15).
Remark 2. The Banach algebra contains several interesting functions which arise naturally in quantum mechanics. Let be the class of -valued countably additive measures on , the Borel class of . For , the Fourier transform of is a complex-valued function defined on by the following formula:
Let be the set of all complex-valued functions on of the form , where is a family from satisfying the following two conditions:(i)for every , is Borel measurable in , (ii).
Let and let be given by for s-a.e. . Then, using the methods similar to those used in , we can show that the function is Borel-measurable and that , , and are elements of . These facts are relevant to quantum mechanics where exponential functions play a prominent role.
Let be a nonnegative self-adjoint operator on and any complex measure on . Then the functional belongs to because it can be rewritten as for . Let be self-adjoint but not nonnegative. Then has the form where both and are bounded, nonnegative, and self-adjoint operators.
In this section we will extend the ideas of  to obtain expressions of the generalized analytic Feynman integral and the GFFT of functionals of the form (19) when is no longer required to be nonnegative. To do this, we will introduce definitions and notations analogous to those in [7, 12, 13].
Let denote the class of all Wiener measurable subsets of the product function space . A subset of is said to be scale-invariant measurable provided is -measurable for every and , and a scale-invariant measurable subset of is said to be scale-invariant null provided for every and . A property that holds except on a scale-invariant null set is said to hold s-a.e. on . A functional on is said to be scale-invariant measurable provided is defined on a scale-invariant measurable set and is -measurable for every and . If two functionals and defined on are equal s-a.e., then we write .
We denote the product function space integral of a -measurable functional by whenever the integral exists.
Throughout this paper, let , and denote the set of complex numbers, complex numbers with positive real part, and nonzero complex numbers with nonnegative real part, respectively. Furthermore, for all , is always chosen to have positive real part. We also assume that every functional on we consider is s-a.e. defined and is scale-invariant measurable.
Definition 3. Let and let . Let be such that for each and , the function space integral exists. If there exists a function analytic in such that for all and , then is defined to be the analytic function space integral of over with parameter , and for we write Let and be nonzero real numbers. Let be a functional such that exists for all . If the following limit exists, we call it the generalized analytic Feynman integral of with parameter , and we write where through values in .
Definition 4. Let and be nonzero real numbers. For and , let For , we define the analytic GFFT, of , by the formula if it exists; that is, for each and , where . We define the analytic GFFT, of , by the formula if it exists.
We note that for , is defined only s-a.e. We also note that if exists and if , then exists and . Moreover, from Definition 4, we see that for ,
Next we give the definition of the generalized Fresnel type class .
Definition 5. Let and be bounded, nonnegative, and self-adjoint operators on . The generalized Fresnel type class of functionals on is defined as the space of all functionals on of the following form: for some . More precisely, since we identify functionals which coincide s-a.e. on , can be regarded as the space of all -equivalence classes of functionals of the form (30).
Remark 6. (1) In Definition 5, let be the identity operator on and . Then is essentially the Fresnel type class , and for and nonzero real numbers and ,
if it exists, where for all and means the analytic GFFT on ; see [6, 12].
(2) The map defined by (30) sets up an algebra isomorphism between and if is dense in , where indicates the range of an operator. In this case becomes a Banach algebra under the norm . For more details see .
Remark 7. Let be given by (30). In evaluating and for and , the expression
occurs. Clearly, for , , for all . But for , is not necessarily bounded by 1.
Note that for each with , , Hence, for with , ,
The right hand side of (34) is an unbounded function of for . Thus , , , and might not exist. Thus throughout this paper we will need to put additional restrictions on the complex measure corresponding to in order to obtain our results for the GFFT and the generalized analytic Feynman integral of .
In view of Remark 7, we clearly need to impose additional restrictions on the functionals in .
For a positive real number , let and let Then, for all
We note that for all real with , , and .
Remark 8. Note that in case and on , the function space reduces to the classical Wiener space and for all . Hence, for all , and for any positive real number , , the Kallianpur and Bromley’s class introduced in Section 1.
Theorem 9. Let be a positive real number and let be an element of . Then for any nonzero real numbers and with , , the analytic GFFT of , exists and is given by the following formula: for s-a.e. , where is given by (32).
Proof. We first note that for , the PWZ stochastic integral is a Gaussian random variable with mean and variance . Hence, using (30), the Fubini theorem, the change of variables theorem and (14), we have that for all and , Let for each . Clearly, for all and . Let be given by (35). Then for all Using this fact and the dominated convergence theorem, we see that is a continuous function of on . For each , is an analytic function of throughout the domain so that for every rectifiable simple closed curve in . By (42), the Fubini theorem and the Morera theorem, we see that is an analytic function of throughout the domain . Finally, using (28) with the dominated convergence theorem, we obtain the desired result.
Proof. Let be given by (35). It was shown in the proof of Theorem 9 that is an analytic function of throughout the domain . In view of Definition 4, it will suffice to show that for each and ,
Fixing and using the inequalities (37) and (39), we obtain that for all , and all , Hence, by the dominated convergence theorem, we see that for each and each and , which concludes the proof of Theorem 10.
Remark 11. In view of Theorems 9 and 10, we see that for each , the analytic GFFT of , is given by the right hand side of (40) for , , , and as in Theorem 9, and for s-a.e. ,
In particular, using this fact and (29), we have that for all ,
For nonzero real numbers and with , , define a set function by where and are related by (30) and is the Borel -algebra of . Then it is obvious that belongs to and for all , can be expressed as for s-a.e. . Hence, belongs to for all .
4. Relationships between the GFFT and the Function Space Integral of Functionals in
In this section we establish a relationship between the GFFT and the function space integral of functionals in the Fresnel type class .
Throughout this section, for convenience, we use the following notation: for given and , let where is a complete orthonormal set in .
Let be an orthonormal set in , let be a Lebesgue measurable function, and let be given by Then in the sense that if either side of (54) exists, both sides exist and equality holds.
We also need the following lemma to obtain our main theorem in this section.
Lemma 12. Let be an orthonormal subset of and let be an element of . Then for each , the function space integral exists and is given by the formula where is given by (52) above and
The following remark will be very useful in the proof of our main theorem in this section.
Remark 13. Let be a positive real number and let be given by (35). For real numbers and with , , let be a sequence in such that
Let for and . Then for , and
for each . Since for , there exists a sufficiently large such that for any , and are in and
Thus, there exists a positive real number such that .
Let be a complete orthonormal set in . Using Parseval's identity, it follows that for all , . In addition for each , for every .
Since and for there exists a sufficiently large such that for any for .
Using these and a long and tedious calculation we can show that for every ,