#### Abstract

A Gel’fand triple is constructed of functionals of , where is an appropriate Bernoulli noise on a probability space . Characterizations are given to both and . It is also shown that a Wick-type product can be defined on and moreover forms a commutative algebra with the product. Finally, a transform named -transform is defined on and its continuity as well as other properties are examined.

#### 1. Introduction

In the constructive quantum field theory, there have been encountered infinite quantities, which originated (from a mathematical point of view) from the product of generalized functions (see [1, 2]). To obtain useful information out of these infinite quantities, Wick [3] first introduced the now so-called Wick renormalization, which was actually a formal manipulation. Hida and Ikeda [4] first rigorously defined the Wick renormalization on functionals of Brownian motion and called it Wick product instead. Meyer and Yan [5] extended the Wick product to cover generalized functionals of Gaussian white noises (also known as Hida distributions). Now the Wick product is widely used as a tool in stochastic differential equations, stochastic partial differential equations, stochastic quantization [2], and many other fields.

Bernoulli noise functionals (also known as functionals of Bernoulli noises) have attracted much attention in recent years. In 2001 Émery [6] discussed the chaotic representation property of a class of discrete-time stochastic processes including Bernoulli noises. In 2008 Privault [7] surveyed his discrete-time chaotic calculus. In 2010 Nourdin [8] considered Rademacher functionals (a special case of Bernoulli noise functionals) by using Stein’s method. The same year, Wang et al. [9] introduced a notion of quantum Bernoulli noises and defined corresponding quantum stochastic integrals, which are actually about operator processes acting on Bernoulli noise functionals. Recently Wang et al. [10] have presented an alternative approach to Privault’s discrete-time chaotic calculus. There are other works devoted to development of a theory of quantum Bernoulli noises (see, for instance, [11, 12]).

As can be seen, most of the existing work concerning the Wick product has been done within generalized functionals of Gaussian white noises. In this paper, we aim to construct generalized functionals of Bernoulli noises and develop a Wick-type calculus on them.

The paper is organized as follows. In Section 2, we fix some necessary notation and recall main notions and facts about Bernoulli noises for our later use.

Sections 3, 4, and 5 are our main work. In Section 3, we first introduce a weight function on the finite power set of , where denotes the set of all nonnegative integers. And then, by using the weight function and the full Wiener integral operator , we construct a nuclear space of functionals of , where is the Bernoulli noise fixed in Section 2. With as the testing functional space, we get the generalized functional space by taking the dual. We also give characterizations to both and . In Section 4, we define the Wick product on the generalized functional space and prove that forms a commutative algebra with the Wick product. Finally, in Section 5, we introduce a transform, called -transform, on the generalized functional space by using the coherent states. We examine its continuity as well as other properties.

#### 2. Preliminaries

Throughout this paper, designates the set of all nonnegative integers. We denote by the finite power set of , namely, where means the cardinality of as a set. We use to mean the usual Hilbert space of square summable real-valued functions on .

Let be a probability space and an independent sequence of real-valued random variables on satisfying that with , , and , namely, generates . We note that such a sequence of random variables does exist. In fact, is a discrete-time Bernoulli stochastic process and if we put then is a martingale. Hence we may call a Bernoulli noise.

In the following, we set , , . In this way, the sequence forms a filtration of -fields over . As usual, we use to mean the usual space of square integrable real-valued random variables on . We denote by the norm of .

The next lemma [9] shows that has the chaotic representation property.

Lemma 1. *The family forms a countable orthonormal basis of , where
**
and .*

Lemma 2. *There exists a unique isometric isomorphism such that
**
where the series on the right-hand side converges in the norm .*

The isometric isomorphism is known as the full Wiener integral operator [10].

Lemma 3. *For each , there exists a bounded operator on such that
**
where and the indicator of as a subset of .*

Lemma 4. *Let ; then , the adjoint operator of , has following property:
**
where .*

The operator and its adjoint operator are referred to as the annihilation operator and creation operator [10], respectively.

#### 3. The Framework of Bernoulli Noise Analysis

In the present section, we mainly construct a Gel’fand triple of functionals of the Bernoulli noise , which serves as the framework where we work.

To start with, we first introduce a weight function on , which will play a key role in stating and proving our main results.

Proposition 5. *Define an -valued function on as
**
and . Then for each , it holds that
*

*Proof. *For , let be the subset of defined by
Writing , we obtain
This, together with the fact , gives (10).

*Remark 6. *For , we can verify that , where denotes the cardinality of .

For , let be the collection of real-valued functions on satisfying Then forms a Hilbert space with the usual linear operations and the inner product defined by We use to denote the norm generated by . Note that for any and . In the sequel, we set

Recall that the full Wiener integral operator is an isometric isomorphism. This, together with the fact , , suggests the next definition.

*Definition 7. *For , one defines as the image of under , namely,
and endows it with the inner product given by
One uses to denote the norm generated by .

Clearly, for each , is a Hilbert space and the restriction of to is a Hilbert isomorphism from to . Additionally, if , then one has

Proposition 8. *For , the system forms an orthonormal basis of the Hilbert space and moreover,
*

*Proof. *It can be verified that forms an orthonormal basis of , where
Clearly , . Thus forms an orthonormal basis of . Now for , we have
This together with the well-known Parseval’s equality yields (19).

*Definition 9. *One sets
and endows it with the topology generated by the norm sequence . One calls the testing functional space and its elements testing functionals.

Note that, for each , is just the completion of with respect to norm . Thus, with inner product norms , , forms a countably-Hilbert space. It is also worthwhile to note that on whenever and, in particular, the inclusion mapping from to is continuous.

Proposition 10. *The testing functional space is a nuclear space, namely, for any , there exists such that the inclusion mapping from to is a Hilbert-Schmidt operator.*

*Proof. *Let . Then there exists such that . By Proposition 8, forms an orthonormal basis of . Thus, we have
where denotes the Hilbert-Schmidt norm of an operator. Therefore the inclusion mapping from to is a Hilbert-Schmidt operator.

Proposition 11. *Let and with . Then if and only if . In that case
*

*Proof. *The first part is easy to verify. We need only to show that for each , the series in (24) converges to in norm . To do so, we take . Since forms an orthonormal basis of , we have
in norm . Note that the series on the right-hand side also converges to in the norm of since . On the other hand, because of , we have
Thus , , which together with (25) implies

*Definition 12. *One defines as the dual of and endows it with the strong topology. One calls the generalized functional space and its elements generalized functionals.

Now, by identifying with its dual, we come to a real Gel’fand triple This justifies naming the generalized functional space. By convention we may call the above triple the framework of Bernoulli noise analysis.

*Remark 13. *For , let be the dual of . Then and moreover
Thus can also be endowed with the inductive limit topology given by space sequence , which, however, coincides with the strong topology because is a countably-Hilbert space.

For , we define as the vector space of real-valued functions on satisfying and endow it with the inner product given by Clearly forms a Hilbert space. We set Note that for , , the series always converges absolutely.

Proposition 14. *For each , there exists a unique such that
**
where . Conversely, given , a generalized functional is defined by (33).*

*Proof. *The proof of the second part is trivial. We are now turning to the proof of the first part. Let . In view of Remark 13, there exists some such that . According to the Riesz representation theorem, there exists a unique such that
Note that . Thus we can define a function on as , . Direct computation gives that
which shows that , in particular . For , using the fact that
and taking the continuity of into account, we have
Finally we show that such an is unique. In fact, if satisfies
with , then for each , by using , we get . Thus .

*Definition 15. *For , one calls the function given in Proposition 14 the Guichardet representation of .

#### 4. Wick Product

In this section, we define the Wick product on the generalized functional space . We prove that forms a commutative algebra with this product. Some other properties are also shown of the Wick product.

*Definition 16. *For , , one defines a function on as
and calls it the Wick product of and .

Proposition 17. *Let and , . Then whenever .*

*Proof. *For , it follows from Proposition 5 that
Thus
which shows that .

From Proposition 17, we easily see that remains in for all , ; namely, is closed under the Wick product.

Note that the generalized functional space can be characterized in terms of . Thus we naturally come to the following definition.

*Definition 18. *For generalized functionals , , their Wick product is defined as
where and , the Guichardet representations of and , respectively.

Proposition 19. *Let and , . Then whenever . In particular, for all , .*

*Proof. *Let , be the Guichardet representations of and , respectively. Then, by Definition 18, we find that just has as its Guichardet representation. Thus by using Proposition 17 we know the claim is true.

Proposition 20. *Let , . Then
*

*Proof. *Let , be the Guichardet representations of and , respectively. Then, by Definition 18, and have and as their Guichardet representations, respectively. Clearly . Thus .

Proposition 21. *Let , , . Then
*

*Proof. *Let be the Guichardet representation of , where , , . Then, for each , we can show that
where is a partition of . Thus
which implies .

The following two propositions can be proved similarly and their proofs are omitted.

Proposition 22. *Let , , . Then
*

Proposition 23. *Let , and , the real numbers. Then
*

*Remark 24. *It follows from the above propositions that forms a commutative algebra with the Wick product.

Let . Then for with being its Guichardet representation, there exists an element, denoted by , in such that has the function being its Guichardet representation. In this way, an operator is naturally defined.

The following proposition offers a link between the Wick product and the operator .

Proposition 25. *Let and , . Then
*

Obviously, for and , their Wick product makes sense and belongs to . This leads us to the following definition.

*Definition 26. *For and , one defines their Wick product as
where , and the Guichardet representation of .

Proposition 27. *Let , and . Then .*

*Proof. *Let and the Guichardet representation of . Then and . Thus
which implies . On the other hand, has as its Guichardet representation. Therefore .

#### 5. -Transform

It is known that the -transform on generalized functionals of Gaussian white noises plays an important role in Hida’s white noise analysis [13]. In the last section, we define the -transform on our generalized functional space and examine its fundamental properties.

Let , where Clearly is a dense linear subspace of . Additionally forms a countably-Hilbert space with the sequence , , of inner products given by For , the coherent state is defined as which belongs to (see [11] for details). It can be shown that whenever and moreover is total in .

In the following, we denote by the canonical bilinear form on given by

*Definition 28. *For , one defines its -transform as a function on given by

Obviously, a generalized functional is completely determined by its -transform since is total in .

*Remark 29. *It can be proved that the coherent mapping is a continuous mapping from to . Thus, for a generalized functional , its -transform is actually a continuous function on .

The next proposition offers a norm estimate to the coherent state .

Proposition 30. *Let . Then for each , it holds that
**
where and .*

*Proof. *Define a function on as and
Then it is easy to show that and moreover , where is the full Wiener integral operator (see Section 2). Thus, by using the following expression:
we have
which gives (57).

Note that the left-hand side of (57) can be strictly less than the right-hand side [11]. Using Proposition 30, we can get a growth estimate of the -transform of a generalized functional.

Proposition 31. *Let . Then there exist constants and such that
*

*Proof. *Since , there exists some such that . Take . Then, by using Proposition 30, we get

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This paper is supported by National Natural Science Foundation of China (no. 11061032) and Natural Science Foundation of Gansu Province (Grant no. 0710RJZA106).