Integrability on the Abstract Wiener Space of Double Sequences and Fernique Theorem

Abstract

The integrability of a function defined on the abstract Wiener space of double Fourier coefficients is explored. The abstract Wiener space is also a Hilbert space. We define an orthonormal system of the Hilbert space to establish a measure and integration on the abstract Wiener space. We examine the integrability of a function defined on the abstract Wiener space for Fernique theorem. With respect to the abstract Wiener measure, the integral of the function turns out to be convergent for . The result provides a wider choice of the constant than that of Fernique.

1. Introduction

We explore an abstract Wiener space that consists of double Fourier coefficients focused on the integrability of functions defined on the space. The space is also a Hilbert space. We define an orthonormal system of the Hilbert space and utilise the system to define a probability measure on the abstract Wiener space of double sequences and develop to integration. Using the probability measure, we examine the integral of , i.e., Fernique theorem in the abstract Wiener space. It is proved that the integral of the function with respect to the abstract Wiener measure converges for . The specified range of that we verified in this paper provides a wider choice of the constant than that of Fernique. We expect that the results can be applied to check integrability of related functions.

The concepts of an abstract Wiener space are known to be first appeared on [1], and we mainly refer to [2] for necessary definitions and theorems of an abstract Wiener space. An analogue Wiener space of sequences is studied in [3]. An abstract Wiener space of sequences is discussed in [4]. Both of them are for the single-indexed sequences. An abstract Wiener space of double Fourier coefficients is defined in the author’s work [5] where detailed development of the space can be found. Fernique theorem is introduced in [6]; we use an English version for the theorem. There is a work [7] that generalises the Fernique theorem to functions having values in the extended real number system.

For background knowledge, we introduce an abstract Wiener space as well as the Hilbert spaces of double Fourier coefficients in the next section. An orthonormal system for the Hilbert space is defined in Section 3. In Section 4, we define an abstract Wiener measure using the orthonormal system and provide the main theorems related to Fernique theorem. In the last section, conclusions and discussions are given.

2. Preliminaries

In this paper, underlying concepts are overlapped with the author’s recent work [5], such as the setting of an abstract Wiener space based on double Fourier coefficients. We briefly introduce definitions, notations, and basic properties of the space; some of which are in common with the author’s previous work. (1)Let and be real numbers. A function is said to be -periodic in variables and provided that holds for all and (2)Trigonometric form [8]: let a function be continuous on and periodic with period . Then, the double Fourier series of is as follows: where  The derivation of double Fourier series can be found in [9].(3)Let and be the space of two-variable functions on . For in , we define as a periodic function having period and to be symmetric with respect to the first argument and symmetric with respect to the second argument within the rectangle ; in and is -periodic in the whole plane (4)The coefficients of are all zero due to the symmetries in and

The underlying spaces of this paper are the following three types of double sequences.

Definition 1 (see [5]). A double sequence is denoted by . (a)For , let (b)Let be the space of all double sequences in with an inner product  For in , we let .(c)Let be the space of all double sequences in such that the limit converges uniformly on .

Remark 2 (see [5]). (a)It is obvious that holds for (b)In this work, represents a double sum. A double sum may not coincide with iterated sums. As known well, if a double series converges absolutely, then the double sum and iterated sums exist and are all equivalent. We deal with sequences in for the paper, and rearranging the order of summation does not affect the convergence of double series in the definition(c)The double summation is carried out by diagonal directions, (d)The space is defined in association with a double Fourier series; the limit of the arithmetic means of partial sums of the series is called a Cesáro mean. The denominator in its limit is the number of terms involved in a partial sum for each diagonal explained in (c). The inner product defined for in (b) is also motivated by this

Proposition 3 (see [5]). If is in , then, the series converges uniformly on .

Proposition 4 (see [5]). The following notations are for the definitions from [2]. (i) be a real separable Hilbert space with norm (ii) is an orthogonal projection on (iii) is the partially ordered set of orthogonal projections of ( means for )

Definition 5 (see [2] p59). A seminorm in Hilbert space is called measurable if for , there exists such that for all , .

Definition 6 (see [2] p63). Let be the completion of with respect to a measurable norm . will denote the inclusion map of into . The triple is called an abstract Wiener space.

Proposition 7 (see [5]). is an abstract Wiener space.

3. An Orthonormal System of the Hilbert Space

From Equation (6), the space of double sequences contains , which is a Hilbert space. We find an orthonormal system in to define an appropriate probability and integration in the space that will be used in the next section.

Definition 8. For nonnegative integers and , we define the following: Here, the argument is for a component of . That is, is a member of whose components are all 0 except for the -th component.

Theorem 9. The set in Definition 8 constitutes an orthonormal system in .

Proof. Let be in . By Definition 1, the inner product of and is as follows: We have for , and Therefore, is a complete orthonormal system in .

Remark 10. By Theorem 9, any in can be expressed as the following limit:

Then, Theorem 9 is also valid for .

We introduce an operator for an expression of an inner product in . Let be an operator defined by the following:

If is in , then obviously. We want to extend the concept of this inner product to members of the larger space or .

Theorem 11. Let be in , in , and is the orthonormal system in Definition 8. The limit exists in the -sense.

Proof. Let be in . We want to show that exists for any in (or in ). We express the double sum as . It suffices to show that is Cauchy. The last inequality comes from Schwarz inequality and the norm defined in Definition 1.

Definition 12. Let be in and be in . We define the following: where the limit is in -sense, which has been verified in Theorem 11.

4. Integrability of in

We defined an orthonormal system in Definition 8 for Fernique theorem in the abstract Wiener space . Each member of the orthonormal system is a double sequence; the components of are all 0 except for the -th component. We define an abstract Wiener measure with the dual of the orthonormal system. Then, we can examine the integrability of in as in Fernique theorem.

Proposition 13 (Kuo and Fernique [2, 6]). There exists such that , where is a Banach space and is an abstract Wiener measure.
Let be a member of , the dual space of , which has been defined using the orthonormal system in Definition 8; as in Equation (8). That is, for in

4.1. Single Indexing for a Double Sequence

The underlying space of integral for Fernique theorem will be . Since the orthonormal system and its dual on the space are double sequences, we need to deal with double indexing. This makes calculation and development of the functional on very complicated. Hence, we convert double indexing of sequences in into single indexing to alleviate the complexity in manipulation.

Definition 14. We define , for . Here,

Remark 15. For Definition 14, we have the following: (1)The function is a bijection(2)We adopt the following notation for a double sequence where denotes summation by diagonals as in Definition 14 for a double sequence.

Using Definition 14, any double sequence can be regarded as a single indexed sequence. For the integrability of a function in relation to Fernique theorem, we use the expression of single indexing in Equation (16) in this section.

4.2. Orthogonal System of Dual Members with Single Indexing

The orthonormal system in in Definition 8 can be regarded as a single indexed sequence using Equation (16); we denote the system by , the new name of . Applying Definition 14 to Equation (14), the -th component of is and all other components are zero: where is the -th component of .

According to this conversion with Equation (16), we rewrite the properties of the orthonormal system in terms of a single indexed sequence . As is , for , and , consists of an orthonormal system in . Here, is the -norm.

Consider the dual space of ; for in , its norm is defined by .

Let , the dual of , be a member of . Then, maps to its -th component divided by ; i.e., . Let us compute the norm of :

Then, the set of consists of an orthogonal system in .

4.3. Fernique Theorem on

In order to be used in Fernique theorem, we need a functional for a measure (density) on the abstract Wiener space . We regard each as a random variable defined on and have the following theorems.

Theorem 16. Each of the orthogonal system is normally distributed; .

Proof. For a real number , we calculate the probability as follows. Let be in . Then, for nonzero , Hence, is normally distributed with mean and variance .

Then, the space is with the measure defined in the proof of Theorem 16.

Theorem 17. is normally distributed; .

Proof. We first show that the random variables ’s are independent to each other using characteristic functions, i.e., Fourier transforms: if is a random variable and , then the Fourier transform of is .
Suppose . Then, . We have , and . As and , we have . We obtained . Hence, and are stochastically independent by Theorem 16.13 in [10].
As is an independent system, . Therefore, follows a normal distribution and we have .

Now, we need the convergence of an infinite product for Fernique theorem in the abstract Wiener space , which will be discussed and provided here.

Lemma 18. The infinite product converges for all .

Proof. The convergence of an infinite product is defined as . We use the theorem that converges if and only if converges. We take a logarithm for the infinite product: Therefore, it suffices to show that converges for all . As the antilogarithm is less than , each logarithmic term of the series is negative. For each of the finite series , we use a Taylor series , where is plugged into of the Taylor series. Then, the finite series is regarded as a double series as follows: We change the order of the two summations in the last equality. The sign of each term of this double series is and it is put into the front of the series in the last equation. We examine the double series for each : and so on. For all , the sum is a partial sum of a -series () which converges to . Then, Now the series is of positive terms and smaller than a geometric series which is convergent for . By a comparison test, the series also converges for . When , the series diverges. Therefore, converges for all . It is obvious that the infinite product converges for .

The function to be integrated in Fernique theorem is for a variable in that is an infinite dimensional space. When a function defined on an abstract Wiener space assigns finite values, its integral on the abstract Wiener space makes sense and can be expressed by a Lebesgue integral. We use a projection for a function of in , where is an ()-dimensional projection; . Then, the integral of for Fernique theorem should be explained via the integral of .

Theorem 19. for all .

Proof. For a sequence in , . As we have shown in Theorems 16 and 17, random variables are independent and the variance of each is . Therefore, We calculate the -th integral in the product of the last equality: For the equality () above, we use the well-known formula . Then, the integral in Equation (23) is expressed by the following: We need to show that the limits of both sides of Equation (25) are equal. Taking the limit on its left side, by the monotone convergence theorem since . Also, taking the limit on its right side, the limit is . The convergence of this infinite product has been shown for all in Lemma 18, and we have the desired result.

5. Conclusions and Discussions

We explored the abstract Wiener space consisting of sequences of double Fourier coefficients for integrability. As it is also a Hilbert space, we found its orthonormal system and used it to define a probability measure on the abstract Wiener space. Then, we examined integrability of a function appeared in Fernique theorem. We verified that the function is integrable with respect to the abstract Wiener measure with a wider choice of a constant than that of Fernique. As this paper provides a specific range of values for the constant , the result can be applied to related functions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

This research was supported by 2019 Hongik University internal fund.