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

Authors: Jeong-Gyoo Kim
Journal: Journal of Function Spaces
 (2021)

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## 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 functioneα·2 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 α<1/2. The result provides a wider choice of the constant α than that of Fernique.


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## Body

## 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 ofeα·2, 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 α<1/2. The specified range of α that we verified in this paper provides a wider choice of the constant α 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 Section3. 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)
LetS and T be real numbers. A function f:ℝ×ℝ⟶ℝ is said to be S,T-periodic in variables s and t provided that fs+S,t+T=fs,t holds for all s and t(2)
Trigonometric form [8]: let a function f:ℝ×ℝ⟶ℝ be continuous on −S,S×−T,T and periodic with period 2S,2T. Then, the double Fourier series of f is as follows:
(1)fx,y=∑m,n=0∞Am,nfcosmπxScosnπyT+Bm,nfsinmπxScosnπyT+Cm,nfcosmπxSsinnπyT+Dm,nfsinmπxSsinnπyT,where
(2)Am,nf=1ST∫−SS∫−TTfx,ycosmπxScosnπyTdxdy,Bm,nf=1ST∫−SS∫−TTfx,ysinmπxScosnπyTdxdy,Cm,nf=1ST∫−SS∫−TTfx,ycosmπxSsinnπyTdxdy,Dm,nf=1ST∫−SS∫−TTfx,ysinmπxSsinnπyTdxdy.The derivation of double Fourier series can be found in [9].(3)
LetR=0,S×0,T and FR be the space of two-variable functions on R. For x in FR, we define x~s,t as a periodic function having period 2S,2T and to be symmetric with respect to the first argument and symmetric with respect to the second argument within the rectangle −S,S×−T,T; x~s,t=xs,t in R and x~ is 2S,2T-periodic in the whole plane ℝ×ℝ(4)
The coefficientsB,C,D of x~ are all zero due to the symmetries in s and t(3)x~s,t=∑m,n=0∞Am,nx~cosmπsScosnπtT,∑m,n=0∞Am,nx~2=1ST∫−SS∫−TTx~s,t2dsdt.The underlying spaces of this paper are the following three types of double sequences.Definition 1 (see [5]).
A double sequence is denoted byam,n.
(a)
For1≤p<∞, let ℓp=am,n∑m,n=0∞am,np1/p<∞(b)
LetH be the space of all double sequences am,n in ℓ1 with an inner product
(4)am,n,bm,n=∑k=0∞∑l=0kkk+12+l+12al,k−lbl,k−l.Foram,n in H, we let am,nH=am,n,am,n.(c)
LetU be the space of all double sequences am,n in ℓ2 such that the limit
(5)xs,t≔limp⟶∞1pp+1/2+q+1∑m=0p∑u=0m∑k=0qak,u−kcoskπsScosu−kπtTconverges uniformly on ℝ×ℝ.Remark 2 (see [5]).
(a)
It is obvious thatℓp⊂ℓq holds for 1≤p≤q(b)
In this work,∑m,n=0∞am,n 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 ℓ1 for the paper, and rearranging the order of summation does not affect the convergence of double series in the definition(c)
The double summation∑k=0∞∑l=0kak,k−l is carried out by diagonal directions, a0,0+a0,1+a1,0+a0,2+a1,1a2,0+⋯(d)
The spaceU 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 pp+1/2+q+1 in its limit is the number of terms involved in a partial sum for each diagonal explained in (c). The inner product defined for H in (b) is also motivated by thisProposition 3 (see [5]).
Ifam,n is in ℓ1, then, the series ∑m,n=0∞am,ncosmπs/Scosnπt/T converges uniformly on −S,S×−T,T.Proposition 4 (see [5]).
(6)H⊊ℓ1⊊U⊊ℓ2.
The following notations are for the definitions from [2].
(i)
H be a real separable Hilbert space with norm ·=·,·(ii)
P is an orthogonal projection on H(iii)
F is the partially ordered set of orthogonal projections P of H (P>Q means PH⊃QH for P,Q∈F)Definition 5 (see [2] p59).
A seminorm· in Hilbert space H is called measurable if for ε>0, there exists P0∈F such that μPx>ε<ε for all P⊥P0, P∈F.Definition 6 (see [2] p63).
LetB be the completion of H with respect to a measurable norm ·. i will denote the inclusion map of H into B. The triple i,H,B is called an abstract Wiener space.Proposition 7 (see [5]).
i,H,U is an abstract Wiener space.


## 3. An Orthonormal System of the Hilbert SpaceH

From Equation (6), the space of double sequences U contains H, which is a Hilbert space. We find an orthonormal system in H to define an appropriate probability and integration in the space U that will be used in the next section.Definition 8.
For nonnegative integersm and n, we define the following:
(7)am,nu,v≔1m+nm+n−1/2+n+1,ifu,v=m,n,0,otherwise.
Here, the argumentu,v is for a component of am,n. That is, am,n is a member of H whose components are all 0 except for the m,n-th component.Theorem 9.
The setam,n in Definition 8 constitutes an orthonormal system in H.Proof.
Letc=cu,v be in H. By Definition 1, the inner product of c and am,n is as follows:
(8)c,am,n=cu,v,au,vm,n=cm,nm+nm+n−1/2+n+1,ifu,v=m,n,0,otherwise.
We haveai,j,am,n=0 for i,j≠m,n, and
(9)am,n,am,n=∑k=0∞∑l=0kkk+12+l+12al,k−lm,nal,k−lm,n=m+nm+n+12+n+121m+nm+n−1/2+n+12=1.
Therefore,am,n is a complete orthonormal system in H.Remark 10.
By Theorem9, any c=cu,v in U can be expressed as the following limit:
(10)c=limm,n⟶∞∑i=0m∑j=0nc,ai,jai,j=limm⟶∞∑k=0m∑l=0kc,al,k−lal,k−l.Then, Theorem9 is also valid for U.We introduce an operator for an expression of an inner product inU. Let Tm,n:U⟶U be an operator defined by the following:
(11)Tm,ndu,v≔dm,nm+nm+n−1/2+n+1,ifu,v=m,n,0,otherwise.Ifd is in H, then Tm,nd=d,am,n obviously. We want to extend the concept of this H inner product to members of the larger space U or ℓ2.Theorem 11.
Letc=cu,v be in H, d=du,v in U, and am,n is the orthonormal system in Definition 8. The limit limm⟶∞∑k=0m∑l=0kc,Tl,k−ldal,k−l exists in the ℓ2-sense.Proof.
Letd=du,v be in U. We want to show that ∑m,n=0∞cu,v,Tm,ndu,val,k−l exists for any d in U (or in ℓ2). We express the double sum ∑m,n=0∞ as ∑k=0∞∑l=0k. It suffices to show that ∑k=0m∑l=0kc,Tl,k−ld is Cauchy.
(12)c,∑k=m1+1m2∑l=0kTl,k−ld=cu,v,∑k=m1+1m2∑l=0kTl,k−ldu,v=∑k=m1+1m2∑l=0kkk+12+l+12cl,k−ldl,k−lkk+1/2+l+1=∑k=m1+1m2∑l=0kkk+12+l+1cl,k−ldl,k−l≤cu,vH∑k=m1+1m2∑l=0kdl,k−l2.
The last inequality comes from Schwarz inequality and the norm defined in Definition1.Definition 12.
Letc=cu,v be in U and d=du,v be in U. We define the following:
(13)c,d=limm⟶∞∑k=0m∑l=0kc,Tl,k−ld=∑k=0∞∑l=0kc,Tl,k−ld,where the limit is in ℓ2-sense, which has been verified in Theorem 11.


## 4. Integrability ofeα·2 in U

We defined an orthonormal systemam,n in Definition 8 for Fernique theorem in the abstract Wiener space U. Each member of the orthonormal system am,n is a double sequence; the components of am,n are all 0 except for the m,n-th component. We define an abstract Wiener measure with the dual of the orthonormal system. Then, we can examine the integrability of eα·2 in U as in Fernique theorem.Proposition 13 (Kuo and Fernique [2, 6]).
There existsα>0 such that ∫Beαx2dμx<∞, where B is a Banach space and μ is an abstract Wiener measure.
Letb∗:U⟶ℝ be a member of U∗, the dual space of U, which has been defined using the orthonormal system am,n in Definition 8; bm,nd=d,am,n as in Equation (8). That is, for d=du,v in U(14)bm,nd≔dm,nm+nm+n−1/2+n+1.


### 4.1. Single Indexing for a Double Sequence

The underlying space of integral for Fernique theorem will beU. 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 U very complicated. Hence, we convert double indexing of sequences in U into single indexing to alleviate the complexity in manipulation.Definition 14.
We defineψ:ℕ∪0⟶ℕ∪0×ℕ∪0, ψp≔qp,rp for p∈ℕ∪0. Here,
(15)qp=maxk∈ℕkk+12≤p,rp=p−qpqp+12.Remark 15.
For Definition14, we have the following:
(1)
The functionψ is a bijection(2)
We adopt the following notation for a double sequencedm,n(16)∑p=0∞dp=∑ψp∈ℕ∪0×ℕ∪0dψp=diag−∑m,n=0∞dm,n,where diag−∑m,n=0∞ denotes summation by diagonals as in Definition 14 for a double sequence.Using Definition14, 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 systemam,n in U in Definition 8 can be regarded as a single indexed sequence using Equation (16); we denote the system by fn, the new name of am,n. Applying Definition 14 to Equation (14), the n-th component of fn is 1/n+1 and all other components are zero:
(17)fnk=1n+1,ifk=n,0,otherwise,where k is the k-th component of fn.According to this conversion with Equation (16), we rewrite the properties of the orthonormal system am,n in terms of a single indexed sequence fn. As is am,n, fn,fm=0 for n≠m, and fn2=fn,fn=∑j=0∞n+12fnj2=n+12/n+12=1, fn consists of an orthonormal system in U. Here, · is the ℓ2-norm.Consider the dual spaceU∗ of U; for b∗ in U∗b=bn, its norm is defined by b∗=supcn=1cn,bn/cn=supcn=1∑n=0∞cnbn.Letfn∗, the dual of fn, be a member of U∗. Then, fn∗:U⟶ℝ maps dk to its n-th component divided by n+1; i.e., fn∗dk=dn/n+1. Let us compute the norm of fn∗:
(18)fn∗=supcm=1fn∗cmcmu=supcm=1cnn+1=1n+1.Then, the set offn∗consists of an orthogonal system in U∗.


### 4.3. Fernique Theorem onU

In order to be used in Fernique theorem, we need a functional for a measure (density) on the abstract Wiener spaceU. We regard each fn∗ as a random variable defined on U and have the following theorems.Theorem 16.
Eachfn∗ of the orthogonal system is normally distributed; fn∗~N0,fn∗2.Proof.
For a real numbera, we calculate the probability wc∈Ufn∗c<a as follows. Let c=cm be in U. Then, for nonzero fn∗∈U∗,
(19)wc∈U:c,fn<a=wc∈U:c,fnfn∗<afn∗=12π∫−∞a/fn∗exp−u22du=12πfn∗2∫−∞aexp−u22fn∗2du.
Hence,fn∗ is normally distributed with mean 0 and variance fn∗2.Then, the spaceU is with the measure w defined in the proof of Theorem 16.Theorem 17.
Km=f0∗+f1∗+⋯+fm∗ is normally distributed; Km~N0,∑k=0mfk∗2.Proof.
We first show that the random variablesfn∗’s are independent to each other using characteristic functions, i.e., Fourier transforms: if X is a random variable and X~Nμ,σ2, then the Fourier transform of X is FXt=eiμt−σ2t2/2.
Supposem≠n. Then, fm∗+fn∗2=supck=1cm/m+1+cn/n+1=1/m+12+1/n+12. We have fm∗+fn∗~N0,1/m+12+1/n+12, and Ffm∗+fn∗t=e−1/m+12+1/n+12t2/2. As fm∗~N0,1/m+12 and fn∗~N0,1/n+12, we have Ffm∗t1·Ffn∗t2=e−t12/m+12+t22/n+12/2. We obtained Ffm∗t·Ffn∗t=Ffm∗+fn∗t. Hence, fm∗ and fn∗ are stochastically independent by Theorem 16.13 in [10].
Asfn∗ is an independent system, Km2=∑k=0mfk∗2. Therefore, Km=f0∗+f1∗+⋯+fm∗ follows a normal distribution and we have Km~N0,∑k=0mfk∗2.Now, we need the convergence of an infinite product for Fernique theorem in the abstract Wiener spaceU, which will be discussed and provided here.Lemma 18.
The infinite product∏k=0∞k+1/k+12−2α converges for all α<1/2.Proof.
The convergence of an infinite product∏k=0∞zn is defined as limm⟶∞∏k=0nzn. We use the theorem that ∏k=0∞zn converges if and only if ∑k=0∞lnzn converges. We take a logarithm for the infinite product:
(20)ln∏k=0∞k+1k+12−2α=∑k=0∞lnk+1k+12−2α=∑k=0∞−12ln1−2αk+12.
Therefore, it suffices to show that∑k=0∞ln1−2α/k+12 converges for all α<1/2. As the antilogarithm 1−2α/k+12 is less than 1, each logarithmic term of the series is negative. For each k of the finite series ∑k=0mln1−2α/k+12, we use a Taylor series ln1+x=∑i=1∞−1i+1/ixi, where 2α/k+12 is plugged into x of the Taylor series. Then, the finite series ∑k=0m is regarded as a double series as follows:
(21)∑k=0mln1−2αk+12=∑k=0m∑i=1∞−1i+1i−2αk+12i=−∑i=1∞∑k=0m2αii1k+12.
We change the order of the two summations in the last equality. The sign of each term of this double series is−1i+1+i=−1 and it is put into the front of the series in the last equation. We examine the double series for each i:
(22)fori=1,∑k=0m2α111k+12=2α11∑k=0m1k+12<2α11π26,fori=2,∑k=0m2α221k+12=2α22∑k=0m1k+12<2α22π26,and so on. For all i, the sum ∑k=0m1/k+12 is a partial sum of a p-series (p=2) which converges to π2/6. Then, ∑i=1∞2αi/i∑k=0m1/k+12≤π2/6∑i=1∞2αi/i. Now the series ∑i=1∞2αi/i is of positive terms and smaller than a geometric series which is convergent for 2α<1. By a comparison test, the series ∑i=1∞2αi/i also converges for 2α<1. When α≥1/2, the series diverges. Therefore, ∏k=0∞k+1/k+12−2α converges for all α<1/2. It is obvious that the infinite product converges for α<0.The function to be integrated in Fernique theorem iseα·2 for a variable c in U 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 Pm for a function of c=cn in U, where Pm:U⟶U is an (m+1)-dimensional projection; Pmc=c0,c1,c2,⋯,cm,0,0,⋯. Then, the integral of eαc2 for Fernique theorem should be explained via the integral of eαPmc2.Theorem 19.
∫Ueαc2dwc<∞ for all α<1/2.Proof.
For a sequencec=ck in U, Pmc=c0,c1,c2,⋯,cm,0,0,⋯. As we have shown in Theorems 16 and 17, random variables fj∗ are independent and the variance of each fj∗ is fj∗2=1/j+12. Therefore,
(23)∫UeαPmc2dwc=12πm+1∏j=0mfj∗∫ℝm+1expα∑j=0muj2exp−∑j=0muj22fj∗2du0du1⋯dum=∏j=0m12πfj∗∫ℝm+1expα−12fj∗2uj2duj=∏j=0mj+12π∫ℝexpα−j+122uj2duj.
We calculate thek-th integral in the product of the last equality:
(24)k+12π∫ℝexpα−k+122uk2duk=k+12π∫ℝexp−k+122−αuk2duk=∗k+12π2k+12−2απ=k+1k+12−2α.
For the equality (∗) above, we use the well-known formula ∫ℝexp−a2u2du=π/a. Then, the integral in Equation (23) is expressed by the following:
(25)∫UeαPmc2dwc=∏k=0mk+1k+12−2α.
We need to show that the limits of both sides of Equation (25) are equal. Taking the limit on its left side, limm⟶∞∫UeαPmc2dwc=∫Ueαc2dwc by the monotone convergence theorem since limm⟶∞Pmc=c. Also, taking the limit on its right side, the limit is ∏k=0∞k+1/k+12−2α. The convergence of this infinite product has been shown for all α<1/2 in Lemma 18, and we have the desired result.


## 4.1. Single Indexing for a Double Sequence

The underlying space of integral for Fernique theorem will beU. 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 U very complicated. Hence, we convert double indexing of sequences in U into single indexing to alleviate the complexity in manipulation.Definition 14.
We defineψ:ℕ∪0⟶ℕ∪0×ℕ∪0, ψp≔qp,rp for p∈ℕ∪0. Here,
(15)qp=maxk∈ℕkk+12≤p,rp=p−qpqp+12.Remark 15.
For Definition14, we have the following:
(1)
The functionψ is a bijection(2)
We adopt the following notation for a double sequencedm,n(16)∑p=0∞dp=∑ψp∈ℕ∪0×ℕ∪0dψp=diag−∑m,n=0∞dm,n,where diag−∑m,n=0∞ denotes summation by diagonals as in Definition 14 for a double sequence.Using Definition14, 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 systemam,n in U in Definition 8 can be regarded as a single indexed sequence using Equation (16); we denote the system by fn, the new name of am,n. Applying Definition 14 to Equation (14), the n-th component of fn is 1/n+1 and all other components are zero:
(17)fnk=1n+1,ifk=n,0,otherwise,where k is the k-th component of fn.According to this conversion with Equation (16), we rewrite the properties of the orthonormal system am,n in terms of a single indexed sequence fn. As is am,n, fn,fm=0 for n≠m, and fn2=fn,fn=∑j=0∞n+12fnj2=n+12/n+12=1, fn consists of an orthonormal system in U. Here, · is the ℓ2-norm.Consider the dual spaceU∗ of U; for b∗ in U∗b=bn, its norm is defined by b∗=supcn=1cn,bn/cn=supcn=1∑n=0∞cnbn.Letfn∗, the dual of fn, be a member of U∗. Then, fn∗:U⟶ℝ maps dk to its n-th component divided by n+1; i.e., fn∗dk=dn/n+1. Let us compute the norm of fn∗:
(18)fn∗=supcm=1fn∗cmcmu=supcm=1cnn+1=1n+1.Then, the set offn∗consists of an orthogonal system in U∗.


## 4.3. Fernique Theorem onU

In order to be used in Fernique theorem, we need a functional for a measure (density) on the abstract Wiener spaceU. We regard each fn∗ as a random variable defined on U and have the following theorems.Theorem 16.
Eachfn∗ of the orthogonal system is normally distributed; fn∗~N0,fn∗2.Proof.
For a real numbera, we calculate the probability wc∈Ufn∗c<a as follows. Let c=cm be in U. Then, for nonzero fn∗∈U∗,
(19)wc∈U:c,fn<a=wc∈U:c,fnfn∗<afn∗=12π∫−∞a/fn∗exp−u22du=12πfn∗2∫−∞aexp−u22fn∗2du.
Hence,fn∗ is normally distributed with mean 0 and variance fn∗2.Then, the spaceU is with the measure w defined in the proof of Theorem 16.Theorem 17.
Km=f0∗+f1∗+⋯+fm∗ is normally distributed; Km~N0,∑k=0mfk∗2.Proof.
We first show that the random variablesfn∗’s are independent to each other using characteristic functions, i.e., Fourier transforms: if X is a random variable and X~Nμ,σ2, then the Fourier transform of X is FXt=eiμt−σ2t2/2.
Supposem≠n. Then, fm∗+fn∗2=supck=1cm/m+1+cn/n+1=1/m+12+1/n+12. We have fm∗+fn∗~N0,1/m+12+1/n+12, and Ffm∗+fn∗t=e−1/m+12+1/n+12t2/2. As fm∗~N0,1/m+12 and fn∗~N0,1/n+12, we have Ffm∗t1·Ffn∗t2=e−t12/m+12+t22/n+12/2. We obtained Ffm∗t·Ffn∗t=Ffm∗+fn∗t. Hence, fm∗ and fn∗ are stochastically independent by Theorem 16.13 in [10].
Asfn∗ is an independent system, Km2=∑k=0mfk∗2. Therefore, Km=f0∗+f1∗+⋯+fm∗ follows a normal distribution and we have Km~N0,∑k=0mfk∗2.Now, we need the convergence of an infinite product for Fernique theorem in the abstract Wiener spaceU, which will be discussed and provided here.Lemma 18.
The infinite product∏k=0∞k+1/k+12−2α converges for all α<1/2.Proof.
The convergence of an infinite product∏k=0∞zn is defined as limm⟶∞∏k=0nzn. We use the theorem that ∏k=0∞zn converges if and only if ∑k=0∞lnzn converges. We take a logarithm for the infinite product:
(20)ln∏k=0∞k+1k+12−2α=∑k=0∞lnk+1k+12−2α=∑k=0∞−12ln1−2αk+12.
Therefore, it suffices to show that∑k=0∞ln1−2α/k+12 converges for all α<1/2. As the antilogarithm 1−2α/k+12 is less than 1, each logarithmic term of the series is negative. For each k of the finite series ∑k=0mln1−2α/k+12, we use a Taylor series ln1+x=∑i=1∞−1i+1/ixi, where 2α/k+12 is plugged into x of the Taylor series. Then, the finite series ∑k=0m is regarded as a double series as follows:
(21)∑k=0mln1−2αk+12=∑k=0m∑i=1∞−1i+1i−2αk+12i=−∑i=1∞∑k=0m2αii1k+12.
We change the order of the two summations in the last equality. The sign of each term of this double series is−1i+1+i=−1 and it is put into the front of the series in the last equation. We examine the double series for each i:
(22)fori=1,∑k=0m2α111k+12=2α11∑k=0m1k+12<2α11π26,fori=2,∑k=0m2α221k+12=2α22∑k=0m1k+12<2α22π26,and so on. For all i, the sum ∑k=0m1/k+12 is a partial sum of a p-series (p=2) which converges to π2/6. Then, ∑i=1∞2αi/i∑k=0m1/k+12≤π2/6∑i=1∞2αi/i. Now the series ∑i=1∞2αi/i is of positive terms and smaller than a geometric series which is convergent for 2α<1. By a comparison test, the series ∑i=1∞2αi/i also converges for 2α<1. When α≥1/2, the series diverges. Therefore, ∏k=0∞k+1/k+12−2α converges for all α<1/2. It is obvious that the infinite product converges for α<0.The function to be integrated in Fernique theorem iseα·2 for a variable c in U 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 Pm for a function of c=cn in U, where Pm:U⟶U is an (m+1)-dimensional projection; Pmc=c0,c1,c2,⋯,cm,0,0,⋯. Then, the integral of eαc2 for Fernique theorem should be explained via the integral of eαPmc2.Theorem 19.
∫Ueαc2dwc<∞ for all α<1/2.Proof.
For a sequencec=ck in U, Pmc=c0,c1,c2,⋯,cm,0,0,⋯. As we have shown in Theorems 16 and 17, random variables fj∗ are independent and the variance of each fj∗ is fj∗2=1/j+12. Therefore,
(23)∫UeαPmc2dwc=12πm+1∏j=0mfj∗∫ℝm+1expα∑j=0muj2exp−∑j=0muj22fj∗2du0du1⋯dum=∏j=0m12πfj∗∫ℝm+1expα−12fj∗2uj2duj=∏j=0mj+12π∫ℝexpα−j+122uj2duj.
We calculate thek-th integral in the product of the last equality:
(24)k+12π∫ℝexpα−k+122uk2duk=k+12π∫ℝexp−k+122−αuk2duk=∗k+12π2k+12−2απ=k+1k+12−2α.
For the equality (∗) above, we use the well-known formula ∫ℝexp−a2u2du=π/a. Then, the integral in Equation (23) is expressed by the following:
(25)∫UeαPmc2dwc=∏k=0mk+1k+12−2α.
We need to show that the limits of both sides of Equation (25) are equal. Taking the limit on its left side, limm⟶∞∫UeαPmc2dwc=∫Ueαc2dwc by the monotone convergence theorem since limm⟶∞Pmc=c. Also, taking the limit on its right side, the limit is ∏k=0∞k+1/k+12−2α. The convergence of this infinite product has been shown for all α<1/2 in Lemma 18, and we have the desired result.


## 5. Conclusions and Discussions

We explored the abstract Wiener spaceU 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 eα·2 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 α than that of Fernique. As this paper provides a specific range of values for the constant α, the result can be applied to related functions.



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