Quaternion-valued positive definite functions on locally compact Abelian groups and nuclear spaces

Applied Mathematics and Computation 286 (2016) 115–125

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Quaternion-valued positive definite functions on locally compact Abelian groups and nuclear spaces Daniel Alpay a, Fabrizio Colombo b, David P. Kimsey a,∗, Irene Sabadini b a b

Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi, 9, Milano, 20133, Italy

a r t i c l e

i n f o

Keywords: Bochner’s theorem Bochner-Minlos theorem quaternionic analysis nuclear spaces

a b s t r a c t In this paper we study quaternion-valued positive definite functions on locally compact Abelian groups, real countably Hilbertian nuclear spaces and on the space R N = { ( x1 , x2 , . . . ) : xd ∈ R } endowed with the Tychonoff topology. In particular, we prove a quaternionic version of the Bochner–Minlos theorem. A tool for proving this result is a classical matricial analogue of the Bochner–Minlos theorem, which we believe is new. We will see that in all these various settings the integral representation is with respect to a quaternion-valued measure which has certain symmetry properties. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Positive definite functions in various settings are an important tool in mathematical analysis with applications to moment problems, operator theory, function theory and many other areas such as probability theory and stochastic processes. In [10], Herglotz characterized positive definite functions on the integers Z via an integral representation with respect to a uniquely determined positive measure on [0, 2π ]. Subsequently, in [5], Bochner characterized positive definite functions on R by an integral representation with respect to a uniquely determined positive measure on R. Bochner’s proof is easily adapted to the multidimensional setting Rd . Moreover, in [16], Weil characterized positive definite functions on a locally compact Abelian group G via an integral representation on the character group of G. See the book of Sasvári [14] for more information of positive definite functions on locally compact Abelian groups. Positive definite functions have also been studied in various infinite dimensional settings, e.g., real Fréchet nuclear spaces. Nuclear spaces were introduced by Grothendieck in [9]. They were later studied also from the point of view of duality theory, see e.g. [2]. The Bochner–Minlos theorem characterizes positive definite functions on a nuclear Fréchet space X by an integral representation with respect to a positive measure on the topological dual of X. See [8, Theorem 2, p. 350] for the theorem and see the historical notes in that book and [1] for more information and connections with the work of Sazonov. The Bochner–Minlos theorem is the starting point of Hida’s white noise space theory and the corresponding infinite dimensional analysis (see for instance [3,4,11,12]). The present paper can be seen as the starting point of these topics in the quaternionic setting. ∗

Corresponding author. Tel.: +972 08 9343573. E-mail addresses: [email protected] (D. Alpay), [email protected] (F. Colombo), [email protected] (D.P. Kimsey), [email protected] (I. Sabadini). http://dx.doi.org/10.1016/j.amc.2016.03.034 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.

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We will denote by H the set of quaternions, i.e., the real algebra generated by the units e1 , e2 , e3 which obey

e21 = e22 = e23 = e1 e2 e3 = −1. Given p = p0 + p1 e1 + p2 e2 + p3 e3 , we will let

p = p0 − p1 e1 − p2 e2 − p3 e3 and

| p| =



pp =



p20 + p22 + p22 + p23 .

We will also let S denote the sphere of purely imaginary unitary quaternions. For any i ∈ S, we may consider the complex plane

Ci = {x + iy : x, y ∈ R}. Note that

H=



Ci .

i∈S

To illustrate that the quaternionic setting indeed introduces new cases consider the function N 

F (s ) =

n=1

 exp R

 (ein s(x) − 1 )dx ,

where i1 , . . . , iN ∈ S all belong to different complex planes, and s varies in the space S of real-valued Schwartz functions. As is well known (see for instance [8, formula (15), p. 282]) each of the functions



 (ein s(x) − 1 )dx

exp R

is positive definite on S. The range of the function F is not contained in a given Ci for some i ∈ S, but it is quaternionic. The principal goal of this paper is to extend various classical results on positive definite functions to positive definite functions which are quaternion valued. The paper is organized as follows. In Section 2, we highlight results on positive definite matrix-valued functions on locally compact Abelian groups which will make use of later. In Section 3, we will product an analogue of Bochner–Minlos for complex matrix-valued functionals on a real countably Hilbertian nuclear space which will be used later. In Section 4, we will characterize quaternion-valued positive definite functions on locally compact Abelian groups via a quaternion-valued measure with special structure (which will generalize a recently obtained result in [10]). In Section 5, we will provide a similar characterization for quaternion-valued positive definite functions on a real countably Hilbertian nuclear space. Finally, in Section 6, we will study quaternion-valued positive definite functions on the space

RN = { ( x1 , . . . , xd , . . . ) : xd ∈ R} endowed with the Tychonoff topology. 2. Positive definite complex matrix-valued functions on a locally compact Abelian group In this section we fix an imaginary unit i ∈ S, and we consider Csi ×s -valued measures. Throughout this paper, G will denote a locally compact Abelian group under addition and  i will denote the group of Ci -valued characters of G, i.e., the set of continuous maps γ : G → Ci such that

|γ (x )| = 1 and γ (x + y ) = γ (x )γ (y ) for x, y ∈ G. Note that if γ ∈  i , then γ ∈ i , where γ is given by γ (x ) = γ (x ). Definition 2.1. Fix i ∈ S. If σ is a positive Csi ×s -valued measure on a locally compact Abelian group G with character group  i with respect to i ∈ S, then σˆ will denote the Csi ×s -valued function on  i

σˆ (γ ) =



G

γ (x )dσ (x ).

We will call the function σˆ the Fourier–Stieltjes transform of the measure σ with respect to i ∈ S. Definition 2.2. Fix i ∈ S. Let G be a locally compact Abelian group with character group  i with respect to i ∈ S. If σ is a positive Csi ×s -valued measure on  i , then σˇ will denote the Csi ×s -valued function on G

σˇ (x ) =



i

γ ( x )d σ ( γ ).

We will call the function σˇ the inverse Fourier–Stieltjes transform of the measure σ with respect to i ∈ S.

D. Alpay et al. / Applied Mathematics and Computation 286 (2016) 115–125

117

Definition 2.3. Fix i ∈ S. Let G be a locally Abelian group. We call a function f : G → Csi ×s positive definite if N 

v∗b f (xa − xb )va ≥ 0

a,b=0

for all choices of {v0 , . . . , vN } ⊂ Csi and {x0 , . . . , xN } ⊆ G and N = 0, 1, . . .. Theorem 2.4. Fix i ∈ S. Let G be a locally compact Abelian group with character group  i . Every Csi ×s -valued measure σ on  i is uniquely determined by σˇ . Proof. If s = 1, then see, e.g., Theorem 1.9.5 in Sasvári [14]. If s > 1, then suppose there exist Csi ×s -valued measures, on  i , σ and ν such that σˇ = νˇ yet σ = ν . It follows from σˇ = νˇ that v∗ σˇ w = v∗ νˇ w, for v, w ∈ Csi . Thus, it follows from the s = 1 case of the theorem that v∗ σ w = v∗ ν w for v, w ∈ Csi , i.e. σ = ν .  The next theorem (which is usually called Bochner’s theorem) is due to Weil [16] for the case when s = 1. The case when s = 1 was proved by Herglotz [10] and Bochner [5] for G = Z and G = Rd , respectively. Theorem 2.5. Let G be a locally compact Abelian group and  i denote the character group of G with respect to i ∈ S. If σ is a positive Csi ×s -valued measure on  i , then σˇ is a continuous positive definite function on G. Conversely, if f : G → Csi ×s is a continuous positive definite function on G, then there exists a unique positive Csi ×s -valued measure σ on  i such that f = σˇ . Proof. If s = 1, then see, e.g., Theorem 1.9.6 in Sasvári [14]. Suppose s > 1. If σ is a positive Csi ×s -valued measure on  i , then

 2   N   v σˇ (xa − xb )va =  σˇ (xa )va  dσ (γ ) ≥ 0.    i a=0 a,b=0 N 



∗ b

Thus σˇ is positive definite. Write σ = (σa,b )sa,b=1 . The fact that σˇ is continuous on G follows from the fact that σˇ (x ) = (σˇ (x )a,b )sa,b=1 and σˇ a,b is continuous on G for all a, b = 1, . . . , s (see, e.g., Theorem 1.9.3 in [14]).

Conversely, if f is a continuous positive definite Csi ×s -valued function on G, then for any v ∈ Csi , fv = v∗ f v is a continuous positive definite function on G. Indeed, for any vector w = (wa )sa=1 ∈ CN , i

⎛

w∗ ⎝

v∗ f ( 0 )v .. .

v∗ f ( N )v

··· .. . ···

⎞ v∗ f (−N )v N  .. ⎠w = (wa v )∗ f (a − b)(wb v ) . a,b=0 v∗ f ( 0 )v =

N 

ξa∗ f (a − b)ξb ≥ 0, ξa = wa v.

a,b=0

Thus, we have the existence of a positive measure σv on  i such that

f v (x ) =



i

γ ( x ) d σv ( γ ) .

(2.1)

Consequently, we may define

1 4

σv,w = (σv+w − σv−w + iσv+iw − iσv−iw )

for

v, w ∈ Cs .

One can check that σv,v = σv for all v ∈ Csi . Indeed, for any Borel set E of  i ,

1 4 1 = {(2v )∗ σ (E )(2v ) + i(v + iv )∗ σ (E )(v + iv ) − i(v − iv )∗ σ (E )(v − iv )} 4 = v∗ σ ( E )v

σv,v (E ) = {σ2v (E ) + iσv+iv (E ) − iσv−iv (E )}

= σv ( E ) . Therefore, we can define a positive Csi ×s -valued Borel measure σ = (σa,b )sa,b=1 on  i which satisfies

w∗ σ (E )v = σv,w (E ) Moreover,

for

v, w ∈ Csi .

(2.2)

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D. Alpay et al. / Applied Mathematics and Computation 286 (2016) 115–125

 i

γ ( x )d ( w∗ σ ( γ )v ) =



γ (x )dσv,w (γ )

  σ (γ ) − σv−w (γ ) + iσv+iw (γ ) − iσv−iw (γ ) = γ ( x )d v+w i

4

i

1 { fv+w (x ) − fv−w (x ) + i fv+iw (x ) − i fv−iw (x )} 4 1 = {(v + w )∗ f (x )(v + w ) − (v − w )∗ f (x )(v − w ) 4 + i(v + iw )∗ f (x )(v + iw ) − i(v − iw )∗ f (x )(v − iw )} =

= w∗ f ( x )v, Therefore,

f (x ) =

 i

for

v, w ∈ Csi .

γ ( x )d σ ( γ ).

The asserted uniqueness follows from Theorem 2.4.



3. Positive definite complex matrix-valued functions on a countably Hilbertian nuclear space Definition 3.1. We will call a locally convex topological vector space X over R a real nuclear space if the topology of X is generated by a family of Hilbert seminorms {}∈F on X such that the completions {X }∈F of X satisfy the following condition: For all  ∈ F , there exists ˜ ∈ F with  ˜ , i.e., the topology generated by ˜ is coarser than the topology by ϱ, such that the embedding

I˜ : X → X˜ is nuclear, i.e., the bounded linear operator I˜ is trace class. Definition 3.2. Let X be a real nuclear space. By X we mean the set of continuous real-valued functionals on X. We shall call X the dual of X. Note that a real countably Hilbertian nuclear space also has the moniker real Fréchet nuclear space. Remark 3.3. Let X be a real nuclear space and X be the dual of X. There are various useful topologies to put on X , e.g., the weak topology and the strong topology, i.e., the topologies of pointwise convergence and uniform convergence on bounded sets, respectively. See [6] and [13] for more details. Throughout this note, we will use the weak topology on X for simplicity. However, in point of fact we could use any topology which is intermediate to the weak and strong topology. Following [13], we will introduce the following notion. Definition 3.4. We will call a locally convex topological vector space X a real countably Hilbertian nuclear space, if X is a real nuclear space whose topology is generated by a countable family of seminorms {}∈F on X. Definition 3.5. Fix i ∈ S. Let X be a real countably Hilbertian nuclear space with dual X . If σ is a Csi ×s -valued measure on X , then σˇ will denote the Csi ×s -function valued on X such that

σˇ (x ) =



X

ei(y,x ) dσ (y ).

We will call the function σˇ the inverse Fourier transform of the measure σ with respect to i ∈ S. Definition 3.6. Fix i ∈ S. Let X be a countably Hilbertian nuclear space with dual X . We call a function f : X → Csi ×s positive definite if N 

v∗b f (xa − xb )va ≥ 0

a,b=0

for all choices of {v0 , . . . , vN } ⊂ Csi and {x0 , . . . , xN } ⊆ X. Lemma 3.7. Fix i ∈ S and a real countably Hilbertian nuclear space X with dual X . Every bounded Csi ×s -valued measure σ on X has the property that σˇ is a (uniformly) continuous Csi ×s -valued function on X. Proof. Write σ = (σa,b )sa,b=1 . It follows from Proposition 9 on page 93 of Chapter IX in [6] that σˇ a,b is a continuous function on X for a, b = 1, . . . , s. Thus, as

D. Alpay et al. / Applied Mathematics and Computation 286 (2016) 115–125

√



σˇ (x ) − σˇ (x˜) ≤ s

119

 max

a,b=1,...,s

|σˇ a,b (x ) − σˇ a,b (x˜)|

for

we have that σˇ is a continuous Csi ×s -valued function on X.

x, x˜ ∈ X,



Lemma 3.8. Fix i ∈ S and a real countably Hilbertian nuclear space X with dual X . Every bounded Csi ×s -valued measure σ on X is uniquely determined by σˇ . Proof. If s = 1, then the assertion is a particular case of Proposition 3 on page 76 of Chapter IX in [6]. Suppose s > 1 and there exist bounded Csi ×s -valued measures σ and ν on X such that σˇ = νˇ and σ = ν . But then for any v, w ∈ Csi , we have w∗ σˇ v = w∗ νv ˇ . As w∗ σ v and w∗ νv are bounded measures on X , it follows from the s = 1 case that σˇ = νˇ . Since v and w are arbitrary in Csi , we have that σ = ν .  The following theorem is usually called the Bochner–Minlos theorem when s = 1. The case when s > 1 seems to be new. Theorem 3.9. Fix i ∈ S and a real countably Hilbertian nuclear space X with dual X . If σ is a bounded positive Csi ×s -valued measure on X , then σˇ is a continuous positive definite Csi ×s -valued function on X. Conversely, if f is a continuous positive definite Csi ×s -valued function on X, then there exists a unique bounded positive Csi ×s -valued measure σ on X such that f = σˇ . Proof. If s = 1, then, upon noting that a countably Hilbertian nuclear space is barreled, the asserted necessity and sufficiency follow from the corollary on p. 97 of Chapter IX in [6]. Suppose s > 1. If σ is a bounded positive Csi ×s -valued measure on X , then N 

v∗b σˇ (xa − xb )va =



a,b=0

X



N 

ei(y,xa ) va 2 dσ (y ) ≥ 0

a=0

for all choices of {x0 , . . . , xN } ⊂ X and {v0 , . . . , vN } ⊂ Csi . Thus, σˇ is a positive definite Csi ×s -valued function on X. The continuity of σˇ on X follows from Lemma 3.7. Conversely, if f is a continuous positive definite Csi ×s -valued function on X, then one can argue in a similar to manner to the proof of Theorem 2.5 to obtain that

f v = v∗ f v is a positive definite function on X for any choice of v ∈ Csi . Thus, there exists a bounded positive measure on X such that

f v (x ) =



X

ei(y,x ) dσv (y )

for

x ∈ X.

Consequently, we may define

1 4

σv,w = (σv+w − σv−w + iσv+iw − iσv−iw ) for v, w ∈ Csi . One can easily adapt the procedure given in the proof of Theorem 2.5 to the present setting to obtain the following conclusions: (1) σv,v = σv for v ∈ Csi . (2) There exists a bounded positive Csi ×s -valued measure σ = (σa,b )sa,b=1 on X such that

w∗ σ v = σv,w

for

v, w ∈ Csi .

(3) σ satisfies



X

ei(y,x ) (w∗ dσ (y )v ) = w∗ f (x )v

and hence

f (x ) =

 X

for

v, w ∈ Csi

ei(y,x ) dσ (y ).

Finally, the uniqueness of σ follows from Lemma 3.8.



4. Quaternion-valued positive definite functions on locally compact Abelian groups Fix i ∈ S and j ∈ S so that 1, i, j, ij is a basis for the quaternions and every quaternion p ∈ H can be written as

p = p0 + p1 i + p2 j + p3 i j = z1 + z2 j, where p0 , p1 , p2 , p3 ∈ R and z1 , z2 ∈ Ci . Thus every quaternion p is identified by the pair z1 , z2 , moreover the map χi : H → C2i ×2 given by



χi ( p) =

z1 −z2

z2 z1



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D. Alpay et al. / Applied Mathematics and Computation 286 (2016) 115–125

is an injective homomorphism which identifies quaternions with the matrices of the above form. The map χ i , strictly speaking, depends on both the choices of i and j. Given a locally compact Abelian group G, let as above  i denote the group of Ci -valued characters of G. Given P ∈ Hs×s , there exist unique P1 , P2 ∈ Csi ×s such that P = P1 + P2 j. Thus, we can generalize the map χ i introduced above, and we define the injective homomorphism χi : Hs×s → C2i s×2s given by



χi (P ) =



P1 −P 2

P2 . P1

(4.1)

Definition 4.1. Fix i and j ∈ S so that 1, i, j, ij is a basis of H. Let G be a locally compact Abelian group and  i be the character group with respect to i ∈ S. Given an Hs×s -valued measure μ on  i , we may always write μ = μ1 + μ2 j, where μ1 , μ2 are uniquely determined Csi ×s -valued measures on  i . We call a measure μ on  i q-positive if the C2i s×2s -valued measure

σ=

 μ1 μ∗2

μ2 μ3



is positive and μ3 and μ2 satisfy dμ3 (γ ) = dμ1 (γ ) and dμ2 (γ ) = −dμ2 (γ )T for γ ∈  i respectively. Definition 4.2. Fix i ∈ S. Let G be a locally Abelian group and  i be the character group of G with respect to i ∈ S. If μ is a Hs×s -valued measure on G, then μ ˆ will denote the Hs×s -valued function on  i such that

μˆ (γ ) =



G

γ (x )dμ(x ).

We will call the function μ ˆ the Fourier–Stieltjes transform of the measure μ with respect to i ∈ S. If ν is a Hs×s -valued measure on  i , then νˇ will denote the Hs×s -valued function on G such that

νˇ (x ) =



i

γ ( x )d ν ( γ ).

We will call the function νˇ the inverse Fourier–Stieltjes transform of the measure ν with respect to i ∈ S. Definition 4.3. Let G be a locally compact Abelian group. We call a function f : G → Hs×s positive definite if N 

v∗b f (xa − xb )va ≥ 0

a,b=0

for all choices of {v0 , . . . , vN } ⊂ Hs and {x0 , . . . , xN } ⊆ G. Theorem 4.4. Fix i ∈ S. Let G be a locally compact Abelian group and  i be the character group of G with respect to i ∈ S. Every Hs×s -valued measure μ on  i is uniquely determined by μ ˇ. Proof. Let ν denote the C2i s×2s -valued measure on  i given by

ν (E ) = χi (μ(E )), where E is a Borel subset of  i . It follows from Theorem 2.4 that ν is uniquely determined by νˇ . Since χ i is injective and νˇ (x ) = χi (μˇ (x )) for x ∈ G, we get that μ is uniquely determined by μˇ .  Theorem 4.5. Fix i, j ∈ S such that 1, i, j, ij is a basis of H. Let G be a locally compact Abelian group and  i be the character group of G with respect to i ∈ S. If μ is a q-positive measure on  i , then μ ˇ is a continuous positive definite function on G. Conversely, if f is a continuous positive definite Hs×s -valued function on G, then there exists a unique (relative to i and j) q-positive measure μ = μ1 + μ2 j on  i such that f = μˇ . Proof. If μ = μ1 + μ2 j is a q-positive measure on  i , then

 μ1 μ∗2

μ2 μ3



is a positive C2i s×2s -valued measure, where

dμ3 (γ ) = dμ1 (γ ), d μ2 ( γ ) = − d μ2 ( γ ) T ,

γ ∈ i ,

(4.2)

γ ∈ i .

Thus, it follows from Theorem 2.5 that σˇ is a continuous positive definite

(4.3) C2i s×2s -valued

function on G. If we let

D. Alpay et al. / Applied Mathematics and Computation 286 (2016) 115–125



121



g11 (x ) g12 (x ) g21 (x ) g22 (x )  γ ( x ) d μ1 ( γ ) = i ∗ i γ (x )d μ2 (γ )

g( x ) =

  γ ( x ) d μ2 ( γ ) i , i γ (x )d μ3 (γ )

then, in view of (4.2),



g11 (x ) =

i

 =

i

γ ( x ) · d μ1 ( γ ) γ ( x ) · d μ3 ( γ )

= g22 (x ). Similarly, it follows from (4.3) that

g12 (x ) = −g21 (x ). Thus,

g(x ) = χi (g11 (x ) + g12 (x ) j ), and hence, as g is a continuous positive definite C2i s×2s -valued function on G, μ ˇ = g11 (x ) + g12 (x ) j is a continuous positive definite function Hs×s -valued function on G. Conversely, if f = f1 + f2 j, where f1 , f2 are Csi ×s -valued, is a continuous positive definite Hs×s -valued function on G, then

χ i (f) is a continuous positive definite C2i s×2s -valued function on G. Thus, we may use Theorem 2.5 to obtain a unique positive C2i s×2s -valued measure



 Cs i σ12 :  σ22 s

σ σ = 11∗ σ12

(χi ( f ))(x ) = It follows from



f 2 (x )

− f 2 (x )

f 1 (x )

γ (x )dσ11 (γ ) =

(4.4)



f 1 (x )

and (4.4) that i



Csi

γ ( x )d σ ( γ ).

i



χi ( f (x )) = 

→

Ci

on  i such that

Csi

 

=

i i

for x ∈ G

γ (x )dσ22 (γ ) γ (x ) dσ22 (γ )

and hence using the uniqueness in Theorem 2.4 we get

dσ11 (γ ) = dσ22 (γ ),

γ ∈ i .

Similarly, one can show that

dσ12 (γ ) = −dσ12 (γ )T ,

γ ∈ i .

Thus, if we put μ1 = σ11 and μ2 = σ12 , then μ is a q-positive measure. It is easy to check that





− jI

2f = I



f1 − f¯2

f2 f¯1

  I jI

and hence (4.4) yields



2 f (x ) = I  =



− jI



 



γ ( x )I

i

γ ( x )d σ ( γ ) 

I jI

−γ (x ) jI dσ (γ )

  I jI

(4.5)

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D. Alpay et al. / Applied Mathematics and Computation 286 (2016) 115–125

 =



γ (x )dσ11 (γ ) +

 

γ (x )dσ12 (γ ) j −

 + γ (x )dσ22 (γ )    = 2 γ (x )dσ11 (γ ) + 2 

=2μ ˇ ( x ).



 

γ (x )dσ12 (γ )T j

γ (x )dσ12 (γ ) j

Note that in the above manipulations we used the fact that jA = A¯ j for any A ∈ Csi ×s . Finally, the asserted uniqueness of μ follows from Theorem 4.4.  5. Quaternion-valued positive definite functions on a nuclear space In this section, we will generalize Theorem 3.9 to case that f : X → Hs×s . Definition 5.1. Fix i ∈ S and let j ∈ S be such that 1, i, j, ij is a basis of H. Let X be a real countably Hilbertian nuclear space with dual X . Given a Hs×s -valued measure μ on X , we can write μ = μ1 + μ2 j, where μ1 , μ2 are Csi ×s -valued measures on X . We will call μ q-positive if



μ2 μ3

μ1 ν= μ∗2



is a positive C2i s×2s -valued measure on X and μ2 and μ1 satisfy dμ2 (−y ) = −dμ2 (y )T and, with dμ3 (y ) = dμ1 (−y ) for y ∈ X , respectively. Theorem 5.2. Fix i, j ∈ S so that 1, i, j, ij is a basis of H and a real countably Hilbertian nuclear space X with dual X . If μ is a bounded positive Hs×s -valued measure on X , then σˇ is a continuous positive definite Hs×s -valued function on X. Conversely, if f is a continuous positive definite Hs×s -valued function on X, then there exists a unique (relative to i and j) bounded q-positive Hs×s -valued measure μ on X such that f = μ ˇ. Proof. If μ = μ1 + μ2 j is a q-positive measure on X , then



μ2 μ3

μ1 ν= μ∗2



is a positive C2s×2s -valued measure on X , where

dμ3 (y ) = dμ1 (−y ),

y ∈ X , y ∈ X .

dμ2 (y ) = − dμ2 (−y )T ,

(5.1) (5.2)

Thus, it follows from Theorem 3.9 that





g11 (x ) g12 (x ) g( x ) = g21 (x ) g22 (x )  ei(y,x ) dμ (y ) = X i(y,x ) 1∗ d μ2 ( y )

X e

  i(y,x ) X ei(y,x ) dμ2 (y ) d μ3 ( y ) X e

is a continuous positive definite C2i s×2s -valued function on X. In view of (5.1)

g22 (x ) =



X

 =

X

e−i(y,x ) dμ3 (y ) ei(y,x ) dμ1 (y )

= g11 (x ). Similarly, in view of (5.2)

g21 (x ) = −g12 (x ) and hence, as g is a continuous positive definite function on X if and only if f = g11 + g12 j is a continuous positive function on X, f is a continuous positive function on X. Conversely, if f = f1 + f2 j, where f1 , f2 are Csi ×s -valued, is a continuous positive definite Hs×s -valued function on X, then χ i (f) is a continuous positive definite C2i s×2s -valued function on X. Thus, we may use Theorem 3.9 to obtain a unique

D. Alpay et al. / Applied Mathematics and Computation 286 (2016) 115–125

measure



 Cs i σ12 :  σ22 s

σ11 ∗ σ12

σ=

on X such that



(χi ( f ))(x ) = It follows from

X



f1 − f¯2

χi ( f ) =

X



Csi

ei(y,x ) dσ (y ).

f2 f¯1

and (5.3) that



Csi

→

Ci

ei(y,x ) dσ11 (y ) =

123

(5.3)

 (5.4)

 X

e−i(y,x ) dσ22 (y ) =

 X

ei(y,x ) dσ22 (−y )

and hence using the uniqueness in Theorem 2.4 we get

y ∈ X .

dσ11 (y ) = dσ22 (−y ),

Similarly, using (5.4) and (5.3), one can show that

dσ12 (y ) = −dσ21 (−y )T . Thus, if we put μ1 = σ11 and μ2 = σ12 , then μ is a q-positive measure on X . It is easy to check that





− jI

2f = I



f1 − f¯2

f2 f¯1

  I jI

and hence (5.3) yields



2 f (x ) = I

− jI

 =



X

e

 



i(y,x )

X

I

e

i(y,x )

−e

d σ (y )

−i(y,x )



I jI

jI dσ (y )

  I jI

  ei(y,x ) dσ11 (y ) + ei(y,x ) dσ12 (y ) j − e−i(y,x ) dσ12 (−y )T j X

X

X

 + ei(y,x ) dσ22 (−y ) X

  =2 ei(y,x ) dσ11 (y ) + 2 ei(y,x ) dσ12 (y ) j 

=

X

X

= 2μ ˇ ( x ). Note that in the above manipulations we used the fact that jA = A¯ j for any A ∈ Csi ×s . Finally, the asserted uniqueness of μ follows from uniqueness of σ (see Theorem 3.9) and the fact that χ i is injective.  6. Quaternion-valued positive definite functions on RN Let RN denote the linear space of all real numerical sequences, i.e.,

RN = { ( x1 , . . . , xd , . . . ) : xd ∈ R}. Throughout this section x = (x1 , . . . , xd , . . . ) will denote an arbitrary element of RN . We will say that x ∈ RN has length L if

xL = 0

and xd = 0

for

d > L.

If such an L does not exist, then we will say that the length of x is ∞ and we shall define the length of 0 = (0, . . . , 0, . . . ) ∈ RN to be 0. We will endow RN with the Tychonoff topology, i.e., the topology generated by sets of the form

Oε,n (0 ) =

n  d=1

{x ∈ RN : |xd | < ε}.

124

D. Alpay et al. / Applied Mathematics and Computation 286 (2016) 115–125

Let A denote the algebra of sets given by

A=



∞ 

∞ 

E×



R : E is a Borel subset of R

d

(6.1)

m=d+1

d=1

and A denote the smallest σ -algebra generated by A. We will call μ : A → H a Borel measure if μ is countably additive function μ : A → H and μ(∅ ) = 0. For more information on RN , see, e.g., the book [15]. As RN has a countable basis, RN is automatically separable. Moreover, convergence in RN is equivalent to coordinate-wise convergence in the usual topology. Finally, RN is a Fréchet space, i.e., RN is locally convex, metrizable and complete. We shall denote the topological dual of RN by RN . It can be easily checked that (see, e.g., item 5 on page 3 of [15]) 0 N RN 0 = {x ∈ R : x has finite length}.

We will now introduce a locally convex topology Lc on RN which is usually called the topology of uniform convergence on 0 compact sets. We let Lc be the topology generated by

Uε,K (0 ) = {y ∈ RN 0 : sup |y (x )| < ε}, x∈K

N where y : RN 0 → R is continuous and K is an arbitrary compact set in R . In this section, we will generalize the main theorem of Subsection 1.2.2 in [15] to the case when the positive definite function on RN is quaternion-valued.

Theorem 6.1. Fix i ∈ S and let j ∈ S be such that 1, i, j, ij is a basis of H. If μ = μ1 + μ2 j is a q-positive probability measure on RN , then 0

f (x ) = μ ˇ (x ) = where (y, x ) =

∞



RN 0

n=1 xn yn ,

ei(y,x ) dμ(y )

for

x ∈ RN ,

(6.2)

satisfies the following:

(1) f is positive definite on RN . (2) f (0 ) = 1. (3) f is continuous at 0 in the Lc topology. Conversely, if f : RN → H satisfies conditions (1)-(3), then there exists a q-positive probability measure μ = μ1 + μ2 j such that (6.2) holds. Moreover, in this case for fixed i and j, the measure μ is uniquely determined. Proof. The strategy of the proof will be to follow the proof of Theorem 1.2.2 in [15] using Theorem 4.5 with G = Rd in place of Bochner’s theorem on Rd for d = 1, 2, . . .. The proof is broken into steps. Step 1. Show that conditions (1)–(3) are sufficient. If f is continuous at 0 in the Lc topology, then

f (x1 , . . . , xd , 0, 0, . . . ) is continuous at 0d = (0, . . . , 0 ) ∈ Rd for d = 1, 2, . . .. Moreover, as f is positive definite RN , it is immediate that f |Rd is positive definite on Rd for d = 1, 2, . . . , i.e., the function g : Rd → H given by

g( x 1 , . . . , x d ) = f ( x 1 , . . . , x d , 0 , 0 , . . . ) is positive definite. Consequently, we may use Theorem 4.5 with G = Rd to deduce the existence of a system of q-positive measures

μ (1 ) , . . . , μ (d ) , . . . , where

μ( d )

(6.3)

is a uniquely determined (relative to i and j) q-positive measure on

f ( x1 , . . . , xd , 0, 0, . . . ) =





···

Rd

ei(y1 x1 +...+yd xd ) dμ(d ) (y1 , . . . , yd ).

Rd

such that (6.4)

In view of the aforementioned uniqueness, the system of q-positive measures in (6.3) is consistent, i.e., for any Borel set E of Rd ,

μ(d ) (E ) = μ(d+1) (E × R ) for d = 1, 2, . . . . Consequently, we can let μ ˜ : A → H, where A was defined in (6.1) be given by



μ˜ E ×

∞ 



R

= μ ( d ) ( E ).

m=d+1

As μ(d) is countably additive, μ ˜ is also countably additive. Write

μ˜ = μ˜ 1 + μ˜ 2 j,

(6.5)

D. Alpay et al. / Applied Mathematics and Computation 286 (2016) 115–125

125

where

μ˜ 1 : A → [0, ∞ ) and μ˜ 2 : A → Ci . It follows from the countable additivity of μ ˜ that μ ˜ 1 and μ ˜ 2 (and its positive measure components via the Jordan decomposition) are countably additive. Thus, we may use the Hahn–Kolmogorov extension theorem (see, e.g., Theorem 1.14 in [7]) to obtain

μ1 : A → [0, ∞ ) and μ2 : A → Ci , where A is the σ -algebra for RN defined in the beginning of the section. If we put

μ = μ1 + μ2 j, then we obtain a unique measure μ : A → H such that

μ|A = μ˜ . Since μ(d) is a q-positive measure on Rd , μ ˜ is q-positive on the algebra A and consequently the extension μ is q-positive on the σ -algebra A . Thus, μ is a q-positive measure on RN . Moreover, as each μ(d) is uniquely determined q-positive measure (relative to i and j) we get that μ ˜ is uniquely determined and thus so is μ. Step 2. Show that conditions (1)-(3) are necessary. The verification of condition (1) can be completed in much the same way as in Theorem 4.5. As μ is a q-positive probability measure, condition (2) holds automatically. The verification of (3) follows from the observation that if we write f = f1 + f2 j, where f1 and f2 are Ci -valued functions on RN 0 , then

| f (x ) − f (x˜)|2 = | f1 (x ) − f1 (x˜)|2 + | f2 (x ) − f2 (x˜)|2 . One can then check the asserted continuity of the component functions f1 and f2 in the same way as appeared on page 8 of [15].  Acknowledgment D. Alpay thanks the Earl Katz family for endowing the chair which supported his research. D. Kimsey gratefully acknowledges support by the ISF within the ISF-UGC joint research framework (grant No. 1775/14). F. Colombo and I. Sabadini acknowledge the Center for Advanced Studies of the Mathematical Department of the Ben-Gurion University of the Negev for the support and the kind hospitality during the period in which part of this paper has been written. References [1] A. Badrikian, Remarques sur les théorèmes de Bochner et P. Lévy, in: Proceedings of the Symposium on Probability Methods in Analysis (Loutraki, 1966), Springer, Berlin, 1967, pp. 1–19. [2] W. Banaszczyk, Pontryagin duality for subgroups and quotients of nuclear spaces, Math. Ann. 273 (4) (1986) 653–664, doi:10.1007/BF01472136. [3] Y.M. Berezansky, Y.G. Kondratiev, Spectral Methods in Infinite-Dimensional Analysis, volume 12/1 of Mathematical Physics and Applied Mathematics, vol. 1, Kluwer Academic Publishers, Dordrecht, 1995, doi:10.1007/978- 94- 011- 0509- 5. Translated from the 1988 Russian original by P. V. Malyshev and D. V. Malyshev and revised by the authors. [4] Y.M. Berezansky, Y.G. Kondratiev, Spectral Methods in Infinite-Dimensional Analysis, volume 12/2 of Mathematical Physics and Applied Mathematics, vol. 2, Kluwer Academic Publishers, Dordrecht, 1995, doi:10.1007/978- 94- 011- 0509- 5. Translated from the 1988 Russian original by P. V. Malyshev and D. V. Malyshev and revised by the authors. [5] S. Bochner, Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse, Math. Ann. 108 (1) (1933) 378–410, doi:10.1007/BF01452844. [6] N. Bourbaki, Integration. II, Chapters 7–9, Elements of Mathematics, Springer-Verlag, Berlin, 2004. Translated from the 1963 and 1969 French originals by Sterling K. Berberian. [7] G.B. Folland, Real Analysis, Pure and Applied Mathematics, second ed., John Wiley & Sons, Inc., New York, 1999. Modern techniques and their applications, A Wiley-Interscience Publication. [8] I.M. Gelfand, N.Y. Vilenkin, Generalized Functions, vol. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964. Applications of harmonic analysis, Translated from the Russian by Amiel Feinstein [1977]. xiv+384 pages. [9] A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Resenhas 2 (4) (1996) 401–480. Reprint of Bol. Soc. Mat. São Paulo 8 (1953), 1–79 [MR0094682 (20 #1194)]. [10] G. Herglotz, Über Potenzreihen mit positivem, reellen Teil im Einheitskreis, Leiziger Berichte, Math.-Phys. Kl 63 (1911) 501–511. [11] T. Hida, H.-H. Kuo, J. Potthoff, L. Streit, White Noise, of Mathematics and its Applications, vol. 253, Kluwer Academic Publishers Group, Dordrecht, 1993. An infinite-dimensional calculus. [12] T. Hida, S. Si, Lectures on White Noise Functionals, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008, doi:10.1142/9789812812049. [13] Z. Huang, J. Yan, Introduction to infinite dimensional stochastic analysis, of Mathematics and its Applications, vol. 502, chinese, Kluwer Academic Publishers, Dordrecht; Science Press, Beijing, 20 0 0, doi:10.1007/978- 94- 011- 4108- 6. [14] Z. Sasvári, Positive Definite and Definitizable Functions, of Mathematical Topics, vol. 2, Akademie Verlag, Berlin, 1994. [15] N.N. Vakhania, Probability Distributions on Linear Spaces, North-Holland Publishing Co., New York-Amsterdam, 1981. Translated from the Russian by I. I. Kotlarski, North-Holland Series in Probability and Applied Mathematics. [16] A. Weil, L’intégration dans les groupes topologiques et ses Applications, Actual. Sci. Ind., no. 869, Hermann et Cie., Paris, 1940. [This book has been republished by the author at Princeton, N. J., 1941.].