1
Research supported by the Air Force Office of Scientific Research under
Grant AFOSR-72-0S96.
~
~OME
~
REMARKS ON THE EQUIVALENCE
OF GAUSSIAN PROCESSES
1
A. Gual tierotti·· and S. Cambanis
Depaptment of Statistios
Vnivepsity of Nopth CapoZina at ChapeZ HiZZ
Institute of Statistics Mimeo Series No. 874
June, 1973
Unclassified
Se curlty CI assificp tion
-
C:>CUMENT COK-ROL QAT A • R&D
(S""u,lty "I".sltlclttt,·n of title, b"d,· of
,. ORIGINATING ACTIVITY (('>'P0,.te autho,)
b.f,,,,,t Hn·d Ind,,_in,: "nnot"tlon
mu~t
be enle,ed when fhe ove,,,1/ ,,,,,o,t ,. cl•••ltled)
a.,
~I:"O~T
SECURITY CLASS'I'ICATION
Unclac::c::i.fiArl
epartment of Statistics
University of North Carolina
abo
G~OUP
3. REPORT TITLE
Some Remarks on the Equivalence of Gaussian Processes
4. DESCRIPTIVE NOTES (TyP" of ,epo,tand incillsive da~R)
Technical Renort
s· AU THORIS) (FI,st name, middle inltla'. 'ast na",e)
A. Gualtierotti and S. Cambanis
e.
REP OR T DA TE
r
b • NO. 0ioEF'S
7a. TOTAL NO. OF' PAGES
June 1973
11
ea. CONTRACT OR GRANT NO.
AFOSR-72-0596
9a. ORIGINATOR'S REPORT NUMBERIS'
Institute of Statistics Mimeo Series
No. 874
b. PROJ EC T NO.
c.
9b. OTHER REPORT Nols, (Any othe, numbe,. that may be ."'lfted
thl. repo,t)
d.
'0. DISTRIBUTION STATEME
~T
The distribution of this report is unlimited.
II. SUPPLEMENTARY NOTE:;
12. SPONSORING MILITARY ACTIVITY
Air Force Office of Scientific Research
'3. ABSTRACT
The norms in a reproducing kernel Hilbert space corresponding to
equivalent Gaussian processes are characterized, and some representations
are obtained for all Gaussian processes equivalent to a fixed Gaussian process.
KEY WORDS:
Gaussian processes, reproducing kernel Hilbert spaces,
representation of equivalent Gaussian processes.
Unclassified
Sl'curltv
Clil~SlhcRlIon
SOME REMARKS ON THE EQUIVALENCE
OF GAUSSIAN PROCESSES
A. GUL\LTI EROTT I
ABSlRACT.
AI\[)
s.
CAMBAN IS1
The norms in a reproducing kernel Hilbert space corresponding
to equivalent Gaussian processes are characterized, and some representations are obtained for all Gaussian processes equivalent to a fixed .
Gaussian process.
AMS (MOS) subject classifications (1970).
Key words and phrases.
1
--
Primary
60Gl5.
Gaussian processes, reproducing kernel Hilbert
spaces, representation of equivalent Gaussian
processes.
Research supported by the Air Force Office of Scientific Research under
Grant AFOSR-72-0596.
2
X = {X ' tET} and Y = {Yt , tET} be real, zerot
mean Gaussian processes with respective covariances R and Ry ' defined
X
on a probability space (Q,A,p), where T is an arbitrary index set.
1.
INTRODUCTION.
Let
T
x and Py the probabilities induced on (RT,C(R )) by
Denote by P
T
P, X and Y respectively, where C(R ) is the a-algebra generated by the
T
cylinder sets of the set R of real functions on T. L (X) and L (Y)
2
2
will denote the subspaces of L (Q,A,p) generated by X and Y respectively,
2
and H(X) and H(Y) the corresponding reproducing kernel Hilbert spaces (RKHS's).
The associated canonical isometries will be denoted by Ux and Uy respectively
(UXX t
=
RX(t,o), tET, and similarly for Uy ).
We say that the processes
and
Yare equivalent, or that the probabilities
PX~
Py ' if Px and
Px
and
X
Py are equivalent,
Py are mutually absolutely continuous.
The following
properties are well known:
(i)
P
x
~
Py if and only if Yt
= FXt ,
tET, where
F is an equivalence
operator from L (X) to L (y) (i.e., F has bounded inverse and I-F*F is Hilbert2
2
Schmidt) [6]; or equivalently if and only if Y = X - AX t , tET, where A
t
t
is a Hilbert-Schmidt operator in L2 (X) and I-A has bounded inverse, and the
equality is in law, i.e.
(ii)
If Px
~
Py=P(I-A)X [9].
Py then
sH(X) = sH(Y), where
s
indicates that what
follows is considered as a set and not as a space [7, p. 181].
The first question considered in this note is the converse of (ii), i.e.,
if
X and
Y have the same RKHS's, under what additional condition on the
RKHS's are they equivalent?
The answer is given in Propositions I and 2 and
results in a characterization of the norms in a RKHS corresponding to
equivalent Gaussian processes.
The fact, mentioned in (i), that all Gaussian processes equivalent to Xt
3
are of the form
X - AX , with A a Hilbert-Schmidt operator in L (X),
2
t
t
raises the problem of expressing AX t in a more explicit way in terms of
the process
2.
X.
This is done in Propositions 4, 5 and 6.
THE RKHS OF EQUIVALENT GAUSSIAN PROCESSES.
Here, and in the next section,
we adopt the notation of the introduction.
P ~ Py if and only if sH(X) = sH(Y) and (a) the identity
x
on sH(X) = sH(Y) is an equivalence operator from H(X) to H(Y), or (b) for
PROPOSITION 1.
J
every f in sH(X)
Ilf
for some
PROOF.
For every
* -1
= <UX J
11~(y)
A~H(X)@H(X)
Ilf 11~(x)
=
+
<f
@f
,1\~(X)@H(X)
which is sYmmetric and such that - RX «
Suppose first that
sH(X) = sH(Y)
~EL2(Y) we have <~'Yt>L
uy~,Xt>L2(x)'
LetF
*
(Y) =
and
J
A.
is an equivalence.
(Uy~)(t) = (J-lUy~)(t) =
2 * -1
= UXJ Uy .
.
Slnce
J
is an equivalence,
so is J-l,and since
Ux.U y are unitary, F* is an equivalence and so is
It now follows from
<~'Yt>L (Y)
.
in
L2 (Y), that
Y = FX t ·
t
Conversely, suppose that
F.
= <F*~,Xt>L (X) = <~,FXt>L (Y)' for all ~
2
2
Thus Px ~ Py .
2
P ~ P
Th
Y
FX
h
X
Y·
en
t = t were F is an
L2 (X) to L (Y). For every f in H(Y) we have
2
equivalence operator from
*
* *
f(t)= <Uyf, Yt>L (Y) = <F U/'Xt>L (X) = [UXF*U~:f] (t) = [JUXF*U;t] (t). Thus
* *2
2
JUXF Uy = IH(y) and J = Uy(F*)-lU~. Since F is an equivalence, so is
* -1
.
(F)
and Slnce UX,U y are unitary, J is an equivalence.
Finally, (b) is equivalent to (a) as it follows from Property (i) and the
fact that Hilbert-Schmidt operators on RKHS's have kernels in the direct
product of the considered RKHS's [1].
The characterization of Proposition 1 is particularly useful if the
.~
elements in the common RKHS can be obtained in the way described in the
following Proposition.
4
PROPOSITION 2.
space and
sH(X)
Suppose there exists a pair (H,L), where H is a Hilbert
L a unitary map from
H to H(X).
Then
P '" Py if and only if
X
= sH(Y) ·and for all h in H,
(1)
where
K is a self-adjoint, Hilbert-Schmidt operator on
REMARK 1.
IILhll~(y) =
(b)
\Lh,Lht)H(Y)
I/hl
/~
+ <Kh,h)H
= \Lh,LH)H(X)
PROOF OF PROPOSITION 2.
equivalence and
~
in
J
(W: H(X)
H there is a
for h in
H,
follows from
cr(K).
or
= <h,h')H
+ <LKh,Lht)H(X)
+ \Kh,h')H
P '" Py ' Then J is an
x
J = UW, with W= (J*J)~ and U
can be decomposed as
+
H(X)
and U: H(X)
g
in
H such that
IIJLhIIH(y)
S
'
Suppose first that
L* WLg, for some
obtained as
<
Condition (1) is equivalent to
(a)
. unitary.
H such that -1
g in
H.
+
H(Y)). Since
Lh
= WLg.
Set
S
W is onto, for every h
Thus every h in
H can be
= L*WL. Then J = ULsi* and thus
= IluLSL*LhIIH(y) = IILShIIH(x) = Ilshll H . Now it
= L*U*JL that S is an equivalence, since it is obtained from
an equivalence operator
- S*S is equal
H
to a self-adjoint, Hilbert-Schmidt operator -K that does not have -1 among its
eigenvalues.
J
by "unitary multiplication."
and (1) follows from
T: H + H(Y)
by
sH(X)
= sH(Y) and
k
T{(IH+K)~h}
'
(1)
holds.
Define a unitary
= JLh. This definition makes sense since T
is obviously onto and by (1), IIT{ (IH+K)~h}
J.
+ \Kh,h)H
(a) of Remark 1.
Conversely, suppose that
But then
I
Hence, as
IIJLhll~(y) = IIshll~ = \(I+K)h,h)H = IIhll~
operator
Thus'
11~(y)
=
IIJLhll~(y)
=
II (IH+K)~hll~
.1- *
= T(I H+K)2L
,where T and L* are unitary and (IH+K) ~ is an
.
5
equivalence.
Hence,
REMARK 2.
is an equivalence and
J
P
x
~
Py.
The existence of the assumed pair (H,L) in Proposition 2 is not
as restrictive as it appears.
In fact, whenever
HeX) (or equivalently
L2 (X))
is separable, this assumption is satisfied and one can take H to be an L
2
space. This follows from Theorem 2 of [3]. Indeed, if H(X) is separable, then,
for an arbitrary measure space
(E,E,~)
L2(~)
is separable and infinite
Xt = J ~t(u)dZ(u) ,where ~t€L2(~)
E
Z is an orthogonal random measure on E such that L (Z) = L (X). This
2
2
dimensional, we have for all
and
such that
t
in
T,
L2 (X) such that A~t = Xt ' tET,
H(X) defined by L = UXA is unitary. The Hilbert space
implies that there is a unitary map
and clearly L:
L2(~) +
H is clearly non-unique.
a natural way.
L2(~) +
A:
However in some
~ecific
cases,
H can be chosen in
In fact, most of the known RKHS's are obtained in this way.
We
list some examples below.
EXAMPLES.
1)
Let
X have orthogonal increments on [0,1] and start almost
surely at the origin, with RX(s,t) = F(SAt),
H(X) =
{J
t
~(u)dF(u), ~€L2(dF)},
° H(X)
. .defined by
L: L2 (dF)
+
the Wiener process (F(u)
2)
Let
~
<J
•
°
J ~dF>H(X)
~dF,
[L~](t) =
u).
X have the covariance
Then
F continuous.
•
Jto~dF
=<~'~>L (dF) , and
2
is an isometry.
This example includes
°
RX(s,t) = F(sAt) G(svt), where
F and
G
are continuous with bounded variation on [0,1], F is strictly positive, except
at the origin, and
G is strictly positive.
Suppose further that
H(u) = F(u)/G(u)
t
is strictly increasing.
Then H(X) = {G(t)J ~(u)dH(u), ~€L2(dH)} and
..
° Thus
<G(o)l ~dH, G(O)l ~dH>H(X) = <~'~>L2[0,1]
t
[L~](t) = G(t) J~dH
3)
Let
is
o
L: L2 (dH)
+
H(X)
defined by
unitary.
X be a linear operation on a stationary process with spectral measure
00
~.
Then
RX(s,t) =
J ~s(A)~t(A)d~(A)
_00
and if H is the closure in
L2(~)
6
00
of the linear span of {<j>t,tETl then
H(X) =
{I
<j>t~d\1' <j>EH} and <I<j>.<j>d\1
_00
I<j>.iiJ"d\1>H(X) =
00
Thus L: H + H(X) defined by [L<j>](t) = I <j>t<j>d\1 is unitary.
=<<j>,1JJ> L ().
_00
2 \1
itA
includes the case where X is stationary (<j>t (A) = e
) and then H = L2 (\1)
This
if
T = R.
4)
Let
X be a linear operation on a harmonizable process with two-dimensional
spectral measure
r.
Then RX(s,t) =
00
II
<j>s(u)~t(v)dr(u,v)
and if H is the closure
_00
in the Hilbert space
A2 (r) [5] of the linear span of
{<j>t,tET} then
00
H(X) =
{II
<j>t(u)~(v) dr(u,v),
<j>EH} and
_00
<II<j>.<j>dr,II<j>.~dr>H(X) = <~'~>A (r) = IIoo<j>(u)ijJ"(v)dr(u,v). Thus L: H +H(X) defined
by [L<j>](t) = II <j>t(u)~(v)dr(u,v) is unitary. This includes the case where X is
00
2-00
-00
harmonizable
REMARK 3.
(<j>t(u) = e
itu
)
and then
H = A (r) if T = R.
2
The existence of the pair (H,L) as described in Proposition 2 is
easily seen to be equivalent to the existence of a representation of the covariance
RX of the form
RX(s,t) = <<j>s,<j>t>H where
{<j>t,tET} 5.. H.
Proposition 2 can be also expressed in terms of covariances and it then
contains as particular cases the results of [10] and [8].
PROPOSITION 3.
space and
Suppose there exists a pair (H,L), where
L a unitary map from
H to H(X).
RX(s,t) = <<j>s,<j>t>H (see Remark 3).
Then
Px
Let
~
Py
H is a Hilbert
<j>t = L*RX(t,·) or
if and only if
(2)
where
M is self-adjoint, Hilbert-Schmidt and such that
PROOF.
Assume first that
operator from
L2 (X)
to L2 (y).
Px
Yt = FXt where F is an equivalence
*
It follows that Yt = FUXL<j>t
and
~
Py •
a(M) < 1.
Then
* .
Ry(s,t) = <L*UxF*FU~L<j>s,<j>t>H . Thus (2) is valid with M = L* UX(I L (X)- F* F)UXL
2
Since
I
and UX,L
L2
(X) - F*F
is self-adjoint, Hilbert-Schmidt
are unitary, it follows that
with a(I
(X)- F*F) < 1,
L2
M is self-adjoint Hilbert-Schmidt with
7
•
a(M)
< 1.
Conversely, assume that (2) is valid.
<McjlS,cjlt>H
*
*U x ,x >L
= <UX'LML
x s t
(X)
and (2) is written Ry(s,t)
=
2
* *U ) \ ,x >L (X) . Since M is self-adj oint, Hilbert-Schmidt
(X) - UXLML
x
t 2
L2
with a(M) < 1, and UX,L are unitary, it follows that U~LML*Ux is self-adjoint,
=
<(I
. bert-Schmidt.
Hll
Fa
square root
* *U
I L (X) - UxLML
x
Also
2
is an equivalence.
*
= UxL(I
H-
M)L *Ux > 0 and hence its
We then have <Ys'Yt>L (Y)
=
Ry(s,t)
=
2
=
<FOXs,FOXt>L2(X) which implies that the map
operator
FOX t
U from
L;(X) to L2 (Y) and then Yt
Thus P ~ Py .
equivalence.
=
Yt extends to a unitary
+
where
FXt
F
=
UFO
is an
x
REMARK 4.
The relationship between the operators
F, K and M of Property
(i) and Propositions 2 and 3 respectively is as follows
F
*
= U[UXL(I
H+
~
=
* ~
K) -1 LUx]
*
= U[UXL(I
H-
L*UX[(F*F)-l - IL2(X)]U~L
*
M = L*Ux(I L (X)- F*F)UXL
=
2
=
M)L *UX] ~
(I H- M)-l_ I H
I H - (I H+ K) -1 •
We also have
Ry(S,t)
PROOF.
Ry(s,t)
sH(X)
=
sH(Y)
= <JLcjls,JL~t>H(Y)
is unitary,
--
we have
(I H- M)cjlt •
F and
M are derived in the proof of
Ry(t,o) E H(X).
By Remark 1 we obtain
.
Since
{cjlt' tET} is complete in
Ry(S,t)
and since
=
Rx(s,t)
=
H and hence
<cjls,cjlt>H
M.
L*Ry(t,o). Then
= <RX(s,o),
Ry(t,o»H(Y)
=
= <ep ,ep
RX(s,t)
<Ry(S,o), RX(t,o»H(X)
Rx(s,t)
~t =
Let
we have
s t >H
{RX:(t,.), tET} is complete in H(X)
Since
=<cjls' (IH+KHt>H'
<eps,(IH+K)~t>H
We now have
=
It then suffices to derive the relationship between K and
= <~s'~t>H'
<eps,cjlt>H =
~t
where
The expressions relating
Proposition 3.
Since
= <~s'~t>H
ep
t
= (IH+KHt
and
L
.
= <L~s,Lcjlt>H(X) = <~s,ept>H =
it follows from (2) by inspection
8
M= I
that
=
3.
(IH-
H
M)<p t
- (I
+
H
K)
-1
.
Hence
K=
(I H- M)
-1
-
I H and also
1);
t
=
(IH+ K)
-1
<P t
=
·
REPRESENTATION OF EQUIVALENT GAUSSIAN PROCESSES.
When a pair (H,L) as
described in Propositions 2 and 3 exists, then one can obtain explicit representations of the process AX
in Property (i). Here it is more appropriate to
t
consider the unitary map V: H + L2 (X) related to L by V = U~L.
PROPOSITION 4. Suppose that there exists a pair (H,V), where H is a Hilbert
space and
V a unitary map from
H to
L (X). If A is a Hilbert-Schmidt
2
' where A is a Hilbert-Schmidt operator in
L2 (X), then AX = VAh
t
t
Hand ht = V* Xt '
- is Hilbert-Schmidt in
PROOF. If -A = V*AV, then A
operator in
= W *AVht = VAht
EXAMPLES.
and AX t
t
= AVht =
.
Let
5)
1
V<p
=I
1_
0
defin~d by
Hand AX
X be as in Example 1.
<p(u)dXu '
Then V: L (dF) + L2 (X) is
2
Consequently ht = It' the indicator function of [O,t]
= l[AI t ] (u)dXu '
type with kernel
Since a Hilbert-Schmidt operator in L2 (dF) is of integral
a(u,v) in L2(dF~F) we finally have
AX
t
= II Ita(u,v)dF(v)dX
o 0
u
This result is obtained for the Wiener process in [9].
6)
RX(S,t)
Consider the case where the covariance of X has the representation
=I
<ps(u)~t(u)d~(u). with (E,E,~)
a finite (for simplicity) measure space
E
and <P t E L2(~)' This includes both Examples 2 and 3. Then there is an orthogonal
random measure Z on E such that Xt = J <Pt(u)dZ(u) [5]. Let H be the closure
E
in L2 (~) of the linear span of {<p , tET}. Then V: H + L (X) is defined by
2
t
V<p = I </>(u)dZ(u). Consequently ht = <P t and AX = J[A<P ] (u)dZ(u). As in Example
t
t
E
E
5, A is of integral type with kernel a(u,v) in L2(~x~) and finally we have
AXt
In the case of Example 2,
= IIa(u,v)~t(v)d~(v)dZ(u) .
EE
H = L (dH) and AX
2
t
1
t
= G(t)J J a(u,v)dH(v)dZ(u).
·0 D
9
7)
Consider the case where the covariance of X has the representation
= II
~s(u)~t(v)dr(u,v) with (E,E) a measurable space, r a twoEE
dimensional spectral measure on ExE and ~t in AZ(r). Then there is a random
RX(s,t)
Xt = I~t(u)dZ(u) [5]. Let H be the closure
E
A2 (r) of the linear span of {~t' tET}. Then V: H + LZ(X) is defined by
Z on
measure
in
I
V~ =
E such that
~(u)dZ(u).
Consequently ht = ~t
is a Hilbert-Schmidt operator in AZ(r).
E
and AX t = I[A~t](u)dZ(u), where A
T
However, no kernel representation of
A seems to be available because
AZ is a more complicated space than LZ' Never. theless, as follows from (i) of the Lemma at the end of this section, if E is
an interval, A is the limit in the operator norm of a sequence of Hilbert-Schmidt
operators
in
with kernels
LZ(~)
{a }
00
n n=l
and we thus have
AX t = lim IffI ~t(v)a(w,u)dr(v,w)}dZ(u)
n+oo E EE
n
where the limit is in LZ(P).
Consider now the important particular case where
there is a measurable process
{Z , uEE} and a measure
u
the Lebesgue measure such that
Z(B) = ~Zud~(U).
IRZ(u,u)d~(u) <
00
~
on
E equivalent to
and for every
BEE,
E
Then
Xt
=
l~t(U)Zud~(U) and
AX t = lim Ig~n) (u)Zud~(u)
n+oo E
where
g~n)(U) = II~t(v)a (w,u)RZ(v,w)d~(v) d~(w) is in LZ(~), the integral is
EE
n
defined almost surely, i.e., on the paths of
Z, and the limit is in L2 (P).
As
particular cases of this example we obtain the following representations.
PROPOSITION 5.
Let
X be mean square continuous,
Hilbert-Schmidt operator on
LZ(X).
T an interval and
Then
AX t = Igt(u)X d~(u) = lim Igt(n) (u)X d~(u)
T.
u
n+oo T
u
where the measure
gtEA2(RX
.~x~),
~
on the Borel sets of
T satisfies (ii) of the Lemma,
(n)
the first integral is defined in quadratic mean,gt
ELZ(~)'
A a
,
10
~
the second integral is defined almost surely, i.e., on the paths of
X, and
the limit is in
L (P).
2
X has a measurable modification which is henceforth considered.
PROOF.
~,
There exist finite measures
JRX(t,t)d~(t) <
that
00
[2].
equivalent to the Lebesgue measure, and such
Let
= RX(u,v)d~(u)d~(v).
dr(u,v)
T
Since
X is
Xt = {~t(U)Xud~(U) with
The result then follows from the last case considered in
mean square continuous, it has a representation
{~t'
tET} c A2 (r) [2].
Example 7.
PROPOSITION 6.
Let
o=0
X be mean square continuous on [0,1], X
RX of bounded variation on [0,1] x [0,1].
Then
J gt(u)dXu = lim
°
n~
gtEA2(d2Rx), ~ corresponds to
are of the form
PROOF.
A be a Hilbert-Schmidt operator on
1
AX t =
where
Let
a.s., and
The
(n)
gt
(u) =
dZR
X
Jgt(n) (u)dXu
as in (i) of Lemma,
l 6an(w,u)d RX(V,w)
t
1_
Z
g~n)ELz(~)
, and anELZ(~x~).
proof is obvious from Example 7 and the observation that
1
Xt =
J It(u)dX
o
u
, It the characteristic function of the interval [O,t], i.e.,
~t = It'
LEMMA.
Let
E be an interval, E its Borel sets, r
spectral measure on ExE, and
~
(i)
~inite,
=R(u,v)d~(u)d~(v),
where
E by
~(B)~
on
LZ(~)'c
L2(~)
R is a covariance and
~
co
and are such that K
n
A (r).
Z
PROOF.
~
a finite measure
J R(u,u)d~(u)<
00,
E
co
AZ(r), and there is a sequence of Hilbert-Schmidt operators {Kn}n=l
with kernels {kn}n=l' that are defined from Az(r) to
[Kn f](u) =
If
Irl (ExB) for all BEE, or if
on E, equivalent to the Lebesgue measure and such that
then
two-dimensional
K a Hilbert-Schmidt operator on AZ(r).
is the finite measure defined on
dr(u,v)
(ii)
a
Both (i) and (ii) imply that
LZ(~)
+
c A2 (r)
L2(~)
by
K in the operator norm in
and that there is a sequence
•
11
00
{fn}n=l in
L2(~)
which is orthonormal and complete in A2 (r).
is shown in [2] and for (i) it is shown as Theorem 2 of [4].
For (ii) this
In the sequel <.,.> and 11'1 Idenote inner product and norm in A2 (r). Since
K is Hilbert-Schmidt we have . Lon
~oo ,m=l I<Kfn' f m>1 2 < 00. For every fEA 2 (r) we have
l<f,f >f and thus Kf = ,00 l<f,f >Kf = ~oo 1{~00 l<Kf ,f ><f,f >}f .
f = ~oo
Ln=
n n
Lon=
n n
Lo m= Lon=
n m
n m
· is in L2 (~x~), it defines
kN(u, v) = IN
n,m= l<Kf,
n f m>fn (u)fm(v). Since kN
a (finite rank) Hilbert-Schmidt operator ~ on L2(~)' K is also defined from
N
N
A2 (r) to L2(~) by
[K f](u) = <f(.),kN(·,u» = I <Kf ,f ><f,f >f .
N
1 n m
n m
00
n,m=
Then
2
2
2
II Kf-KNf 11 ~ II f 11
2 1< Kfn , f m> 1
n,m=N+l
which implies that K + K in the operator norm in A2 (r).
N
Define
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1.
N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68
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2.
S. Cambanis, Representation of stochastic processes of second order and
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4.
5.
6.
7.
J. Neveu, Processus Aleatoires Gaussiens, Les Presses de l'Universite de
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8.
W.J. Park, On the equivalence of Gaussian processes with factorable covariance
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9.
10.
~
S. Cambanis and B. Liu, On harmonizable stochastic processes, Information and
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H. Cramer, A contribution to the theory of stochastic processes, Proc. Second
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DEPARTMENT OF MATHEMATICS AND STATISTICS, CASE WESTERN RESERVE UNIVERSITY,
CLEVELAND, OHIO 44106
DEPARTMENT OF STATISTICS, UNIVERSITY OF NORTH CAROLINA, CHAPEL HILL, N.C. 27514