Interpolation and prediction problems for connected compact abelian groups

Integr. equ. oper. theory 40 (2001) 212-230 0378-620X/01/020212-19 $1.50+0.20/0 �9 Birkh/iuser Verlag, Basel, 2001

I Integral Equations and Operator Theory

I N T E R P O L A T I O N A N D P R E D I C T I O N P R O B L E M S F O R C O N N E C T E D C O M P A C T A B E L I A N G R O U P S .

MARISELA DOMINGUEZ

Extensions of the Nehari theorem and of the Sarason commutation theorem are given for compact abelian groups whose dual have a complete linear order compatible with the group structure. As a special case a version of the classical interpolation theorem due to Carath~odory - F~jer is obtained.

For these groups an extension of the Helson - Szeg5 theorem and integral representations for positive definite generalized Toeplitz kernels are given.

I. INTRODUCTION

Several classical questions concerning interpolation theory, moment problems, Toeplitz and Hankel operators, weighted inequalities for the Hilbert transform and pre- diction theory are closely related. And the theory of analytic functions in the circle gives the natural environment for these problems.

This theory can be extended in several ways. In one type of extension the unit disk is replaced by other plane domains, by domains on Riemann surfaces, and by domains in spaces of several complex variables. Another type of extensions is based on the following remark: Much function theory in the circle T ~ [0, 27r] is considered depending on group properties of the circle and its dual Z. Following this idea the theory of analytic functions can be generalized to compact abelian groups whose dual is ordered (see [i], [2], [3], [18], [20], [21], [26]).

In the present paper some of the mentioned classical problems are studied for a connected compact abelian group whose dual has a complete linear order compatible with the group structure.

Partially supported by the CDCH of the Universidad Central de Venezuela, and by CONICIT grant G-97000668.

Dominguez 213

Let G be a compact abelian group. A complex function "/on G is called a character of G if h,(x)l = 1 for all z E G and if V(x -5 y) -- 7(x)v(y) for all x, y C G.

The set of all continuous characters of G forms an abelian group F, the dual group

of G, if addition is defined by the usual product of functions on G: (7172)(x) = 71(z)V2(x)

for x c G and 3% "72 E F. Since G is a compact abelian group, its dual F is a discrete group

(cf. [26], Theorem 1.2.5). Let e be the neutral element of F.

Let dx be the Haar measure in G. For 1 _< p < oo let LP(G) be the Haar - Lebesgue space in G and let II-Ilp be the n o r m in LP(G).

For f E L 1 (G) and 3' E F consider the Fourier coefficients:

f'y(-x)f(x)dx= f'y- (x)f(z)dz. G G

Recall that an abehan group (E, .), with neutral element e, is said to be an ordered group if there exists a closed set E1 C E such that:

where

{o" E E : o" = "/6,7 E E1,6 E El} =- El:

z l n (s -b = {4,

Ei -1 = {~ E E : ~-i e El}.

Under these conditions if a, 6 E E we write cr < 5 when a-15 E El.

It is well known that F can be ordered if and only if G is connected (see [26], Sec.8.1). Suppose this is so, and suppose that a certain order has been chosen for F.

A function f C LI(G) is said to be of analytic type if f (7) -- 0 for all 7 < e. Let 1 < p < oo, the set of M1 functions of analytic type which belong to LP(G) will be denoted by H p. Observe that e E H p. The class H p depends on G and on the order we are considering. These spaces are ordered versions of the classical Hardy spaces.

Also let H 2- be the set of functions f E L2(G) such that f (7) = 0 for all V - e. Let P+ and P - be the orthogonal projections in L2(G) with range H 2 a n d / / 2 - respectively.

Theorems 1 and 2 are ordered versions of the Nehari theorem [23] and the interpo- lation theorem of Sarason [27] using these spaces instead of the classical Hardy spaces.

For g C L~176 let Mg denote the operator on L2(G) of multiplication by g.

214 Dominguez

The shift operators S~ on L2(G) are given by S.~f = 7 f for all f E L2(G) and 7 C F.

T H E O R E M 1. L e t T : H 2 ~ H 2- be an operator such that

P - S ~ T = TS~IH2 for every 7 >- e.

I f T is bounded then there exists g E L~(G) such that T = P-Mg and

Jlrtl = inf{l lg - 5 l I ~ : 5 c H ~} -- IM I~ ,

so the infimum is attained.

For any 7 E F, we define the operator S~ on H 2 by STf = 7 f -

A linear subspace D of H 2 is said to be invariant if S~D C D for all 7 > 0. A linear subspace K of H 2 is said to be coinvariant if S.~K C K for all 7 > 0.

We call a function %b E H 2 an inner function if Ir = 1 (almost everywhere on G). We call a function g C H 2 an outer function if

f log]g(x)ldx = log I / g(zDdxJ.

G G

Let r be a nonconstant inner function. Then K = H 2 O r 2 is a coinvariant subspace (where O stands for orthogonal difference).

For every closed subspace K of H 2 let PK be the orthogonal project ion in L2(G) with r a n g e / ( .

T H E O R E M 2. Given a nonconstant inner function r let K be the closed sub- space of H 2 given by K = H 2 @ r 2.

I f X is a bounded linear operator on K that commutes with PKSTtK for all 7 >- e then there is a function r e g ~ such that X = PKMr and ][r = [[Z[[.

Theorem 8.5.3. of [26] states that if D is an invariant closed subspace of H 2 which

contains a function g with ~(e) ~ 0 then there is an inner function r such that D is the

smallest closed invariant subspace of H 2 which contains ~. In the proof of this theorem it

is also obtained that D -- r 2. This representation also holds if there exists a minimal

% E F such that g(7o) # 0 for some g C H 2.

Remak 8.5.4. of [26] states that if F contains no smallest positive element and if

D is the set of all f E H 2 with f'(e) = 0, then D is an invariant closed subspace which

cannot be represented as H 2 G r 2, So this is not a general form of a coinvariant subspace.

However H 2 O r 2 is ~almost I~ a general form for a coinvariant subspace.

Dominguez 215

The classic problem of Carath~odory - F4jer [10] can be solved using the interpola- tion theorem of Sarason. An ordered version of this problem is solved:

T H E O R E M 3. For 7" > e set I = {2, E F : e < 2, <_ T} let K be the span of ~/ E I. Suppose F has a minimal positive element. Given a function c : I --+ C let X be the operator on K given by

: { if > < Then the following conditions are equivalent

(a) IIXll _< 1. (b) There exists r e H ~ such that IIr -< 1 and "~(~) : 42,) for evem 2, e •

One of the main applications of Nehari theorem concerns spectral estimation of stationary processes based on measured correlation coefficients and the related linear pre- diction. Helson and Szeg5 proved their theorems [19] in prediction theory and conjugate functions following the ideas of Nehari. Theorem 4 and Theorem 5 are ordered versions of these theorems. Related results can be found in [15] and [16].

The trigonometric polynomials are functions f : G --~ C, such tha t

f ( x ) = E "f(7) 7(x) ~/6F

and supp'f is finite. Let 7 ~ be the space of all trigonometric polynomials. Since G is compact, P is a dense subalgebra of the space of all continuous complex functions on G. 7) is also dense in LY(G), for 1 < p < oo (see [26], Sec. 1.5).

With ~ we will indicate the harmonic conjugate of a function v C L2(G):

= - i P + v + i P - v + i~(e).

The ordered Hilbert transform H is introduced using the Fourier coefficients, for f C 1) set

(HI)^( ') ') = - i sgn 2, f(2,)

where sgn "y = 1 if 2, ~ e and sgn 2' = - 1 if 7 < e.

If f C P then

(H f)^(2,) = - i (P+f)^(7) if 2, > e,

(Hf)A(2,) = i ( P - f ) ^ ( 2 , ) if 2, < e.

Therefore

Hf = - i P + f + i F - f ,

i.e., the conjugate function ] differs from H I only by a constant.

216 Dominguez

T H E O R E M 4. Let d# = wdx + d#~ where #, is singular to the Haar measure on G and w E LI(G). There exists a > 1 such that for every f E ~P

f <_ f I:(x)"d.(x) G G

if and only if p , = 0 and there exist u, v E L~ ( G) such that w = exp(u + ~:) and ]]v[]~ < ~/2.

R E M A R K 1. The results of the present paper show that for a connected compact abeIian group G the space H 2 is of special interest in the theory of integral representations of Hankel forms on G. These results also suggest that some other properties of classical Hardy spaces may hold for the ordered versions of Hardy spaces. It is natural to think that for G there must exist HankeI operators of finite type and that there must hold versions of the Kronecker and Adamjan - Arov - Krein theorems.

Also in this spirit some classical positive extension results of functions of one vari- able have recently been generalized to two or more variables (see [7], [9], [17]).

R E M A R K 2. Related topics for the PaIey - Wiener space and for weighted Bergman spaces with exponential type weights can be found in [24] and [25].

R E M A R K 3. For n >_ 1 let H2(T ~) denote the space o f f E L2(T ~) such

that f ( k l , . . . , k~) = 0 for all kj < O j = 1 , . . . ,n. In [12 t and [13] bidimensional versions of Nehari's theorem and Sarason's theorem are obtained for H2(T2), considering liftings of invariant forms defined in semi-invariant subspaces of the Hilbert spaces of scattering systems with two evolution groups. For H2(T=), with n > 1, the theory of Hankel operators in the torus T ~ presents striking differences with that on the circle T , it leads to versions of the Nehari, Adamjan - Arov - Krein and Kroneeker theorems and provides conditions for the existence of solutions of product Pick problems through finite Pick - type matrices (cf. [14]).

A very interesting approach to classical interpolation problems and the work of Sarason is being developped in [6] for the polydisk, it is based on recent results of Agler.

2. MAIN L E M M A S

The main analytic fact needed in the proofs of this paper is the version for groups of the classical theorem of F. Riesz on the factorization of H 1 functions given by Lamina 1.

For 1 <__ p < c~ let Ho p be the subspace of H v consisting of the functions with mean value zero.

LEMMA I.

(a) I r e e H~o and []r < 1 then there exist r e H 2, r e H2o with t[r -< 1, [[r < 1 such that r = r162

Dominguez 217

(b) The products r162 where r e H 2, r e Ho 2, I]r _< 1, 1tr <_ 1 are dense in the unit ball of Hlo .

Proof. To prove (a), take r e Ho 1 with I1r < 1. Define w = Ir where ~ > 0 is small enough, that I1r -< 1. Then w e LI(G),

w > 0 and

Thus By Theorem 8.4.3.

f log w(x)dx > -oo .

G

of [26] there is r E H 2 with ~ ( e ) r 0 such that w = Ir ~.

11r = (11r + e)v2 _< 1. Let r = r162 Clearly, r = r162 and

1r = 1r162 = 1r162 + e ) - ~ < Ir

therefore r e H 2 and 11r -< 1. Approximating r and r by trigonometric

r = r162 imply

Since r = 0 it follows that &(e) = 0. Part (b) follows immediately from (a).

polynomials in H 2, the equation

[]

I f /~ = (#~)~,z=1,2 is a 2 x 2 hermitian matrix of complex finite Borel measures defined on G then it is said tha t /~ is an hermitian complex finite Borel matricial measure on G. In this paper all the matricial measures considered are of this kind and they will simply be called matricial measures.

Set

and

For c~ = 0, 1, 2 consider

Pl = {~ e r : 7 > e},

I'2 = {V e P : 7 < e}

For every matricial measure tt = (#~)a,~=1,2 on G the following coefficient is intro- duced

= pl/flTd,121 , 2

G

the supremum is taken over all f l C ~1, f2 C 7~2 such that Ilfl]lml = 1, IIf211~= = 1 where [I . I]~ is the norm in L 2 ( G , # ~ ) for a = 1,2.

218 Dominguez

The following lemma is a part icular case of the general dual i ty principle for analo- gous extremal problems based on the well known Hahn - Banach theorem.

L E M M A 2. Let tt = (#~)~,p=1,2 be a matricial measure on G, such that d#~. = w~.dx for ~ = 1, 2. I f R(tt) <_ 1 then:

(a) #12 is absolutely continuous with respect to Haar measure on G. (b) There is a real valued function 0 such that

R ( t , ) = i n f { l l e x p ( - ~ - i0) w12 - ~11~: ~ �9 H ~}

where ~z = log ~ .

Proof. Suppose for the moment tha t logw~, �9 LI(G) for a = 1, 2. Theorem 8.4.3. of [26]

guarantees the existence of a function h~ �9 H u such tha t w ~ = [h~] 2 and h~-~(e) ~ 0. By theorem 8.5.2. of [26] we have tha t ha = r where r is an inner function

and g , is an outer function. Thus go �9 H 2 and w ~ = [g,[2. Let ~ = log w ~ and let 0~ be a real valued function such tha t g~ = e x p ( ~ + ion).

Set

It follows tha t

~0 = (~D 1 -F ~D2)/2 and 0 = (01 -I- 02)/2.

Therefore

= log v/-~w22 and gig2 = exp(p + iO).

f

R(t,) = su;] / exp(-v - iO) f lg~T~g~d,l~l * l

G

where A E Pz, f2 e 7)2, IIAJt~oo = 1, Its = 1.

Since l o g w ~ E LI(G) we have that

f log Ig.(x)ld~ >

G

so by the Corollary of Theorem 8.5.2. in [26] the set {fig1 : f l e 7)1} is dense on H 2 and the set {f-2292 : f2 e 7)2} is dense on Ho 2. Thus

sup[ f exp(-~ - iO) r162 R(~) I ]

G

where r E H 2, r E Ho 2, I1r = 1, I1r -- 1. Since the products r162 considered above are dense in the unit ball of H 1 (see

Lemma 1) it follows tha t

R(~) = s~pl [ e x ; ( - W - iO) r * 1

G

where r C HL I[r = I.

Dominguez 219

Let l : Ho 1 --~ C be defined by

z(r = / ~ x p ( - ~ - i0) r G

for every r C Ho 1. Because of ]l (r < R(/~) < 1 = []r we have that l is a linear functional on H 1. Hence exp ( -W)wl2 E H ~~ and d.12 = wz2dx.

Let L : L 1 (G) --+ C be the linear functional defined by

(r = / ~ x p ( - ~ - i0) Wl2r L

G

for every r C L 1 (G). From the Hahn - Banach theorem it follows that

R(/~) = [Ill[ = []n[~o~[]

= inf [ ] e x p ( - ~ - iO) w 1 2 - ~][oo. ~EH ~

Now consider the general case. For 5 > 0, ]og(w~ + 5) e L I (G) for a = 1, 2. L e t / ~ --- (.~a)~,a=l,2 be the matricial measure given by

and

"152 : #25"---~ : "12"

Then R(/~,) _< 1. So by the preceding there is a function 0~ such that

R(Iz , ) -- i n f { H e x p ( - ~ - i~,) w12 - ~Hoo: ~ E H ~176

where d.12 = w12dx and

,,~ = log v/(~i~ + 5)(~o22 + 5). Set qo = log ~ .

Taking L ~ weak-star limit point when 5 --+ 0, we obtain a real valued function 0 such that e x p ( - ~ - iO) E H ~ and

R(tz) = i n f { I l e x p ( - ~ - io) w12 - ~ct[~: r E H ~ } .

[]

L E M M A 3. Let /z = (#~z)~,z=I,2 be a matricial measure on G such that d # ~ = w ~ d x for a = 1, 2. Then:

R ( # ) <<_ 1 i f and only if/ '12 is absolutely continuous with respect to Haar measure on G and there exists h E H 1 such that

]w12 + hi <_ x/wnw22 a.e.

where w12 = d#12/dx.

220 Dominguez

Proo5 Suppose R(tt) _< I. Set ~ = i o g ~ and let ~ > 0. By Lemma 2, #~ is

absolutely continuous and there is ~ E H ~ such that

Iexp(-~- iO) w12 + ~I <- I + ~ a.e.

Taking L ~ weak-star limit point of ~e, when ~ --~ 0, we obtain ~ E H ~~ such that

l e x p ( - ~ - iO) w12 + 5] -< 1 a.e.

Define h : exp(~ + iO) ~ then h 6 H 1 and

1~2 + hi = l e x p ( - ~ - i0) (~1~ + h)lexp(~) exp(~) = v/-w-~lw22 a.e.

The other par t follows immediately. []

3. A N E X T E N S I O N O F T H E T H E O R E M O F N E H A R I

Recall tha t 7)1 is the space of all tr igonometric polynomials f such tha t s u p p f C F1.

P R O P O S I T I O N 1. L e t T : H ~ --+ H 2- be a bounded linear operator such that

P - S ~ T = TS~I~ for every ~/E F1.

Then there exists g E L2(G) such that

T f = P - M g f for all f E 7)1.

Proof. Let g = Te. The result follows since for all ~/E F1

P-Mg'~ : P-S,~Te : T%

then

Even more

Proof.

and

P R O P O S I T I O N 2. L e t T : H 2 --+ H 2-, i f there is 9 E L2(G) such that

T f P - M g f for all f E 7)1

P - S ~ T = TS~IH~ for every ~ E F1.

Take 5 E F1, a E Fo, 9' E Fo. The result follows from

(P-S~TS, or-') = (P-g5, 7-~ -~) = (7g5, cr-~} : <TSvS, cr-~}

[]

Dominguez 221

PROOF OF THEOREM i.

By homogeneity we can suppose IITIi = I. From Proposition I, there is go E L2(G) such that

T f = P - M g o f for all f e 7)1.

Consider the matricial measure It = (#~e)a,~=l,2 given by

d # n : d # = : dx

and

d~l 2 ~-- ~ : --godx. Then R ( , ) = IITI[.

If IOTII _< 1, from Lemma 3 it follows that there exists h C H 1 such that

I 9 o + h l < IITII a e.

Define g = go + h. Thus g E L ~ ( G ) and

s -1) = ~(7 -1) for every 7 6 F1.

From Proposition 2 we have that

<ra, : o(a-l -1 ) =

for every 6 6 F1, a C Fo. Therefore T = P -My. It is clear that

i n f { l lg - ~lbo: g e H a } < tbl[oo -- Ibo + hll~ <_ IIT[I. On the other hand if ~ E H a then

IITII- -IIP-M~<I[-< I[Mg-(fl = I b - g i l a -

4. GENERALIZED INTERPOLATION

PROOF OF THEOREM 2.

First we prove that

for all f C L2(G).

Let h 6 K, since

P K f = C P - r

K c ( r l = C H 2-

we have that ~ h E H 2-. And the equality follows since

(h, r 1 6 2 = (r C f ) = (h, f ) .

Suppose HX[i < 1. Let T : H 2 --+ H 2- be defined by

T : %bXPK

then IITiJ < 1.

222 Dominguez

then

Let 3' �9 Pl and let f �9 H 2, since ~PKf = P - ~ f and X commutes with PKS~IK

T S J = -~XPKS~f = FPKS~XPKf

= P - - S & x P ~ : f = P - & ~ x P K f

= P-S~Tf.

Theorem 1 shows tha t there exists g �9 L~(G) such

Ilrll = Ilgll~. Set r = g r then IIXII

Let f �9 K then

tha t T = P-Mg and

= I1r That r �9 H o~ follows from

P - r = T r = ~ X P K r

= ~ X C P - I r 2 = O.

X f = CTf = CP-g f

= PKCgf = PKMcf.

PROOF OF THEOREM 3.

Suppose tha t I lXll _< 1. Let el be the minimal positive element of F. Take ~ = r e > Then

K = H 2 ~ r ~.

It is easy to prove tha t X commutes with PKS.r[K for all 7 E P1. By Theorem 2, it follows that there is a function r E H ~ such tha t X = PKMe~

and IlXII = I1r Therefore I1r <- 1 and

~(~) = <x~, ~> = e~ for all ~ �9 • The other par t of the theorem follows easily.

which

5. WEIGHTED INEQUALITIES

In this section we give a characterization of those positive measures # on G for

/I(Hf)(x)124~(x) _< af is(~)l~d.(~) G G

for every f E 7 ). This of course means that H extends to a bounded operator on L2(G, IX).

Let # be a finite positive measure on G. Define

= sup] f flT~d,I p(#)

G

where f l E 7)1, f2 e 7)2, I l f l l l . = I l f d l . = 1 and II.ll, is the norm in L2(G,#) .

It is clear that 0 <_ p(#) _< i.

Dominguez 223

Consider the L2(G, ]2) closures ~ and P--22. Observe that:

(a) The spaces 7)1 and 7)2 are orthogonal in L2(G, ]2) if and only if p(]2) = 0. (b) If there is a nonzero vector in both ~ and ~ then p(]2) = 1.

Recall tha t the closed subspaces 7)--~ and 7)--~ of L2(G, ]2) are said to be at positive angle if p(]2) < 1. The angle between the subspaces is arcosp(]2).

P R O P O S I T I O N 3. Let r E (0, 1) and let d]2 = wdx + d]2~ where ]2~ is singular to the Haar measure on G and w E LI(G). Then:

(a) p(]2) < r if and only if for every f C 7)

/I(Hf)(x)12d]2(x) - 1 - r < 1 + r /If(x)12d]2(x)" G G

(b) I f p(]2) < r then ]2. : O.

Proof. Let f E 7 ) then f = f l + f2 where f l E 7)1 and f2 E 7)2 therefore

' (S,,1,'... § Sin.. § § ) 1 - r G G G

- 1-rl + r i Ifl'd]2- S IH(f)12~]2' G G

And part (a) follows easily. To prove part (b), let ~ = (]2~n)~,~=1,2 be the matricial measure given by

]211 : /222 = r]2

and

]212 : ]2

then R(tt) < 1. Lemma 3 shows that # is absolutely continuous with respect to Haar measure on G, so ]2s : 0. []

Set p(w) = p(]2) when d]2 = wdx.

LEMMA 4. Let r e (0, I) and let w ~ L~(G). Then: p(w) < r if and only if there exists h E H 1 such that

w = ex p (u ) lh t = e x p ( ~ + ~) where v = - a r g h,

Ibrloo <_ ~ / 2 - a r c o s r < ~ / 2

and

luf < a r c o s h ( c o s . / ~VV:-~-~ a.e.

224 Dominguez

Proof.

and

It is clear that w > 0 a.e. Suppose p(w) < r. Let ~ = (#~)~,z=~,2 be the matricial measure given by

d/tll = d#22 = r w d x

d#12 = d#21 = wdx

then R(tt) < 1. From Lemma 3 it follows that there exists ho C H 1 such that

I h o - w[ ~ <_ r2w 2 a.e.

Let v = - a r g ho and for k = 1, 2 consider

~ _ ~ \ ~ + (_1)~ eos~ ~ / 1 - r 2

_ Ihol ((- / orcosh / eos P \ ~ / /

Thus 'W 1 ~ W <~ W 2.

Let

u = log ( l-r2/jho,) then the last inequality in the lemma follows easily.

Set h = ho/v/1 - r 2

then w = ]h]exp(u). Since v = - a r 9 h we have that w = exp(u + ~).

From

I1/r - ho/(r~)l <_ 1

and

we obtain Ivl

= -arg(ho / ( r~) )

< ~r/2 . Therefore

s in lv ] < r.

For the converse, we just notice that at all steps of the proof we have equivalent

statements. []

T H E O R E M 5. Let d# = wdx + dtts where tt~ is singular to the Haar measure

on G and w E L I (G) . The subspaces 7)1 and 7)2 are at positive angle i f and only i f #s = 0 and there exist u, v C L ~ ( G ) such that w = exp(u + ~) and Hvl]~ < ~/2.

Pro@ By Proposition 3 we can assume that # is absolutely continuous with respect to

Haar measure on G. Under this assumption the result follows from Lemma 4. []

Dominguez 225

PROOF OF THEOREM 4.

By Proposition 3, H(f) is bounded on L2(G, #) if and only if the subspaces ~ and are at positive angle in L2(G, p). And the result follows from Theorem 5.

6. PREDICTION OF SOME ORDERED PROCESSES

The prediction problem considered in this section is: Describe the spectral measure of the processes such that the future and the past of the process are at positive angle.

Let (~, 5 ~, P) be a probability space and let E denote the mathematical expectation. Let Z = {Z(7)}~r be a second order, weakly stationary and centered process with complex values, that is a family of random variables Z(~/) defined on (~, 9 v, P) such that:

(i) E(IZ(7)[ 2) is finite (ii) E(Z(~/)) = 0 for all ~, e F,

(iii) there is a function k : F x P ~ C such that

E ( Z ( f f ) Z ( a ) ) : h(~-17)

for all 7, ~ �9 F.

Let 7/be the Hilbert space of the random variables generated by the process, that is the closure of

7/0 = { Y = A1Z('~I) -~- -~" ANZ(~/N) where N �9 N, 71, . . . , 7N �9 F and A1,.. �9 , AN �9 C}

relative to the distance IIYlIp = ~ .

In [18] Helson and Lowdenslager extended several one-variable linear prediction results to the multivariate case after recognizing an appropriate definition of the "future" and the "past" considering the subspaces:

span {Z(ff) : 7 e F2} and span { Z ( ' J : 7 �9 F1}

which are known as the past and the future of the process respectively.

The Herglotz - Bochner - Weft theorem (see [26], Sec. 1.4.3.) assures that there exists a positive Borel measure # defined in G such that

E (Z(7 )Z(~ ) ) = [ ( ~ - b ) ( x ) d p ( x )

G

for every 7, ~ E F. This measure # is called the spectral measure of the process.

The latter formula can be interpreted as expressing an isomorphism between the Hilbert spaces 7/and L2(G, #) in which Z(7 ) is made to correspond to 7.

226 Dominguez

The spectral measure of the process enables us to investigate stationary processes

using analytic tools.

C O R O L L A R Y 1. The past and the future of a process are at positive angle if and only if there exist u, v E L~(G) such that the spectral measure # of the process is given by d , = exp(u + ~)dx and llvl]~ < ~/2.

7. I N T E G R A L R E P R E S E N T A T I O N S F O R S O M E K E R N E L S

A kernel on F is a function k : F • F --+ C. k is hermitian if

for all % ~ E F. All the kernels considered in this paper will be hermitian.

k is said to be a positive definite kernel on F if

N

> 0 i,j=l

whenever N is a positive integer, 71, . - . , 7N C F and A~, . . . , AN E C.

k is called a Toeplitz kernel on F if there exists a function r that

for every % ~ C F.

Let # be a complex Borel finite measure on G.

: F --+ C is given by

= / G

for all 7 E F.

Its Fourier-Stieltjes Transform

The Herglotz - Bochner - Weil theorem (see [26], Sec. 1.4.3.) states that if k is a

positive definite Toeplitz kernel on F then there exists a Borel measure /~ definided on G

such that k(V, ~) = ~(~-1~)

for every 7, a C F. This t~ is called the spectral measure of the kernel k.

In this paper a similar integral representation for another class of kernels is given.

For a , ~ = 1, 2 set

ulr = {5 s r : 5 = o--1~ / a n d ('-~, o-) ~ r a x r /~} .

Domingaez 227

D E F I N I T I O N 1. A generalized Toeplitz kernel on F is a function k : F • F -+ C such that there exist four functions k ~ : F~IF~ -+ C such that

k( , , ~) = k , , ( ~ - l ~ )

for every (7, ~) E F~ x F~ for a,/3 = 1, 2 and

if 7 E F~FI.

When F is the set of the real numbers or is the set of the integers analogous kernels were introduced in [ii] and several problems in analysis lead to the consideration of those kernels. For those cases integral representations of generalized Toeplitz kernels have been given in [4], [5], [8], [II]. Theorem 6 gives an integral representation for some generalized Toeplitz kernels, so it is a generalization to the case of ordered groups of some results of Arocena, Cotlar and Sadosky.

P R O P O S I T I O N 4. Let tt = (#,p)~,z=l,2 be a matricial measure on G and let k be generalized Toeplitz kernel given by

for every (7, o-) E F , • F~ for a, j3 = 1, 2. Then: k is positive definite i f and only if R(t t ) <_ 1.

The proof follows from an easy calculation.

A matricial measure tt = (#~)~,~=a,2 is said to be positive when for every Borel set A on G the numerical matrix i t(A) = (#~(A))~,~=1,2 is definite positive.

T H E O R E M 6. I f k is a positive definite generalized Toeplitz kernel on F then there exists a positive finite Borel matricial measure tt = (#~Z)~,p=l,2 defined on G such that

k(7, , ]

G

for every (7, ~r) E F~ x F m for a,/3 = 1, 2.

Proof.

For ct = 1, 2 we have that k~= gives rise to a Toeplitz kernel. Let # ~ be the spectral measure of the corresponding kernel. Then k ~ ( 7 ) = ~ ( 7 ) for every V E F.

Let Lll : P --+ C be defined by

Lll (7) = kll (7)

for every 7 E F.

Recall that 7)o is the subspace of 7)1 consisting of functions with mean value zero. Let L12:7)o -+ C be defined by

512(7) : h2( ) for every 7 E Fo.

228 Dominguez

Set L22 -- Ln. In the same way that Proposition 4 is proved, we obtain that

IL~2(/i~)] 2 <_ Lix(Ifll 2) L22(If~?) for all (f l , f2) e 7)1 x 7)2.

Given f l E 7)0 set

then

IL12(A)I 2 _< L11(I(~jI)AI2)L22(I~J~I 2) k~2(e)ll 2 = f lN .~ .

The Hahn - Banach theorem assures that L12 can be extended to a bounded linear

functional on L2(#n), that we will denote by LI2 too. From the Riesz theorem we obtain

that there exists g E L2(#11) such that

,/

G

for every f C L2(#n) . Set

dyl2 = 9d#n then u12 is a finite Borel measure on G. It follows tha t for every 7 E Fo

k12(7) = L12(7) = / ' T g d V l l 1 2

G

= / ~ d u l 2 = ~ ( ~ ) . G

Let Un = # n , u22 = #22 and ix = (~,Z),,~=1,2. From Proposi t ion 4 it follows that R(ix) < 1.

For a, /3 = 1, 2 let dv,,~Z = w<~zdx + du ~ where ~ ~Z ' is singular to the Haar measure

on G and wan E LI(G). Then

for all f l E 7)1 and f2 C 7)2.

2 2

o o~=I ~=I G

Proposition 1 in [22] guarantees 2 2

a=l i7=I G

for all f l 6 7)1 and f2 ~ P2. Taking w = (w~clx)~,~=l,2 it follows tha t R(w) < 1. Using Lemma 3 we can obtain h ~ H i such tha t

Iw12 + hL _< w11 a.e.

Dominguez 229

Set

d~12 = (wl~ + h)dx.

Then It : (#~)~,~=I,2 is positive and

for every (7,~) E Fa • FZ for a,~ = 1,2. []

Note: I would like to thank the referee for all his valuable comments.

REFERENCES

[1] ARENS, R. A Banach algebra generalization of conformal mappings of the disc. Trans. Am. Math. Soc., 81, 1956, pp. 501 - 513.

[2] ARENS, R. The boundary integral of log Ir for generalized analytic functions. Trans. Am. Math. Soc., 86, 1957, pp. 57- 69

[3] ARENS, R. AND SINGER, I.M. Generalized analytic functions. Trans. Am. Math. Soc., 81, 1956, pp. 379 - 393.

{4] AROCENA, R. Generalized Toeplitz kernels and dilations of interwining operators. Integral Equations and Operator Theory, 6, 1983, pp. 759 -778.

[5] AROCENA, R. AND COTLAR, M. A generalized Herglotz-Bochner theorem and L 2 weighted inequalities with finite measures. Conference on Harmonic Analysis in honor of A. Zygmund (Chicago 1981) I, Wadsworth Int. Math. Series, 1983, pp. 258 - 269.

[6] BALL, J., W.S. LI, D. TIMOTIN AND T. TaENT. In preparation (oral communication). [7] BAKONYI~ M., L. RODMAN, I.M. SPITKOVSKY AND H.J. WOERDEMAN Positive matrix functions on

the bitorus with prescribed Fourier coefficients in a band. Preprint. [8] BRUZUAL, R. Local semigroups of contractions and some applications to Fourier representation theo-

rems. Int. Eq. and Op. Theory, 10, 1987, pp. 780-801. [9] BRUZUAL, R. AND DOMINGUEZ, M. Extensions of operator valued positive definite functions on an

interval of Z 2 with the lexicographic order. To appear in Acta Scientiarum Mathematicarum. [10] CARATHEODORY, C. AND FEJER, L. Uber den Zusammenhang der Extremen yon harmonischen Funk-

tionen mit ihren Koeffizienten lind uber Picard-Landauschen Satz. Rend. Circ. Mat. Palermo, 32, 1911, pp. 218 - 239.

[I1] COTLAR, M. AND SADOSKY, C. On ~he Helson-Szeg5 theorem and a related class of modified Toeplitz kernels. Proc. Symp. Pure Math. AMS., 35-I, 1979, pp. 383 - 407.

[12] COTLAR, M. AND SADOSKY, C. Two parameter lifting theorems and double Hilbert transforms in commutative and non-commutative settings. J. Math. Anal. and Appl., 150, 1990, pp. 439 - 480.

[13] COTLAR, M. AND SADOSKY, C. Transference of metrics induced by unitary couplings, a Sarason theo- rem for the bidimensional torus and a Sz.-Nagy-Foias theorem for two pairs of dilations. J. Functional Analysis, 111 N2, 1993, pp. 473 - 488.

[14] M. COTLAR AND C. SADOSKY. Two distinguished subspaces of product BMO and Nehari-AAK theory for Hankcl operators on the torus. Integral Equations and Operator Theory, 26, 1996, pp. 273 - 304.

[15] DOMINGUEZ, M. Weighted inequalities for the Hilbert transform and the adjoint operator in the contin- uous case. Studia Mathematica, 95, 1990, pp. 229 - 236.

[16] DOMfNGUEZ, M. Mixing coefficient, generalized maximal correlation coefficients and weakly positive measures. Journal of Multivariate Analysis, 43-1, 1992, pp. 110 - 124.

[17] DOMINGUEZ, M. Ordered lifting and interpolation problems for the bidimensional torus. Preprint. [18] HELSON, H. AND LOWDENSLAGER, D. Prediction theory and Fourier series in several variables. Acta

Math., 99, 1958, pp. 165 - 202. [19] HELSON, H. AND SZEG(~, G. A problem in prediction theory. Ann. Math. PurL Appl., 51, 1960, pp.

107- 138.

230 Dominguez

[20] HOFFMAN, •. Boundary behavior of generalized analytic funcyions. Trans. Am. Math. Soc., 87, 1958, pp. 447 - 466.

[21] HOFFMAN, K. AND SINGER~ I.M. Maximal subalgebras of C(F). Am. J. Math., 79, 1957, pp. 295 - 305. [22] NAKAZI, T. AND YAMAMOTO, T. A lifting theorem and uniform algebras. Trans. Amer. Math. Soe.,

305, 1988, pp. 79 - 94. [23] NEHARI, Z. On bounded bilinear forms. Annals of Mathematics, 65-1, 1957, pp. 153 - 162. [24] PENG, L. AND ROCHBERG, R. Trace ideal criteria for Toeplitz and Hankel operators on the weighted

Bergman spaces with exponenetial type weights. Pacific J. Math., 173, 1.996, pp. 127 - 146. [25] ROCHBERG, R. Toeplitz and Hankel operators on the Paley Wiener space. Integral Equations and

Operator Theory, 10, 1.987, pp. 187 - 235. [26] RUDIN, W. Fourier analysis on groups. Interscience, 1962. [27] SARASON, D. Generalized interpolation in H ~176 Trans. Amer. Math. Soc., 127, 1967, pp. 179 - 203.

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Submitted: August 17, 1998 Revised: July 25, 2000