SERIES IN SEVERAL

PREDICTION THEORY AND FOURIER VARIABLES

BY

H E N R Y H E L S O N Berkeley, Califor~tia

AND

DAVID L O W D E N S L A G E R (1~

Urbana, Illinois

SERIES IN SEVERAL

l . I n t r o d u c t i o n

The theory of analytic functions of a complex variable extends only with difficulty and incompletely to functions of several variables. Because the Riemann M a p p i n g Theorem fails in several variables, the description of domains of holomorphy and their analytic transformations has been a major concern. Nevertheless function theory in the bicylinder hardly exists beside the elegant theory of functions in the unit circle.

This circumstance is related to the singular fact, never observed so far as we know, t h a t analytic function theory divides into two distinct disciplines in higher dimensions.

The theory of analytic functions in several variables has been concerned with functions defined locally and consistently b y power series in a domain, whereas much function theory in the circle can be made to depend on group properties of the circle, and generalizes in quite a different way. The study of multiple Fourier series from this point of view is one objective of this paper. The discussion of analyticity in a group-theoretic context was begun by Mackey [13], and recently has been continued with great ingenuity by Arens and Singer [2, 3, 4]. While our work has points of con- tact with t h a t of Arens and Singer, the methods are different, and we have attained a certain completeness at the expense of generality.

(1) The work of the second-named author was supported by contract Nonr-222 (37) with the Office of Naval Research.

I n the work of Kolmogoroff [11] and Wiener [18] on the prediction of second- order stationary stochastic processes, certain theorems a b o u t analytic functions in the circle play an essential part. The analytic difficulty is e x a c t l y m e t b y a theorem of SzegS; and indeed Szeg6's Theorem can be used to prove the various function-theoretic results which would otherwise be used in the proof of the prediction theorem. The second section of this paper is devoted mainly to a generalization of SzegS's Theorem to two or more variables, and this furnishes the solution of a certain prediction problem in several variables. This is not the multiple prediction problem mentioned b y Doob [8, p. 594] and treated recently b y Wiener [19], of which we shall speak presently.

I n the third section we exploit the methods and results of the second section in order to prove a n u m b e r of theorems in multiple Fourier series generalizing ele- m e n t a r y properties of analytic functions of one variable. We obtain an inequality in place of Jcnsen's Formula, under hypotheses slightly different from those of Arens in a paper not y e t published. Then we extend the characterization of functions

w(e "~)

defined on the unit circle having a representation w = I/12 almost everywhere,

where / is analytic and of class H 2 inside the circle. A related theorem states t h a t every function analytic in the circle and of class H is the product of two functions in H 2. Finally we extend the theorem of H a r d y and Littlewood about functions of class H in the circle:

If ] (z) = ~ an z n,

0

then

~ ]anl/(n-F

1 ) < co.

0

For simplicity we t r e a t functions of only two variables in this section. I n each case the class of functions to which our theorem applies is not the double power series, b u t rather the functions defined on the. torus whose Fourier coefficients am n vanish for all (m, n) belonging to a

hall-plane

(in a sense which m u s t be made precise). The proofs depend on this division of the group of lattice points into disjoint semi-groups, rather t h a n on the local properties of functions defined on the t o m s . For functions of one variable the theorems are generally proved b y removing the zeros of an analytic function in the circle. Of course this technique is not available for functions of several variables, and instead our method depends on the fact t h a t every closed convex set in Hilbert space possesses a unique element of minimal norm.

I n section four we discuss Bochner's generalization of a well-known theorem of

P R E D I C T I O N T I t E O R Y A N D F O U R I E R S E R I E S I N S E V E R A L V A R I A B L E S 1 6 7

F. and M. Riesz [15]. I n the form of interest to us, the Riesz Theorem states: i]

/~ (x) is a complex /unction o/ bounded variation on the circle such that f e -~nxd/~(x)=0 for n = l , 2 . . .

then la is absolutely continuous. The obvious analogue in several dimensions is trivially false; nevertheless Bochner [6] has found a generalization for set functions # on the torus. The Riesz Theorem is a convenient tool in proving SzegS's Theorem [1, p. 263] ; but some accounts of prediction theory (for example [8]) do not mention it. We have tried to clarify the relation betwen these theorems b y giving a new proof of Boch- ner's Theorem based on the results of preceding sections. I t is of methodological interest t h a t our proof does not depend on theorems a b o u t analytic functions, as have all the published proofs of the Riesz Theorem.

I n section two we generalized SzegS's Theorem to functions of several variables.

I n section five we consider another kind of generalization: we study functions defined on the unit circle whose values are matrices. Wiener [19] was led to the study of matrix-valued functions b y a prediction problem different from the one treated in section two. After seeing Wiener's paper we succeeded in extending our method to this case. The fundamental result, as before, is a generalization of SzegS's Theorem.

F r o m it flow the solution of a prediction problem, and a number of theorems a b o u t matrix-valued analytic functions defined in the circle. Recently Masani and .Wiener have completed a paper [14] carrying Wiener's work much further. I t is likely t h a t there is a good deal of duplication in our results, although their version of SzegS's Theorem is different from ours. We are h a p p y to accord Masani and Wiener the right of precedence, and to acknowledge our debt to Wiener's paper. We hope nevertheless t h a t the systematic development presented here, as well as our new results, will justify the publication of this section.

I n the last section we extend these theorems to their natural degree of generality.

We consider functions defined on a compact abelian group whose dual is linearly ordered b y a relation consistent with the group structure. The functions m a y t a k e matrices as values. Then SzegS's Theorem and most of our other results can be ex- tended to this setting, and the proofs are word for word the same as proofs of cor- responding theorems in the body of the paper. The torus groups are the best examples of groups to which the analysis applies, b u t there is no restriction in dimension. I n particular, the Bohr compactification of the line (whose.dual is the group of real numbers in the discrete topology) is of the t y p e considered.

2. Doubly Stationary Series

L e t xm, be a n element of a Hilbert space for each integer m a n d n. W e say t h a t { x ~ } is doubly stationary if for all m, n, r, a n d s we have

(Xm+r. ,+~, Xm,) = (xrs, Xoo). (1)

I n this case we define 9 (r, 8 ) = (xrs, x00). (2)

T h e n r is a positive definite function on the group of lattice points of t h e plane.

T h a t is, for a n y complex n u m b e r s ~1 . . . :ok a n d integers r 1 .. . . . rk, s 1 .. . . . s~ we have

k

E o~se(r,-rJ, 8,-s~)>10. (3)

t,i=1 Indeed, using (1) this a m o u n t s to

E 0~t aJ (Xrtst, Xrts t) >~ O,

or (Z oq x~,8,, E oq x,,~,) >10.

T h e last inequality is obvious, a n d so (3) holds.

The t h e o r e m of Herglotz, Bochner, a n d Weil on positive definite functions states t h a t there is a non-negative measure # defined for Borel sets on t h e torus

0~<x~<2n, 0 < y ~ < 2 n e(r, 8)= f e-~('~+~U)d~(x, y) such t h a t

for all integers r a n d 8.

N o w let S be a n y set of lattice points (m, n) in the plane n o t containing (0, 0), a n d let {amn} be a set of n u m b e r s defined for (m, n) in S, vanishing except for a finite set of indices. T a k i n g (1) a n d (2) into a c c o u n t we find

8

= (xoo, xoo) + ~ dmn (xoo, xm n) + ~ am,~ (Xmn, Xoo) + ~ ~ am n ars (Xmn, xr s)

= e ( 0 , 0 ) + ~ a,~,~(m,n)+ ~. % , e ( m , n ) + ~ ~ am, a , , e ( m - r , n - s )

= f l I + Y a , . . e-'('~+'~')l~ d,u (x, Y).

S

(4)

T h u s the problem of a p p r o x i m a t i n g x00 b y a linear c o m b i n a t i o n of elements Xm~ with (m, n) in S is equivalent to minimizing t h e integral at the end of (4). A n explicit

PREDICTION THEORY AND F O U R I E R SERIES .IN SEVERAL VARIABLES 169 evaluation of the infimum is given for the corresponding expression in one variable by the following theorem of Szeg5 [16]:(1)

1/ bt is a /inite non-negative measure defined on the Borel sets o/ the circle l z l = 1 whose absolutely continuous part is w ( e ~ ) d x / 2 ~ then we have

e x p { l ~ f l o g w d x } = i n f f l l + P ( e " ~ ) 1 2 d t t ( x ) ,

where P ranges over the trigonometric polynomials o/ the /orm p (e i x) = ale ~ x + a2 e2~ + ... + an e his.

The left side is to be interpreted as zero i/

f log w ( e ~ ) d x = - oo.

The solution of the prediction problem for any set S of lattice points requires an appropriate generalization of Szeg6's Theorem. We shall find such a generalization for a very special class of sets S. Before stating our theorem we make some ob- servations which do not require hypotheses on S.

Trigonometric polynomials of the form

e - i ( m x + n y )

1 + 5 amn (5)

s

form a convex subset of the Hilbert space of functions square-summable with respect to /z. The closure of this subset will be called S. If S contains the null function, then x00 lies in the manifold spanned by {Xmn}: for (m, n ) i n S, and we say t h a t prediction is perfect. Otherwise (and this is the interesting c a s e ) a n y sequence of elements Qn of S such t h a t

lim IIQ ll=i fllGll (Ges)

is a Cauchy sequence, and converges to the unique element 1 + H of S having mini- real norm. w e have therefore

inf f l l §

_S

amne-'(m += 'lZdb = fll+H[2dt~>O.

⁽⁶⁾

(1) Szeg6 s t a t e d t h e t h e o r e m for a b s o l u t e l y c o n t i n u o u s m e a s u r e s . I t w a s c o m p l e t e d h y K o l - m o g o r o f f a n d K r e i n ; r e f e r e n c e s a r e g i v e n i n [1]. W e s h a l l n e v e r t h e l e s s r e f e r t o t h e f u l l r e s u l t a s S z e g 6 ' s T h e o r e m .

12 - 665064 Aeta mathematiea. 99. Imprim~ le 10 juin 1958

F o r a n y complex n u m b e r ;t a n d (m, n) in S the function 1 + H ( e ~x, e ~) + ~ e -~(mx+n~) belongs to $, a n d therefore

has a unique m i n i m u m at ; t = 0. H e n c e for every (m, n) in S we have

f [1 + H (e fx, dr)] e t(~x+"~) d # = 0. (7)

I f S is closed u n d e r group addition, so t h a t (m,n) E S a n d ( m ' , n ' ) E S

i m p l y (m + m', n + n') E S,

t h e n there is a second o r t h o g o n a l i t y relation. F o r each complex ~ a n d each (m, n) in S the function

[1 + H ( e ix, e'~)] [1 +~te -~(~x+~y)]

belongs to $, a n d its n o r m is minimized at 2 = 0. The conclusion is n o w

f ] 1 + H(e '~,

r ~

~,,~x+~y,

d ~ = 0. (8) B y t a k i n g t h e complex c o n j u g a t e of (8) we see t h e same is true if ( - m , - n ) is in S.

I t is easy to prove t h a t (7) characterizes t h e minimal element of S. Indeed, sup- pose t h a t (7) holds b u t 1 + G is t h e minimal element. T h e n

f i l + H + a ( G - H ) ] 2 d l ~ = / [ I + H I ' d ~ + I ~ . ] ~ / I G - H I 2 d #

for every complex xl. This expression is obviously smallest for ~ = 0 ; b u t it is a t least as small for ~l = 1 if G is t h e minimal element. Since the minimal function is unique, we conclude t h a t G = H.

D E F I N I T I O N . S is a hal/-plane of lattice points if 1 ~ (0, 0) r

2 ~ (m,n) E S if a n d only if ( - m , - n ) r unless m = n = 0 3 ~ ( m , n ) E S a n d ( m ' , n ' ) E S imply ( m + m ' , n §

If S is a half-plane a n d (8) holds for all (m, n) in S, t h e n b y the second eondi.

t i o n (8) holds for all m a n d n except m = n = 0. T h a t is, the Fourier-Stieltjes eoeffi-

P R E D I C T I O N T H E O R Y A N D F O U R I E R S E R I E S I N S E V E R A L V A R I A B L E S 1 7 1

cients of the measure I I+HI2d/~ all vanish e x c e p t ' t h e central one. Therefore this measure is a multiple of Lebesgue measure. I t follows t h a t 1 + H must vanish almost everywhere with respect to the singular component of d/~, and (7) can be written

f [ l + H ( e ~x, e~Y)]et(mz+nY)d/~a=O ((m,n) ES) (7') where /~a is the absolutely continuous part of ju.

We can now state the first generalization of SzegS's Theorem.

THEOREM 1. Let S be a hal/-plane o/ lattice points and let # be a linite non- negative measure on the torus. Let # have Lebesgue decomposition

d # (x, y) = w (e tz, d y) d a + d l~s (x, y),

where w is non-negative and summable /or the measure d a = d x d y / 4 g ~, and i~s is singular with respect to d a. Then

exp {f log wda} = i n f f l i +PI2d/~, (9)

P

where P ranges over /inite sums o/ the /orm

P ( e fx, e~)= ~ atone -i(mx+ny). (10)

s

The le/t side o/ (9) is to be interpreted as zero i/

f log w d a = - oo. (11)

Proo/. If the infimum in (9) is positive, we have seen t h a t it is equal to

fll+Hl d ,,

where 1 + H belongs to S and vanishes almost everywhere for #s. Hence (7') holds.

Moreover 1 + H belongs to the convex set $ formed with the measure w d a instead of d/~, and (7') implies t h a t 1 + H is the minimal function relative to this measure:

inf f I1 +

P]~wd,r= f I1 + Hl~wda= f I1 + Hl~d/~.

P

Therefore it will suffice to prove

exp {flog wda} =inf f[1 + Pl~wda.

(12)

P

On the other hand, even if the infimum in (9) is zero it is enough to prove (12).

F o r in t h a t case the infimum in (12) surely vanishes, and having proved (12), ob- viously (9) .holds. So we shall prove (12) for an arbitrary non-negative summable function w. I t will be convenient to establish two lemmas.

L S M M A 1. I / W i8 a

non-negative summable /unction,

exp If log

wda}

= i n f

fe~wda,

(13)

where ~o ranges over the real summable /unctions such that

f wda=O.

(14)

The geometric and arithmetic means of w are related by the well-known inequality exp {flog

wda} ~ f wda.

The same remark applies to

eVw,

where ~0 is a n y summable function which satisfies (14), and we find therefore

exp [ f l ~

wda}

^~<inf

f eVwda.

⁽¹⁵⁾

The opposite inequality will be established first assuming log w is summable. Define

2r flogwda;

~0 = ~ - log w. (16)

Then y) satisfies (14), a n d we have

f eVwda=

f e a d a = e x p [ f l o g w d a

I.

Therefore the inequality in (15) m u s t be equality, and the minimal function is given by (16). We shall have to refer to the form of the minimal function again.

If log w is not summable this a r g u m e n t does not apply, and except in trivial cases no minimal function exists. B u t log (w + e) is summable for each e > 0, and b y w h a t we have just proved

exp

{flog

( w + e ) da} = i n f

f e~(w+e)da>~inf f eVwda.

As e tends to zero we obtain by the monotone limit theorem exp {flog

wda]

=O~>inf

f evwda>>'O,

from which the statement of the l e m m a follows.

P R E D I C T I O N T H E O R Y A N D F O U R I E R S E R I E S I N S E V E R A L V A R I A B L E S 1 7 3

L ~ M M A 2. For any non.negative summable /unction w we have

exp {flog =inf f (17)

y~

where ~? ranges over the real t r i g o n o m e t r i c p o l y n o m i a l s satis/ying (14).

I t suffices t o p r o v e t h e l e m m a a s s u m i n g t h a t log w is s u m m a b l e ; for as in L e m m a 1, t h e g e n e r a l ease can be t r e a t e d b y a l i m i t process. D i v i d e w b y a c o n s t a n t , if n e c e s s a r y , so t h a t

f log w d a = O. (18)

N o w let u a n d v be t h e p o s i t i v e a n d n e g a t i v e p a r t s of log w, r e s p e c t i v e l y , so t h a t u, v~>0; log w = u - v .

Choose a sequence u 1, u2, ... of b o u n d e d n o n - n e g a t i v e f u n c t i o n s i n c r e a s i n g p o i n t w i s e t o u, a n d a sequence vl, v 2 .. . . of b o u n d e d n o n - n e g a t i v e f u n c t i o n s i n c r e a s i n g t o v.

B y t h e m o n o t o n e l i m i t t h e o r e m ,

]im f u n d a = f u d a = f v d a = l i m f v,d(~.

C o n s e q u e n t l y for each n t h e r e is a n m such t h a t

I n case t h e i n e q u a l i t y is strict, m u l t i p l y vm b y a c o n s t a n t s m a l l e r t h a n one so t h a t e q u a l i t y o b t a i n s , a n d r e n a m e t h e f u n c t i o n v n. W e h a v e t h e n

O<.u,<.u; O<~v,<v; f u , d a =

fv.da.

M o r e o v e r t h e sequence u~ increases m o n o t o n i c a l l y t o u, a n d i t is e a s y t o see t h a t vn t e n d s p o i n t w i s e t o v. F r o m t h e c o n s t r u c t i o n i t follows t h a t

0 <~ e (u-un)-(v-~'n) <~ m a x (1, w).

T h e r e f o r e t h e L e b e s g u e d o m i n a t e d c o n v e r g e n c e t h e o r e m a p p l i e s t o g i v e lim f e ~',-u,~ w d a = l i m f e(U--U~)-(~'-~'n) d a = 1.

Since t h e f u n c t i o n ~ o = v n - u n satisfies (14), we h a v e p r o v e d t h a t

inf feVwda<<. 1, (19)

where now y) ranges over bounded functions satisfying (14). E v e r y bounded function y) is boundedly the limit, of Fejdr means of its Fourier series (in one or several di- mensions); each approximating function is a trigonometric polynomial which is real if ~ is real, and satisfies (14) if ~p does. Therefore (19) continues to hold if W is restricted to real trigonometric polynomials with vanishing integral. In view of (18) and Lemma l, the inequality of (19) must be equality, and the proof is complete.

We return now to the proof of (12) itself. The most general trigonometric poly- nomial ~ satisfying (14) can be written, on account of the second property of half- planes, in the form

e _ i ( m x + n y )

amn + ~ dmrte '(mz+nv). (21)

s s

If P denotes the trigonometric polynomial (10), we have

~ o = P + P = 2 Re (P).

Therefore the result of L e m m a 2 can be restated

exp {flog wd~}

= i n f f r

P

where P ranges over trigonometric polynomials of the form (10).

On account of the third property of half-planes, it is clear t h a t eP=I+Q,

where Q is a continuous function with vanishing integral and having Fourier series of the form {10), although of course Q is not a trigonometric polynomial. Therefore we have

exp {flog w da} >~inf f l l + P I 2 w d a , (23)

P

where P ranges over all continuous functions with Fourier series (10). The infimum is not increased if P is restricted to the class of trigonometric polynomials of the form (10), and so we have proved the first half of (12).

The opposite inequality can, paradoxically, be deduced from (23) itself. Replace w in that formula by I I + Q I 2, where Q is any polynomial of the form (10):

exp {f log ll +Ql~d(r} >~inf f ll + e +Q+ PQl~d(r>~ l,

P

making use once more of the semi-group property of S. Hence log I1 + Q [ 2 is sum.

mable and

f l o g I1 +Q['da>>-O.

P R E D I C T I O N T H E O R Y A N D F O U R I E R S E R I E S I N S E V E R A L V A R I A B L E S 1 7 5

Therefore we can write

II+Ql'=ke~;

^k~>l, f ~ d a = 0 . (24) Now if w is an arbitrary non-negative summable function and Q is a polynomial of the form (10) we have by (24)

f l l + Q l 2 w d a = l c f e ~ w d a > ~ i n f f e ~ w d a = e x p {flogwda}. (25) But this inequality is exactly the opposite of (23) if we pass to the infimum over Q, and so (12) has been proved. This completes the proof of the theorem.

Theorem 1 is a full generalization of Szeg6's Theorem. We have already pointed out its connection with prediction t h e o r y ; in the next section we shall apply it to multiple Fourier series.

3. Multiple Fourier Series

The first application of Theorem 1 is a partial generalization of Jensen's formula.

T H E O R Er~ 2. Let / be summable on the torus with Fourier series

](r e~U),.~b+ ~ b,~ne-t(~x+n~), (26)

s

where S is any hall-plane. Then

f l o g ]/[da>_.log Ib]. (27)

Proo/. B y Theorem 1,

exp {flog

It1~}

= i n f f

I1

+ P l ' l I l ~ a ,

P

where P ranges over the trigonometric polynomials of the form (10). I f ] is square- summable, we can replace Ill in the last formula by I tl ~ and then take the square root of both sides:

exp {flog

I/Ida} =inf [f I(1 + P)/12do] '". (28)

If we set in the Fourier series (26) for / we obtain in the product ( I + P ) / a con- stant term b, since P has no constant term. B y the Parseval equality, the right side of (28) is at least {b I, so t h a t (27) holds.

If / is not square-summable, let {/n} be the Fejdr means of ]. Each / , is a trigonometric polynomial with constant term b, and the sequence converges to / in L.

For any e > 0 and each n we have

f l o g [l/=[+e]da>~ f l o g I f . I d a . > l o g Ibl.

P a s s i n g t o t h e l i m i t in n w i t h e fixed,

f l o g [ l / l + e J d a > ~ l o g Ibl.

T h e r e s u l t follows b y l e t t i n g e t e n d t o zero.

F o r a n a l y t i c f u n c t i o n s of one v a r i a b l e t h e d e f i c i e n c y of t h e r i g h t side of (27) c a n n o t be e v a l u a t e d w i t h o u t s o m e s t r o n g e r h y p o t h e s i s a b o u t t h e f u n c t i o n . (Some consequences of t h i s f a c t a r e e x p l o r e d in [5].) I t w o u l d be i n t e r e s t i n g t o r e p l a c e (27) b y a n e q u a t i o n a n a l o g o u s t o g e n s e n ' s f o r m u l a if, for e x a m p l e , I is a t r i g o n o m e t r i c p o l y n o m i a l .

C O R O L L A R Y . I1 / is summable on the torus, has Fourier series o I the form (26), and has mean value different from zero, then log [/] is summable.(1)

T h e proof is i m m e d i a t e . I n one d i m e n s i o n t h e c o r r e s p o n d i n g t h e o r e m r e q u i r e s n o h y p o t h e s i s on t h e m e a n v a l u e b, b u t h e r e s o m e such c o n d i t i o n is i n d i s p e n s a b l e . To see t h a t t h i s is so, c o n s t r u c t a sequence of f u n c t i o n s of one v a r i a b l e , gl, g2 . . .

e a c h v a n i s h i n g on a f i x e d i n t e r v a l (~, fl) in (0, 2~t), w i t h F o u r i e r s e r i e s

gm(e'~) = ~ a,nne-'n~; ~

lamnl<l/m ~-

n n

Define

l(e'X, etY)= ~

~ atone -i(mx+nu).

m = l n = - c r

T h e n f h a s a b s o l u t e l y c o n v e r g e n t F o u r i e r series a n d so is s u m m a b l e ; m o r e o v e r i t s coefficients a r e r e s t r i c t e d t o a h a l f - p l a n e . B u t / v a n i s h e s on t h e set

0 ~ < x ~ < 2 ~ ; o~ <<. y <<. fl, so t h a t

log Ill

c a n n o t be s u m m a b l e .

I f t h e c o e f f i c i e n t s of f are r e s t r i c t e d t o a s e c t o r of o p e n i n g smaller than ~, t h e n t h e conclusion of t h e c o r o l l a r y h o l d s w i t h o u t a n y r e s t r i c t i o n o n t h e m e a n v a l u e o f / , p r o v i d e d f is n o t t h e null f u n c t i o n . T h e p r o o f is like t h a t of T h e o r e m 2, m a k i n g use of a c o n s t r u c t i o n u s e d a g a i n i n t h e proof of B o c h n e r ' s T h e o r e m i n s e c t i o n four.

I n p a r t i c u l a r , t h e conclusion holds if / is a n a n a l y t i c f u n c t i o n o f t w o v a r i a b l e s , as one c a n also show e a s i l y u s i n g J e n s e n ' s f o r m u l a for a n a l y t i c f u n c t i o n s of one v a r i a b l e . (1) A similar theorem has been proved by Arens, even without the hypothesis that / has mean value different from zero. He assumes instead that ] is defined on a compact group whose dual has an Archimedean order, corresponding here to the case of a half-plane bounded by a line of irrational slope. In the example which follows, the half-plane is bounded by a vertical line, so that the order relation defined by taking S as the set of positive elements is not Archimedean.

PREDICTION THEORY A N D FOURIER SERIES IN SEVERAL VARIABLES 177

T H E O R E M 3. Let w be non-negative and summable on the torus, and let S be any hall.plane. A necessary and su//icient condition /or w to have a representation

w(r x, cy)=lb+ bmn -"mx nY)l ; b.o, Ib nl < (29)

S

is that

f l o g w d a > - oo. (1) (30)

Proo/. If w has the form (29), then as in the proof of Theorem 2 exp

If log

w d a } - i n f f ](1 + P ) ] ~ w d a ~ ] b ] 2 > O .

P

Thus (30) holds.

Conversely, suppose (30) is true. Then there is a unique function H such t h a t

lim f ] H - P n ] * w d a = O (31)

for a sequence of trigonometric polynomials Pn of the form (10), and satisfying exp ( f l o g w d a } =e~--~]l + H ] ~ w .

We shall prove t h a t the obvious equality

I

^{e~t/2 2}

w = I ~ (32)

is a representation for w in the form (29).

B y (32), (1 ~-H) -1 is square-summable. Its Fourier coefficients are

fe-'(m*+'u) 1

l + g d a =

fe-*r247

[ ~_~-~1~

According to (7) this integral vanishes for every (m, n) in S. Therefore the Fourier series of (1-t-H) -1 has the form

b+ ~ bmne -i(mx+n~).

S

If b = 0 , so t h a t

f (l + H)wd,~=O,

we should have

f (1 + P ) (1 + H ) w d a = 0

for every trigonometric polynomial P of the form (10). From this fact and (31)would follow

(1) For functions of one variable this was first proved by Szeg6 [17], using exactly the p r e s e n t

method.

fll+Hl'wda=O,

which contradicts (30); therefore b * 0, and the proof is finished.

L e t R be a set of lattice points containing the origin and closed under addition.

I t is an interesting problem, suggested to the authors b y Mr. G. Weiss and Professor A. Zygmund, to determine whether every summable function / with Fourier series

l(d ~, d~)~ ~.

atone -i(mz+ny) R

can be represented as a product g. h, where g and h are square-summable and, like /, have coefficients restricted to R. E v e n if _R is t a k e n to be the set of lattice points in the first quadrant, so t h a t the problem concerns analytic functions of two variables, the answer seems not to be known. Our next theorem treats the case of a half-plane.

T H E 0 R • M 4. Let S be a hall-plane and / a summable/unction with Fourier series

e - i ( m x + n y ) ;

a + ~ amn a * O .

8

There exist square-summable /unctions 9 and h with Fourier series o/ the same /orm such that / = g. h.

Proo]. Since the leading coefficient of ] is not zero, the corollary of Theorem 2 states t h a t log I/I is summable. B y Theorem 3 a n d its proof,

where

I/l=lgl

g = l + H , . , b + C ~ b,,ne-t(m'~+'~); b, c*O.

S

I f we set h = g - 1 / , it is clear t h a t g and h are square-summable, and g at least has Fourier series of the required kind. B u t we can write

h = g - 1 / = c - l ( l + H ) / = c -1 lim (1 + P , ) / ,

where each P~ is a trigonometric polynomial of the form (10), and the limit is t a k e n in the norm of the space L. I t follows t h a t the Fourier coefficients of h are restricted to S (aside from the constant term), and this completes the proof.

We can now derive an analogue for multiple Fourier series of the classical theo- r e m of H a r d y and Littlewood which states [10; 20, p. 158]: i/ / is summable on the circle with Fourier series

/ (e'Z) .,~ ~ an ei nz,

0

P R E D I C T I O N T H E O R Y A N D F O U R I E R S E R I E S I N S E V E R A L V A R I A B L E S 1 7 9

then /or a certain absolute constant ]c

lanl/(n+ 1) k f It( 'x)l

This theorem follows from Hilbert's inequality and a factorization theorem for analytic functions. The same method of proof works in higher dimensions;first we shall quote an extension of Hilbert's inequality from work of CalderSn and Zygmund, and then Theorem 4 will furnish exactly the factorization theorem we need.

I n the paper [7] of Calderdn and Zygmund, Theorem 14 states: let K be a / u n c - tion de/ined on the lattice points, except the origin, and have the /orm

K (m, n) = ~ (eio) r 2 ; . m + i n = r e ~o.

Suppose that ~ is continuous on the circle, satis/ies a Lipschitz condition o/ positive order, and also

2 ~

] ~ (e i~ d 0 = O.

0

Then there is a constant k depending only on K such that /or any square-summable sequences {Xmn} and {y,s} we have

IZ' K(m+r, n+8)x, y,I Ix,, P Z lY-P]'".

(33)

(The summation on the left is extended over all indices for which the summand is defined.)

Let S be a n y half-plane. There is an angle g, uniquely determined up to mul- tiples of 2 z , such t h a t every lattice point m - F i n = r e ~ in S satisfies a ~ 0 ~ < ~ + ~ . Define a function ~ to be one for ~ < 0 ~ < a + ~ ; then extend ~ to the rest of the circle so as to be continuously differentiable and have mean value zero. The corre- sponding function K is a kernel to which the theorem of CalderSn and Zygmund applies. Suppose t h a t x,,~=O unless (m,n) is in S, or m = n = 0 , and the same for the y,~. Then the only terms which contribute to the sum on the left side of (33) are those for which

K (m, n) = 1 / ( m 2 + n2).

We have therefore the following analogue of Hilbert's inequality:

I ' ~ (m + r) 2x'nn yr

(n + s) ~ I < k [~ l x~" l~ ~ l y'` P]'/'"

(34)

T H E 0 R E M 5. Let / be summable on the torus and have Fourier series

9 e_i(mx+ny).

l(e% e'Y)~a+ ~ amn

s There is an absolute constant k' such that

lal+ ~ lam~l/(m~+n2+l)<<.k'fl/(e '~, e'")lda.

(35)

S

Proo/. B y c o n t i n u i t y i t suffices t o c o n s i d e r t h e case a ~= 0. T h e o r e m 4 s t a t e s t h a t / can be w r i t t e n as t h e p r o d u c t of t h e s q u a r e - s u m m a b l e f u n c t i o n s

g = b + ~ bmne-i(mx+n~); b=vO s

e-i(mx+ny)

a n d h = c + ~. cm~ ; c~=O.

s

T h e n we h a v e amn= ~ b . . . -sCr~, r,s

so t h a t _la""l < y:'

Ibm-r.,,-,cr,I

s ~ m 2 + n 2 m 2 + n 2

I bran Crs I (m + r) ~ + (n + s) ~

I f we set Xm n = ]bm n I,

Yr,=ler,I,

t h e n (34) a p p l i e s t o give

lamnl/(m2+n~)<<-k[~ Ibm~l ~ ~:

le.12] '".

s

From the proof of Theorem 4 we know t h a t I g l 2 = l h l ~ = l f l ; by the Plancherel T h e o r e m t h e r e f o r e

Y la,.~l/(,n'+n')<kfl(e '',

e'")l d,:,.

S

T h e s t a t e m e n t of t h e t h e o r e m follows t r i v i a l l y f r o m t h i s f o r m u l a , w i t h k ' = k + 1.

T h e r e is no d i f f i c u l t y in p r o v i n g a n a n a l o g o u s t h e o r e m for t o r i of a n y f i n i t e d i m e n s i o n (since b o t h T h e o r e m 4 a n d t h e t h e o r e m of C a l d e r 6 n a n d Z y g m u n d a r e t r u e in general). W e d o n o t k n o w w h e t h e r t h e r e is a g e n e r a l i z a t i o n t o t h e class of c o m p a c t g r o u p s discussed in t h e l a s t section.

T h e o r e m 5 a p p l i e s t o a l a r g e r class of f u n c t i o n s t h a n t h e d o u b l e p o w e r series.

H o w e v e r , as B o c h n e r h a s r e m a r k e d , a s t r o n g e r r e s u l t h o l d s for t h e d o u b l e p o w e r series, a n d c a n be p r o v e d e a s i l y f r o m t h e t h e o r e m of H a r d y a n d L i t t l e w o o d . T h e t h e o r e m is as follows: i/ / is summable on the torus with Fourier aeries

amn et(mx+ny),

mpn=O

P R E D I C T I O N T H E O R Y A N D F O U R I E R S E R I E S I N S E V E R A L V A R I A B L E S 181

then ~ lam. I/(mn + ])<k f JfJd(~.

97t, n = O

T h e l a s t t h e o r e m of t h i s s e c t i o n generalizes a t h e o r e m of B e u r l i n g [5].

T H E O R E M 6. Let H be the linear subspace of L ~ consisting o/ the functions whose Fourier series have the form

a+ Z amn e-i(mx+ny)*

S

For any / in H let C r be the smallest closed linear manifold containing (b+ P ) /

for all constants b and trigonometric polynomials P o/ the form (10). We have C r = H i/ and only if

f log l/ld~=log lf /dcr > - ~ .

I t is e a s y t o see t h a t C I = H if a n d o n l y if t h e r e is a n o n - z e r o c o n s t a n t f u n c - t i o n in t h e closure of t h e c o n v e x set of f u n c t i o n s (1 + P ) / . T h e p r o o f t h a t t h i s is e q u i v a l e n t t o t h e c o n d i t i o n of t h e t h e o r e m is e a s y t o c a r r y o u t using T h e o r e m s 1 a n d 2 a n d t h e P a r s e v a l e q u a l i t y .

T h e p r o b l e m s discussed here for t h e case w h e r e S is a h a l f - p l a n e b e c o m e m u c h m o r e difficult when S is, for e x a m p l e , t h e set of l a t t i c e p o i n t s c o n t a i n e d in s o m e s e c t o r of o p e n i n g s m a l l e r t h a n ~. T h e r e is no l o n g e r a n y a n a l o g y w i t h a n a l y t i c f u n c t i o n s of one v a r i a b l e . I t seems to us t h a t t h e s e new p r o b l e m s a r e difficult a n d i n t e r e s t i n g .

4. Theorem of Riesz and Bochner

T h e t h e o r e m of F . a n d M. Riesz [15] ( a l r e a d y r e f e r r e d to in t h e I n t r o d u c t i o n ) s t a t e s : if u is a bounded complex Borel measure on the circle whose Fourier-Stielt~es coefficients vanish /or positive indices, then /~ is absolutely continuous with respect to Lebesgue measure. B o c h n e r [6] o b s e r v e d t h a t n o t e v e r y m e a s u r e on t h e t o r u s w i t h coefficient~ r e s t r i c t e d t o a h a l f - p l a n e is a b s o l u t e l y c o n t i n u o u s w i t h r e s p e c t to t h e in- v a r i a n t m e a s u r e on t h e t o m s ; b u t B o c h n e r p r o v e d t h a t t h e conclusion holds if t h e n o n - v a n i s h i n g coefficients a r e all in a s e c t o r of o p e n i n g less t h a n ~. T h e m a c h i n e r y of B o c h n e r ' s p r o o f is v e r y e l a b o r a t e . I n t h i s s e c t i o n we shall give a new p r o o f of B o c h n e r ' s T h e o r e m w h i c h shows its close c o n n e c t i o n w i t h p r e d i c t i o n t h e o r y . On t h e w a y we p r e s e n t a n e x a m p l e a n d s o m e l e m m a s of i n d e p e n d e n t i n t e r e s t .

A certain p a r t of t h e Riesz T h e o r e m survives in two dimensions, a n d gives t h e following preliminary result.

L EMMA 3. Let # be a complex measure on the torus without absolutely continuous part. I / t h e Fourier-Stielt~es coe//icients

cmn= f e-i('~x+n~) d/~ (x, y) vanish /or all (m, n) in a hall-plane, then also Coo = 0.

D e n o t e t h e total variation of # b y v. T h e n ~ is also singular with respect to d a, a n d so b y T h e o r e m 1

inf f [1 + P ] ~ d v = O ,

P

where P ranges over the trigonometric polynomials of t h e f o r m (10).

L e t P1, P2 . . . . be a sequence of such polynomials for which lim f [ l + P n ] 2 d v = O .

T h e n c l e a r l y lira f (1 + P~) d # = O.

B y hypothesis, for each n f Pn d~u = 0,

a n d so Coo = f d/~ = 0.

I n t h e one-dimensional case, h a v i n g shown in this w a y t h a t c o is zero we can translate t h e coefficient sequence a n d prove in t u r n t h a t c_1, c 2 .. . . all vanish. I n t w o dimensions we c a n n o t conclude a n y t h i n g more from the fact t h a t Coo = 0 ; indeed there exist singular measures # whose coefficients vanish in a half-plane b u t n o t everywhere.

The trivial example, which is m e n t i o n e d b y Boehner, is given b y t h e p r o d u c t of a singular measure d~(x) on the interval with t h e measure e-iYdy. T h e p r o d u c t measure d # is clearly singular with respect to two-dimensional Lebesgue m e a s u r e ; its coefficients are given b y

a n d t h u s vanish for all n >~ 0.

I t is less obvious t h a t there are singular measures whose coefficients vanish on a half-plane b o u n d e d b y a line l: of irrational slope, a n d t h e following c o n s t r u c t i o n of such a measure m a y be of interest.

P R E D I C T I O N T H E O R Y A N D F O U R I E R S E R I E S I N S E V E R A L V A R I A B L E S 183

Project each lattice point onto the real line in the direction parallel to /:. Dis- tinct lattice points have distinct projections, since /: has irrational slope. Moreover, the vector sum of two lattice points is projected onto the ordinary sum of their separate projections. So the lattice points arc isomorphic as a group with a denumer- able dense subgroup of the line, which we endow with the discrete topology and call G. Now • determines two half-planes; the points of one half-plane are projected into the positive r a y of the real line, and the points of the other half.plane into the negative ray.

Now let / be the function on the line equal to one at the origin, a n d decreasing linearly to zero a t 1 and - 1. I t is well-known t h a t / is positive definite. A ]ortiori, ] is positive definite as a function on G. B y .the general theorem of Herglotz, Boeh- ner, and Weil on positive definite functions, / is the transform of a positive measure on the dual group of G, which is the torus. I f we set

g ( x ) = / ( x + l) for x E G ,

then g is the transform of a complex measure # on the torus, and g vanishes for x >~ 0. Considered as a function on the group of lattice points, g vanishes on a half- plane bounded b y s

I f be were absolutely continuous, b y the general Riemann-Lebesgue L e m m a its transform g would tend to zero outside compact sets of G. Since G is discrete, this would mean t h a t Ig(x) l>~e only for a finite set of x, for a n y e > 0 . Obviously this is not the case, so # cannot be absolutely continuous. I t will follow from the n e x t theorem t h a t the singular p a r t of be (in case be is not itself singular) has coefficients vanishing on the same half-plane as the coefficients of be, and this is the example we wanted to find.

T rrEOREM 7. Let be be a measure on the torus whose coe/~icients vanish on a hall.

plane S. Then the coe/]icients o/ its singular and absolutely continuous parts vanish separately on S.

Proo]. Let v be the total variation of be. After adding to be a multiple of Le- besgue measure if necessary, we m a y assume t h a t

inf / l 1 + P I g g y > O,

P

where P ranges over trigonometric polynomials of the form (10). Choose a sequence of polynomials /~ Pz . . . . such t h a t

lira f l l + P , , 1 2 d v = i n f f l l + P l ' d v ,

P

and denote b y 1 + H the limit of 1 + Pn in the space of functions square-summable for the measure d~. L e t P a n d Q be a n y trigonometric polynomials of the form (10) ; using the hypothesis on the coefficients of /~ we have

f P ( 1 + Q ) ( 1 + H) d # = l i m f P ( 1 + Q ) ( I + P n ) d # = O .

Since 1 + H vanishes almost everywhere for the singular p a r t of /~, we have also

fp(l+

Q)(1 + H ) / d a = O , (37) where / d a is the absolutely continuous p a r t of in.

The theorem will be proved if we can show t h a t

f P / d a = O (38)

for arbitrary P of the form (10), for this means t h a t the Fourier coefficients of / vanish on S. We shall need the following relations:

( I + H ) -1 belongs to L2;

( l + H ) - l ~ b + ~ bm~e -t(m~+n~) with b * 0 ;

s

(1 + H ) " / belongs to L 2.

Choose a sequence of trigonometric polynomials (1 + Q~), with each Q~ of the form (10), converging in L ~ to b -1 (1 § -I. B y (37), for each n

f P ( l + Q n ) ( I + H ) / d a = O ;

since ( I + H ) . / is square-summable we can pass to the limit in n and obtain (38).

This completes the proof.

B o c H N S ~'s T H ~ 0 R ~ M. Let T be a sector o/ the plane with opening greater than 7~ radians. Suppose # is a measure on the torus whose coe//icients vanish on T. Then

# is absolutely continuous with respect to Lebesgue measure.

Proo/. I t suffices to consider a sector T with center at the origin, so t h a t T contains the union of different half-planes S and S'. Let /us be the singular p a r t of

# ; b y Theorem 7, the coefficients of #s vanish on S and also on S'. I f a n y coeffi- cient of /us is different from zero, it is easy to see t h a t the coefficient set can b e translated so as to bring a non-zero coefficient to the origin, still leaving a half-plane free of non-zero coefficients. (Indeed, find a line 1: through the origin with irrational slope, lying between the lines bounding S a n d S' in such a way t h a t T contains

P R E D I C T I O N T H E O R Y A N D F O U R I E R S E R I E S I N S E V E R A L V A R I A B L E S 1 8 5

one of t h e half-planes b o u n d e d b y s I f we t r a n s l a t e 1: we e n c o u n t e r a first lattice point at which /~ has non-zero coefficient, a n d t h e inverse translation is the required one.) B u t the result of this c o n s t r u c t i o n is a coefficient set belonging t o a singular measure, vanishing on a half.plane, b u t n o t at the origin. This contradicts L e m m a 3, a n d so /~s is the null measure. This completes t h e proof.

5. Matrix Valued Analytic Functions

F o r each point e ~ on the circle let A (e ~) be an n b y n m a t r i x with entries ajk (e ~) 0", k = l, . . . , n). The normalized trace of A is t h e scalar function

t r A (e L) = 1 ~ a ~ (e~).

n k

The trace a n d d e t e r m i n a n t functions are related b y t h e f o r m u l a d e t e ~ = e n tr 4,

where e A m a y be defined b y its power series.

T h e normalized trace has the following properties (and, in fact, is d e t e r m i n e d b y them): for a n y matrices A, B a n d scalars a, b

t r ( a A + b B ) = a t r A + b tr B t r (AB)= tr (BA), or equivalently

t r ( U - 1 A U ) = t r A if U is u n i t a r y (39) t r A*A >1 O, f r o m which follows t r A * = t r A

t r I = 1, I t h e u n i t m a t r i x .

I f A is a positive definite m a t r i x , there is a unique H e r m i t i a n m a t r i x B satisfying

e ~ = A, (40)

a n d we define B--- log A.

B y a trigonometric polynomial, in t h e c o n t e x t of m a t r i x functions, we shall m e a n a finite s u m of t h e f o r m

Ak e%

where each A , is a c o n s t a n t m a t r i x . The t r i g o n o m e t r i c p o l y n o m i a l is analytic if A , = 0 for n < 0 .

I f each c o m p o n e n t function aju of t h e m a t r i x f u n c t i o n A is s u m m a b l e , we shall s a y t h a t A is summable, or belongs to L. More generally, L v is to consist of t h e

13 - 6 6 5 0 6 4 A e t a m a t h e m a t i c a . 99. I m p r i m ~ le 10 j u i n 1 9 5 8

matrix functions A whose scalar components ajk all belong to the ordinary class L p.

A summable matrix function A has Fourier series A (e'X)'~ 7f Ak e ~ ,

where each Ak is the constant matrix defined by the n 2 scalar equations

A~= (

A (e ~) e-~kX d a (x).

(In this section, d a ( x ) is the measure d x / 2 ~ on the circle.) More generally, if M is a completely additive matrix-valued function of Borel sets (in other words a matrix whose entries are complex measures), we shall write

d M (e u) -~ ~ Ak e ~k~

with the Ak defined as

Ak = f e -ikx d M (e~).

I t follows from definition that a measurable matrix function A is in L ~ if and only if tr ( A ' A ) is summable. We shall also need the fact t h a t a measurable positive semi-definite matrix function W is summable if and only if tr W is a summable scalar function.

The ring of constant matrices possesses the natural inner product

(A, B ) = t r ( B ' A ) . (41)

We can extend this definition to the class of matrix functions in L ~ by setting (A, B ) = ( t r (B* A ) d a = ~ f ajk$j~d(~, (42)

9 j , k

where ajk and bjk are component functions of A and B. The Parseval equality holds for square-summable functions A and B with Fourier coefficients A~ and Bk:

(A, B) = ~ (A~, Bk); (43)

in this formula the inner product on the left is defined by (42), and those on the right by (41).

The main theorem of this section is an extension of SzegS's Theorem to matrix- valued functions defined on the circle.

J

T H E O R E M 8. Let M be a matrix-valued measure de/ined on the circle such that M (E) is Hermitian and positive semi-de/inite /or every Borel set E. Let M have Le- besgue decomposition

d M (e ~) = W (e ~) d a + d M~ (d~),

I ~ R E D I C T I O N T H E O R Y A N D F O U R I E R S E R I E s I N S E V E R A L V A R I A B L E S 187 where W is a summable matrix /unction and Ms is singular with respect to d(~. Then exp If tr log W d o} = inf f tr [(A 0 + P)* (A 0 + P) d M], (1) (44)

Ao, P

where A o ranges over the matrices with determinant one, and P over trigonometric poly- nomials o/ the /orm

P (e ~) = ~ Ak e% (45)

k > 0

The left side o/ (44) is to be interpreted as zero i/

f tr log W d a = - oo. (46)

Proo/. In outline we can follow the proof of T h e o r e m l, meeting each new complication as it arises. L e t L ~ be the set of functions A for which

li A 1[2M = f t r (A* A d M) < oo ; (47) the norm so defined is positive semi-definite. After identifying functions which differ only on a null-set of d M, L~M is a H i l b e r t space with inner p r o d u c t

(A, B)M = f tr (B* A d M ) . (48)

If the infimum on the right side of (44) is positive, choose and fix A 0 with deter- m i n a n t one, and let H be t h a t element of L ~ which is the limit of polynomials P of the form (45) and satisfies

f t r [ ( A 0 + H ) * (Ao+ H) d M ] = i n f f t r [ ( A 0 + P ) * (Ao+ P) d M ].

P

The a r g u m e n t leading to (7) and (8) gives analogous o r t h o g o n a l i t y relations hel~e.

If n > 0 and G is a n y non-zero constant matrix, the expression IJ'4o+ H + lIM

has a unique m i n i m u m at ~ = 0. I t follows t h a t

( A o + H , Ge~nx)M=O ( n = l , 2 . . . . ). (49) A n d the expressions

][ (A o + H) (I + ~ G e ~nx) JIM = [[ (A o + H) + ~ ( i o + H) G e '"x JIM, [[ (I + 2 Ge 'nx) (A o + H)JIM = [[ (A0 + H) + 2 G (A 0 + H) e ~ JIM

(1) The pedantic reader can easily write this symbolic integral literally in terms of the scalar component measures of ~/.

have m i n i m a at ~t = 0, u n i q u e l y unless

(A 0 + H) G = 0 or G (A o + H) = 0.

I n a n y case we have

((A o + H), (A o + H) G e~nx)M = 0 (50)

( n = l , 2 . . . . ).

a n d ((A o + H), G (A o + H) e~nx)M = 0 (51)

T h e definition (48) means t h a t (51) can be written

f e -*~x t r [(A o + H)* G* (A o + H) dM] = 0. (52) T a k i n g the complex c o n j u g a t e of (52) a n d m a k i n g use of (39),

f e ~ tr [(A o + H)* G (A o + H) d M] = 0.

These formulas hold for all G, a n d for n = l, 2 . . . . ; it is easy t o see t h e n t h a t (52) is valid for negative as well as positive integers n. Hence

tr [G (A o + H) d M (A 0 + H)*]

is a c o n s t a n t multiple of scalar Lebesgue measure for each G, so t h a t every com- p o n e n t of the m a t r i x measure

(A o + H) d M (A o + H)*

is a multiple of Lebesgue measure. This fact c a n be written ( A o + H ) d M ( A o + H ) * = C d a (C constant).

Therefore we have

(A o + H) d Ms (Ao + H)* = 0, (53)

(A o + H) W (A o + H)* = C. (54)

I t follows f r o m (53) t h a t A o + H vanishes almost everywhere for dMs, so t h a t (49) takes t h e alternate f o r m

( A o + H , Ge~nX)w=O (G a r b i t r a r y ; n = l , 2 . . . . ), (49') where the inner p r o d u c t refers to t h e Hilbert space of m a t r i x functions square-sum- mable for W d a.

As in t h e scalar case, we conclude f r o m (49') t h a t A o + H has t h e same m i n i m a l p r o p e r t y in L2w t h a t it enjoys in L ~ , a n d so t h e i n f i m u m on t h e right side of (44) is n o t reduced if we replace d M b y W da. Assuming, then, t h a t this i n f i m u m is positive, t h e t h e o r e m will be p r o v e d if we show

exp {ftr log Wda} = inf f t r [(Ao+P)* (ao+P) WJda. (55)

Ao, P -

P R E D I C T I O N T H E O R Y A N D F O U R I E R S E R I E S I N S E V E R A L V A R I A B L E S 1 8 9

On the other hand, if the infimum in (44) is zero, it still suffices to prove (55), b y the same argument as in the scalar case.

L EMMA 4. Let W be Hermitian, positive semi-de/inite and summable. Then exp {f tr log W d a} = inf f tr (e ~" W ) d a, (56) where ~1 r ranges over the Hermitian matrix ]unctions with summable trace /or which

f t r Ut~da=O. (57)

The trace of a Hermitian matrix is the average of its proper values; and the determinant is the product of the same numbers. Using the inequality of the arith- metic and geometric means twice we obtain

e x p { f t r l o g W d a } = e x p { l f l o g d e t W d a } < ~ f ( d e t W ) i l ~ d a < ~ f t r W d a . (58) I n order to have continued equality it is necessary and sufficient t h a t

tr W----(det W ) l / ~ c o n s t a n t ,

which is to say t h a t W is a constant multiple of the identity matrix.

L e t uF be a H e r m i t i a n matrix function with s u m m a b l e trace satisfying (57);

whether or not the positive semi-definite matrix function W' = e ~ W e ~"

is summable, we have as in (58)

exp { f t r l o g W ' d a } < ~ f t r W ' d a < , < ~ . The properties of the trace and d e t e r m i n a n t functions give

n tr log W' = log det W' = log det (e ~ W) = n t r LF + n tr log W, tr W' = t r (e ~" W).

B y (57) we have for every function LF

exp l f t r log W d a } ~ < f t r (e r W ) d a . (59) I f t r log W is summable, define

~ F 0 = 2 I - l o g W; 2 = f t r log W d a . (60) Then ~F 0 is H e r m i t i a n and satisfies (57), and obviously reduces ( 5 9 ) t o equality. This completes the proof if t r log W is s u m m a b l e ; otherwise a limiting process has to be carried out as in the scalar case.

L E ~ M A 5. The statement (56) is still true i/ ~ ranges only over the class o/

trigonometric polynomials which are Hermitian and satis/y (57).

Let A and B be commuting Hermitian matrices. They have a common complete set of proper vectors. Define max (A, B) to be the Hermitian matrix with the same proper vectors and with proper values the larger of the corresponding proper values of A and B. For a n y Hermitian matrix A, the positive part of A can be defined as max (A, 0). This construction makes it possible to carry through the proof of L e m m a 2 unchanged for the matrix case.

B y analogy with the proof for scalar functions, we should like to factor each function e ~r of L e m m a 5 into a product

(Ao + A l e~ § ...)* (Ao § A l e~ + -..),

and then show t h a t it suffices to consider trigonometric polynomials in place of the infinite series. The non-commutativity of matrices introduces a difficulty which must presently be met.

LEMMA 6. Let W be a summable positive semi-definite matrix /unction /or which the infimum o/ (55) is positive. Then W has a /actorization

W = B B* (61)

where B is a matrix /unction in L 2 with analytic Fourier series:

B (e ~) ~ ~ Bn e in~ and det B 0 * 0, (62)

0

I n applying this lemma, we shall need a stronger result than (62) for a narrow class of functions W. I t will be convenient to refer later to the proof as well as the statement of the lemma.

By hypothesis, the convex set of trigonometric polynomials of the form I + ~ Ak e ~x

k>0

is bounded from zero in L2w. Let I § H be the unique element of minimal norm in the closure of this set. From (54) we have

(I + H) W (I + H)* = C, (63)

where C is a constant matrix. We assert that C is non-singular. Indeed, otherwise we could find matrices A with determinant one for which

tr ( A C A * ) = f t r [ ( A + A H ) W ( A + A H ) * ] d a

P R E D I C T I O N T H E O R Y AI~D F O U R I E R S E R I E S I N S E V E R A L V A R I A B L E S 191 is as close to zero as we please, whereas this q u a n t i t y is b o u n d e d f r o m zero b y hypothesis. Clearly t h e n C is H e r m i t i a n a n d positive definite, a n d so has a non- singular square root. The factorization (63) can be p u t in the f o r m

( A o + A o H ) W ( A o + A o H ) * = I ( A 0 = C - t ) (64)

or W = B B * ; B = (A o + AoH) -1. (65)

F r o m (65) it follows t h a t B is square-summable. We shall prove the l e m m a b y showing t h a t its Fourier series is of a n a l y t i c t y p e :

(Ao + AoH)-I,,~ Bo + BierS+ .... (66)

I t suffices to establish t h a t

f t r [ G ( A o + A o H ) - X ] e i ' X d a ( x ) = O ( n = 1, 2 . . . . ) (67) for e v e r y c o n s t a n t m a t r i x G, since t h e n e v e r y c o m p o n e n t function of ( A o + A o H ) -1 is analytic. Making use of (64), t h e left side of (67) is equal to

f tr [O W (A 0 + A 0 H)*] e *n~ d a (x) = f tr [(A 0 + A o H)* O W] e *n~ d a (x) = (A~ O e *n~, I + H)w.

This inner p r o d u c t vanishes b y (49') for n = 1, 2 . . . so t h a t (67) holds.

L e t ~I ~ be a trigonometric p o l y n o m i a l satisfying (57). Then there exists a factori- zation

e ~ = A* A ; A (e ~) N ~ An e *n~, d e t A o = 1. (68) 0

To prove this fact, we consider the positive definite weight f u n c t i o n W = e -~e.

The eigenvalues of W are b o u n d e d f r o m zero a n d from i n f i n i t y ; it follows t h a t t h e spaces L ~ a n d L 2 have e q u i v a l e n t norms. B y a simple calculation we can show t h a t

[[A[[a= f t r ( A * A ) d a ~ l

for each analytic t r i g o n o m e t r i c p o l y n o m i a l A whose leading coefficient A 0 has deter- m i n a n t one. Therefore the i n f i m u m of (55) is positive for this function W, a n d b y L e m m a 6

e - ' r = B B * ; B(et~),~ ~ Bn e~nx. (69)

0

I f t h e scalar c o m p o n e n t s of B(e ~z) are d e n o t e d b y b~j(e t~) for i, j = 1, 2 . . . n, we have

n tr e - ~ ' = ~ lb.[ 2,

i , t

f r o m which it is clear t h a t t h e functions b~j are bounded. N o w the d e t e r m i n a n t of B is a s u m of p r o d u c t s of these functions, a n d since t h e b~i are b o u n d e d , the F o u r i e r

series Of det B can be computed formally from the Fourier series of the component functions. Each bit is analytic, and we find

det B (e ~) ~ det B 0 + c 1 e u + c 2 e 2ix + . . . .

The one-dimensional version of Theorem 2 (or J e n s e n ' s formula) gives

f log I det B Is d a/> log I det B 01 s. (70) The inverse of B is the function A = A o + A o H , obtained as the limit in L~w of a sequence of analytic trigonometric polynomials each having constant t e r m A 0. With the present choice of W the sequence converges in L s as well, so t h a t A is an analytic element of L2:

A,~Ao+Ale~X+ ... ; Ao=C -89

F r o m (69) we have e~'=A*A, (71)

so the components of A, like those of B, are bounded functions; exactly as for B t h e n we have

f l o g [det A I ' d a > ~ l o g Idet Aol s. (72) Now A and B are bounded functions with analytic Fourier series, so the Fourier series of their product is obtained b y formal multiplication of the series for A and B and consequently AoBo=I. I t follows t h a t the right side of ( 7 0 ) a n d of ( 7 2 ) i s finite. Adding these inequalities gives zero on b o t h sides. Therefore (70) and (72) are actually equalities.

B y a s s u m p t i o n ~F satisfies {57). Making use of (71) we have 0 = f log d a t e ~ d a = f log ]det A I s d a = log I det A , I s,

The determinant of A 0 is at a n y rate positive, and therefore is equal to one. Thus (71) is a factorization of the kind we wanted.

Now we can prove (55). L e t W be a H e r m i t i a n positive semi-definite sum- mable m a t r i x function, and let ~F be a H e r m i t i a n trigonometric polynomiM satisfying (57). F r o m the result just proved,

ftr

( e ~ W ) d a = f tr (A*AW)da,

where A is a bounded analytic function and det A 0 = 1. Therefore b y L e m m a 5 exp { f t r l o g W d a } > ~ i ~ a f t r ( A * A W ) d a ,

where A ranges over all bounded functions of analytic t y p e such t h a t det A 0 = 1.

Each component function ats of A is boundedly t h e limit of Fej~r means of its Fou-

P R E D I C T I O N T H E O R Y A N D F O U R I E R S E R I E S I N S E V E R A L V A R I A B L E S 1 9 3

rier series, f r o m which it follows t h a t t h e last i n e q u a l i t y r e m a i n s t r u e w h e n .4 ranges m e r e l y o v e r trigonometric polynomials of the s a m e kind. This is one half of (55).

T h e opposite inequality follows as in the scalar case, or can be deduced directly f r o m t h e one-dimensional version of T h e o r e m 2.

T H E o R E M 9. Let W be a Hermitian positive semi-de[inite summable m a t r i x / u n c . tion defined on the circle. A necessary and su//icient condition /or W to have a /actori- zation

W = B B * , (73)

where B is in L 2 with Fourier series o/ the /orm

B (e ~x) ~ ~ B n e ~nx (det B o * 0)

o

is that

f t r l o g W d a > - ~ . (74)

I / this condition is saris/led we can choose B so that

f l o g ]det B I d a = l o g I d e t Bol. (75) Proo/. Suppose first t h a t W has a factorization of the required kind.. F.or a n y t r i g o n o m e t r i c p o l y n o m i a l of t h e f o r m

A (e ~x) = A o + A I e i~ + . . . , det A o = I we h a v e clearly

f t r (A* A W) d a = f t r [(A B)* (A B)] d a ~> t r [(A o Bo)* (A o Bo) ] ~>

~>ldetAoB0121n=ldetB0121n>0. (76) T h e n (55) shows t h a t t r log W is s u m m a b l e .

Conversely, suppose t h a t (74) holds. T h e n (55) shows t h a t t h e h y p o t h e s i s of L e m m a 6 is satisfied, so there is a factorization (73). W e shall show t h a t the factori- z a t i o n furnished b y L e m m a 6 has the p r o p e r t y (75).

F r o m (76) a n d (55) once m o r e we see t h a t for a n y factorization of the f o r m (73) we h a v e

e x p

{ftr log Wd(r}>~ldetBol 2"~,

which is e q u i v a l e n t to one i n e q u a l i t y in (75). N o w for a n y positive definite m a t r i x C, it is a n e l e m e n t a r y fact t h a t

(det C) 1/n = i n f t r (A o CA~),

A,