PREDICTION THEORY AND FOURIER SERIES IN SEVERAL VARIABLES. II
BY
H E N R Y H E L S O N Berkeley, California
AND
D A V I D L O W D E N S L A G E R Princeton, New Jersey O)
1. Introduction
T h i s s e c o n d p a p e r o n P r e d i c t i o n T h e o r y , like o u r first one [10], is d i v i d e d i n t o t w o p a r t s : t h e first, consisting of t h e first e i g h t sections, t r e a t s c o m p l e x - v a l u e d f u n c t i o n s d e f i n e d on r a t h e r g e n e r a l groups, a n d t h e s e c o n d p a r t d e a l s w i t h m a t r i x - v a l u e d f u n c t i o n s d e f i n e d o n t h e u n i t circle. I n b o t h p a r t s we a r e c o n c e r n e d w i t h d e g e n e r a c i e s w h i c h were e x c l u d e d b y our h y p o t h e s e s before, b u t w h i c h t u r n o u t to be i n t e r e s t i n g f r o m b o t h t h e f u n c t i o n - t h e o r e t i c a n d t h e p r e d i c t i o n - t h e o r e t i c p o i n t s of view.
U n l i k e t h e first p a p e r , t h i s one h a s t o d o w i t h difficulties w h i c h do n o t e x i s t a t a l l for t h e classical case of s c a l a r f u n c t i o n s d e f i n e d on t h e circle g r o u p . B o t h p a r t s of t h e p a p e r l e a v e i n t e r e s t i n g p r o b l e m s u n s o l v e d . I n t h i s i n t r o d u c t i o n w e shall t r y t o p r e s e n t t h e q u e s t i o n s of t h i s p a p e r a n d our c o n t r i b u t i o n t o t h e i r s o l u t i o n in b r o a d t e r m s .
L e t / = / ( e ~x) be a s u m m a b l e f u n c t i o n d e f i n e d o n t h e circle whose 1%urier series h a s t h e f o r m
/
(dO ~ Y ~~x.
n~>O
I t is i m p o r t a n t a n d w e l l - k n o w n t h a t
f l o g I / 1 ~ ( ' )
(:) The authors acknowledge the support of the National Science Foundation, and in the case of the second-named author, of the United States Air Force.
(2) d a always denotes normalized invariant measure on the compact group being considered, ttere it is dx/2zc on (0, 2~z).
176 H E N R Y H E L S O N A N D D A V I D L O W D E N S L A G E R
is finite, unless ] is t h e null-function. I n particular, [ c a n n o t v a n i s h on a set of positive measure.
I n our first p a p e r we o b t a i n e d a generalization of this t h e o r e m to t h e class of c o m p a c t abelian groups whose duals are linearly ordered, b u t u n d e r t h e h y p o t h e s i s t h a t a0, t h e m e a n value of f, is different f r o m 0. F o r t h e circle g r o u p this is no restriction a t all, because we can consider e -v~x] (where p is a positive integer chosen appropriately) in place of [, b u t this device fails if t h e ordered g r o u p which generalizes t h e integer g r o u p has no least positive element. Indeed, trivial examples show t h a t if a 0 = 0 t h e f u n c t i o n m a y vanish on a non- e m p t y open set w i t h o u t vanishing identically.
U n d e r t h e hypothesis t h a t the order relation is archimedean t h e counter-examples just referred to are defeated, a n d Arens [3] has p r o v e d t h a t t h e finiteness of t h e integral follows f r o m t h e continuity of ]. W e shall prove in t h e same direction t h a t ] c a n n o t vanish on a set of positive measure unless it vanishes identically, w i t h o u t assuming t h a t a 0 # 0 or t h a t ] is continuous. I t m a y be u n e x p e c t e d therefore t h a t even bounded functions exist for which t h e logarithmic integral diverges. (This negative result is a corollary of a prediction t h e o r e m of different character.) W e h a v e additional information, b u t no definitive result, on t h e question which n o n - n e g a t i v e functions are t h e m o d u l u s of some f u n c t i o n ] of t h e class considered.
I n t h e second p a r t of t h e paper we reconsider t h e problems of [10] for positive semi- definite m a t r i x functions W = W (e *x) of less t h a n full rank. I n prediction-theoretic terms, we s t u d y a process of less t h a n full r a n k whose covariance m a t r i x is absolutely continuous.
F o r a process with co~r m a t r i x W (of full r a n k or not) these conditions are k n o w n t o be equivalent: t h e process has no r e m o t e past; t h e process is a m o v i n g average; W has t h e form A A * , where A is an analytic m a t r i x function. F o r processes of full rank, these properties are equivalent to this analytic condition on W:
f l o g d e t W d a > - ~ .
B u t this integral always diverges if W is singular, for example if W is A A * for a singular analytic f u n c t i o n A; in this f o r m t h e integral is t h u s t o o crude to give i n f o r m a t i o n a b o u t processes of less t h a n full rank.
A t a l m o s t e v e r y p o i n t W(e ~x) is a positive semi-definite m a t r i x which operates as a non-singular t r a n s f o r m a t i o n on its range ~ (e*X). D e n o t e the d e t e r m i n a n t of this trans- f o r m a t i o n b y A W (e'x). If we replace the d e t e r m i n a n t function in the integral b y A, t h e n t h e finiteness of t h e integral is the first condition for t h e process to be a m o v i n g average.
There is an obvious second necessary condition. I f W = A A* for some a n a l y t i c rune-
P R E D I C T I O I ~ I T H E O R Y A N D F O U R I E R S E R I E S II~ S E V E I ~ A L V A R I A B L E S . IT 177 tion A, then ,~ (e ~x) coincides at almost every point with the range of A (e~X). I n order for W to have this form it is evidently necessary for ~ (e ~x) to be the range of some analytic function.
We prove that these conditions together are sufficient as well as necessary for the process to be a moving average, or for W to have the form A A * for some analytic function A. Of course in application it m a y be difficult to decide whether the range of W coincides with the range of an analytic function, and the criterion found by Masani and Wiener [17] for W to be a square when it is of order two therefore has independent interest.
The theorem on the factoring of W depends on structure theorems for analytic matrix functions which are proved first. We follow L a x [14] in taking Beurling's notions of inner function and outer function as fundamental, b u t we offer our own definition for matrix functions which leads to a new description of singular analytic functions.
I n connection with the first part of the paper, we draw attention to the related work of Arens, Hoffman, and Singer [2, 3, 4, 5, 12]. At a crucial stage of our research we h a d the good fortune to talk at ]ength with Professor P. Malliavin, and we are grateful for his permission to incorporate his ideas in this paper. As far as possible we shall identify his contributions in context.
As with our first paper, this one overlaps the work of Masani and Wiener [16] to some extent. Following several notes, a third paper has been published recently b y Masani [15].
These authors refer in their various papers to Russian work of which we have taken no ac- count.
2. NIise e n s e ~ n e
I n this section we lay down notation and recount known results which will be used throughout the first part of the paper.
R denotes the real line, and Ra the same group in the discrete topology. The elements of R~ will always be called A or ~. The character group of R a is a compact abelian group B also obtained as the Bohr compactffication of R, with elements x, y . . . . , and ]-Iaar measure d a (normalized to have unit total mass). For each A in R a let Z~ be the character of B defined by
z~(x) =x(~)
(allxCB). (1)
The correspondence of ~ to )/~ is an isomorphism of R d with the character group of B, and we have for example
Z0(x ) = 1, z~(x).Z~(x)-Z~+~(x). (2)
178 I-IEIffl~Y H E L S O N A N D D A V I D L O W D E N S L A G E R
W e shall nevertheless preserve t h e n o t a t i o n a l distinction between elements of Ra a n d characters on B, for convenience r a t h e r t h a n for purity.
T h e Fourier series of a f u n c t i o n / which is defined a n d s u m m a b l e on B has t h e f o r m
/ (x) -~ ~ a (2) Z~ (x), (3)
2
with coefficients defined b y
a (2) = f (2) = f Zx (x) / (x) d (r (x). (4)
The Fourier-Stieltjes series of a finite complex m e a s u r e / z is given b y similar formulas.
I t is k n o w n t h a t every discrete abelian group with art a r c h i m e d e a n order relation is isomorphic (with preservation of order) to a subgroup of R a. All t h e results of t h e first p a r t of t h e paper could be stated a n d p r o v e d for a r b i t r a r y archimedean-ordered discrete groups a n d their duals, which means for subgroups of Rd a n d the dual quotient groups of B.
I n d e e d the a r c h i m e d e a n hypothesis is n o t used until section five, so t h a t our first theorems are true in the more general conditions of [10]. W e prefer, however, to write explicitly a b o u t Rd a n d B in order to avoid n o t a t i o n a l difficulty.
A m o n g the characters of Ra are some h a v i n g the f o r m
eu (2) = e ~u~ (5)
for some real n u m b e r u. The m a p p i n g f r o m u to e~ carries real n u m b e r s into B. I t is k n o w n t h a t this m a p p i n g is a one-one continuous isomorphism of R o n t o a dense subgroup B 0 of B. Sometimes it is convenient to identify u with % a n d t h i n k of R as a subset of B. W e shall be concerned later with t h e ergodic properties of B 0 in B.
As generalizations of the H a r d y spaces of analytic functions defined in t h e circle, we consider the spaces H p (1 ~< p ~< oo) of functions defined on B, belonging to L p, whose Fourier coefficients (4) vanish for all 2 < 0. W h e n the value of p is u n i m p o r t a n t we m a y call such a function analytic. I n particular an analytic trigonometric polynomial is a finite sum of t h e form
a (2) Za (x). (6)
2~>0
F o r each p, H~ is t h e subspace of H " consisting of those functions whose m e a n value a (0) vanishes.
A function / in some H " is called outer [6] if
f log I / I d (r = log l f / &r I > - ~ - (7)
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 . I I 179 Beurling proved the fundamental result t h a t / in H e is outer i/ and only i / t h e set o/ /unctions P/, where P ranges over all analytic trigonometric polynomials, is dense in H a. (Beurling's Theorem, originally proved for the circle group, was extended in [10] to compact groups with ordered duals.)
F r o m (7) it follows t h a t every function w of the form I/I 2 for some outer function / in H~ m u s t satisfy
f l o g w d a > - ~ . (8)
Conversely, every non-negative summable /unction w satis/ying (8) is such a square, and moreover ] is unique up to a constant/actor o/ modulus 1. This theorem was also proved (not quite explicitly) in [10]; the unicity of / will be reconsidered in section seven, where the notion of outer function is further developed.
If / is in H 1 and has mean value di//erent/tom O, then [10] there is an outer function g such t h a t ]/] = [gl almost everywhere. I f we write / = g. h, then the function h is analytic also. An analytic function with modulus 1 almost everywhere is called an inner/unction b y Bcurling.
This notation and these results will be used in the first p a r t of the p a p e r without further reference.
3. The Wold decomposition
L e t # be a non-negative finite measure defined on the Borel subsets of B. We form the Hilbert space L~ with inner product
(/, g) = f / ~ d # . (9)
The functions Z~ ( - ~ < 2 < co ) belong to L~ and form a complete set. Moreover t h e y form a stationary stochastic process in the sense t h a t the inner products
(Z~, Z ) =fV~Z_~ d ~ = @ (T - A) (lO)
depend only on the difference T - 2 . We know t h a t a n y stationary process depending on a real p a r a m e t e r (without a n y continuity hypothesis) is isomorphic, with respect to all g i l b e r t space properties, to the process of characters in L~ for a suitable measure #.
For each ~ in R~ we define a unitary operator S~ in L~ b y setting
S~/ =Z~./. (11)
A closed subspace ~ of L~ is called invariant if it is carried onto itself b y each S~.
180 H E N R Y H E L S O N A N D D A V I D L O W D E N S L A G E I ~
L]~MMA 1. Let ~ be a closed invariant subspace o/ L~. The orthogonal projection P on has the/orm
r / = e . / (12)
/or a/unction e in L~ taking the values 0 and 1. Moreover e = P Xo.
The proof, which offers no difficulty, is omitted.
F o r each real 3, let ~ be the smallest closed subspace of L~ containing the functions Zx with ~ > v. The subspaces 7~/~ are nested, and decrease as ~ increases; we set
~8 = [q ~/~. (13)
Then @3, which is clearly closed and invariant, is called the remote past of the process {Z~}. Denote the corresponding projection function of the l e m m a b y ea.
I f Z0 does not belong to ~ 0 , let Y0 be the p a r t of Z0 orthogonal to ~ 0 , so t h a t
Z0 =Y0 + Zo (YoL~lo, Zoe~lo). (14)
I t is easy to see t h a t
y~ = S~y e (15)
is the p a r t of Z~ orthogonal to ~ . Evidently the y~ form an orthogonal set in L~, and their linear combinations span a closed subspace 5~1 which is invariant and orthogonal to
~a. L e t e 1 be the function realizing the orthogonal projection on ~1. (If ~ 0 contains Z0, t h e n ~ is defined to be {0} and el the null function.)
The linear sum ~ 1 0 ~ 3 m a y not be all of L~; its orthogonal complement is a third closed invariant subspace ~2 with projecting function e~. Then b y definition L~ is the orthogonal sum of ~1, ~2, a n d ~a, and
e l + e 2 + e a = 1 , e j e k = 0 a . e . (]~:/c). (16) If only one s u m m a n d ~ j is different from {0}, the process is said to be pure, and of t y p e 1, 2, or 3 depending on whether ~1, ~2, or ~ is non-trivial. More descriptive names have been invented: a process purely of t y p e 1 is an innovation process, with y~ its innovation at time 3; one of t y p e 3 is deterministic; and we suggest the adjective evanescent for a process of t y p e 2.
Set d/~j = ej d# (j = 1, 2, 3); (17)
these measures are m u t u a l l y singular and their sum is #. We have obviously
(e, X,, e, = f e, d = d , , , (lS)
l a 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 . I I 1 8 1
from which we see t h a t (ejX~} is a stationary stochastic process in L~, isomorphic with the canonical process (S~} in L ~ for ] = 1, 2, 3. Each process (ejZ~) lies in ~j. These compo- nent processes have the following fundamental property.
T ~ o , ~ E ~ 1. The process (ejY,~) in L~ is purely o] type ],/or j = 1, 2, 3.
This theorem is a generalization of the well-known result of Wold [21] for the circle group (where only processes of types 1 and 3 occur). A generalization of Wold's Theorem, for processes depending continuously on a real parameter, was given b y Harmer [8], and his method of proof will be a d a p t e d to prove our theorem, b u t in reality his result is quite different from ours and lies deeper.
The theorem contains three statements, which we prove in order.
Statement 1: ] = 1 . - - T h e functions y~ are an orthogonal set fundamental in ~1, and so we have the decomposition
So = F_a(4)y~ + w ( w • ~ ) .
(19)
Unless @, is trivia], it is clear that a(~) = 0 for 4 < O, but a(O) :~0. First projecting (19) into @i and then applying S~ gives
el S~ = ~ a (A) y~+~. (20)
~ 0
This shows t h a t the linear subspace spanned b y (elZ~}~>o is contained in the subspace spanned b y (Y~}~>o.
The converse holds also. F o r Y0 can be a p p r o x i m a t e d b y linear combinations of Z~ (4 i> 0), and since
yOE~l,
b y the same linear combinations of elS ~ (A/> 0). The same a r g u m e n t applies to y~ if ~ > 0.Combining these results, and translating b y T, we see t h a t (elS~}~> v and {y~}~>~
span the same subspace of L~, and indeed of ~1, for each v. The y~ are orthogonal and so form an innovation process with innovation y~ at time T; therefore (elSe} is an innovation process with the same innovation y~. This proves the first p a r t of the theorem.
Statement 2: 7" = 2 . - - W e have to prove t h a t the process (e~S~ } has neither innovation nor remote past. As to innovation, we can form finite sums
S0 + ~ b (~) S~ (21)
2>0
approximating Y0, which belongs to ~i, and so the corresponding sums
e~ (S0 +~>b (~) S~) (22)
a p p r o x i m a t e e~y o = 0 . I n other words, e2Z o is the limit of sums of the form
- ~ b ().) e~S~.
(23)
182 H E N R Y H : E L S O N A N D D A V I D L O W D E N S L A . G E R
This is exactly the s t a t e m e n t t h a t {e2Z~} has no innovation.
To consider the remote p a s t we need a result which will be referred to again, and which is therefore stated formally.
L~MMA 2. ej ~ 0 is a closed subspace o / ~ o / o r j = 1, 2, 3.
L e t / be an analytic trigonometric polynomial with m e a n value zero. Writing / as the sum of its projections and using (20) we have
/=~a(2)y~+w+z ( w e ~ , z e ~ ) . (24)
A>0
Since y~ belongs to ~ 0 for 2 > 0, the first sum, equal to ell, lies in ~ 0 . Now z = % / i s in
~3 which is entirely contained i~ ~ 0 . Hence the remaining t e r m w = e J is in ~ 0 as w e l l This set of functions / is dense in 7~/0, and so e ~ 0 is contained in ~ 0 for j = 1, 2, 3.
Now in the decomposition
~ o = el ~ o | e~ ~ o | e3 ~ o (25) the summands are m u t u a l l y orthogonal and contained in ~ 0 ; it follows t h a t each one is closed, as we h a d to show.
We return to the remote past of {e2Z~ }. Translating (25),
:/~/~ = el"/~/~Oe2~4~eaT/~. (26)
We have to show t h a t the projection of e2z 0 on e 2 ~ ~ has n o r m as small as we please, if z is large enough. F r o m (26), this projection is the same as its projection on ~ itself. B u t the intersection of all ~ is ~3, to which %Zo is orthogonal, and it is a simple exercise ia geometry to verify then t h a t e2z o has small projection in ~ for large T. This completes the proof for j = 2.
Statement 3: j = 3 . - - W e are to show t h a t {e~Z~} is deterministic. I t suffices to prove t h a t linear combinations of %Zz (2 > 0) are dense in the manifold spanned b y %Z~ (all 2), since b y translation the same will hold for the span of e~Zz (2 > T), no m a t t e r how large ~.
I n other words, if
f/(%Zx) d# = 0 (27)
for all 2 > 0, then the same relation is to hold for all 2. B u t (27) can be written
f(e3/)Ld~ = 0, (2s)
and this for 2 > 0 means t h a t e s / i s orthogonal to :/~0 in L~. Then % / i s orthogonal to ~3 contained in ~ 0 ; b u t e J itself belongs to ~3, and therefore m u s t be zero. Hence (28) holds for all 2.
I ) R E D I C T I O I ~ T H E O I ~ Y AI~D F O U R I E R SEI~XES I N S E V E I ~ A L V A I ~ I A B L E S . 1-I 183 This completes the proof of Theorem 1.
We mention here a problem a b o u t processes of t y p e 2, whose solution would contribute a good deal to our knowledge a b o u t function theory on B. L e t / z be absolutely continuous with respect to o, so t h a t d/z = wdc; for some summable function w. Assume the canonical process in L~ is evanescent. We know from definition t h a t
U ~ (29)
1>0
is dense in ~ 0 , b u t it is conceivable t h a t the subspace
Uo=
),<0 (30)is larger t h a n :t/t0. I f T/0 is larger t h a n ://t0, let R0 be the complement of ~ 0 in 7/o- The same definition a t ~ gives a subspace ~ a t each T, and it is easy to see t h a t ~ = S~ ~o. These subspaees are m u t u a l l y orthogonal. We can prove, using theorems presented later in this paper, t h a t each subspace R~ is one-dimensional, and t h a t together t h e y span L~. I f / is a representative of R0, then Z~/is in ~ , and the set {Z J } is a complete orthonormal system in L~. F u r t h e r m o r e ~ 0 is exactly the closed linear span of the elements X~/(4 > 0).
So in a new sense the canonical process in L~ is an innovation process with innovation / at time 0.
The problem is to decide whether 7/o is sometimes or always larger t h a n 7?/0, and what properties of w decide between the two cases if t h e y can both occur. I f 7/o is larger t h a n ~ 0 , then w has the form ] g [ 2, where g belongs to H 2 and is outer in a generalized sense, although the logarithmic integral of (7) evidently diverges. As we shall show, there are functions g in H 2 for which t h a t integral diverges; for w = ]g]2, the canonical process in L2w is clearly evanescent, b u t we do not know whether it can be or m u s t be an innovation process in the new sense.
I n the case of a general measure #, the evanescent p a r t of the Wold decomposition divides further into a p a r t which is continuous from the left as well as from the right, in the sense t h a t (29) is dense in (30), and a p a r t which is an innovation process in the new sense. We shall not be interested in this refinement.
4. Analysis of the decomposition
The decomposition theorem itself does not tell how to find the component measures /~1, ~u2,/~3, or equivalently the subspaces ~1, ~2, ~a from knowledge of/x. The purpose of this section is to set down w h a t we can from known results or standard arguments.
184 H E N R Y H E L S O N A N D D A V I D L O W D E N S L A G E R
I n the case of the circle group (with its dual, the integers) only ~1 and :~3 appear ([7], Chapter X I I ) . Write/~ (now a measure on the circle) as a sum
d # = wd(~ + d#~ (31)
of absolutely continuous and singular parts. The main result is this: i / f log wd(~ > - ~ , then d/~ 1 = wd(r and d#a = d/~s; otherwise d # l = 0 and d/~ a = d[~.
We shall prove a natural generalization of this theorem to the group ~; b u t whereas the result is complete on the circle, it cannot differentiate between/z 2 and #a on B.
T H E O R E ~ 2. I f d~ = w d ~ + df~s, where w is a non-negative summable /unction on B and/~s is singular with respect to cr, and i/
/ log w d ~ > - ~ , (32)
then d[~ 1 = w d c f . I / ( 3 2 ) is/alse, d/~ 1 = O.
Proo/. The p a r t of Z 0 orthogonal to 7~0 was called 1 + H in [10], and was shown not to be the null function under the hypothesis (32). Indeed 1 + H was different from zero almost everywhere for a, b u t vanished for/~s. Therefore e I is the function equal to 1 almost everywhere for g and zero almost everywhere for/~s. This proves t h a t d/~ 1 = wdc;.
I f (32) fails, then Z0 can be a p p r o x i m a t e d b y linear combinations of Za with 2 > 0, and the process has no innovation. Therefore d # l = 0.
B y Theorem 2, if # has absolutely continuous component satisfying (32) t h e n this p a r t of # is exactly the s u m m a n d / ~ l of the Wold decomposition. Moreover ~1 is naturally identified with the Hilbert space L~, and the process {elZz ) in L~ is isomorphic with the canonical process {Z~} in L~. The orthogonality and invariance of the spaces ~ j can be expressed informally b y saying t h a t second-order questions a b o u t the prediction of the canonical process in L~ decompose under the Wold decomposition into the analogous ques- tions for the orthogonal subprocesses; in the case we are discussing, this means t h a t the absolutely continuous and the singular parts of # can be treated separately, provided t h a t (32) is true.
The next theorem shows t h a t this simplification does not depend on (32), and t h a t it suffices to study w d a and d#s separately even when (32) fails.
T H ~ O R ] ~ 3. Let e and e' be/unctions in L~ eatis/ying e = l a . e . (da); e ' = l a . e . (dins); e . e ' - 0 . T h e n eT~T and e ' 7 ~ are closed subspaces o / 7 ~ / o r each ~.
(33)
PREDICTION THEORY A N D FOURIER SERIES IN SEVERAL VARIABLES. II 185 Proo/. F o r simplicity consider T = 0. Suppose t h a t / belongs t o L~ a n d is orthogolzal to
~0:
fZ~fd/x = 0 (all A > 0). (34)
This expresses the fact t h a t [dju has Fourier-Stieltjes series of analytic type. B y Theorem 16 of [10], t h e same is true separately of [ w d # a n d [d/zz, so t h a t e / a n d e'/are separately o r t h o g o n a l to ~/0. This is the same as to s a y t h a t / is o r t h o g o n a l to e ~ 0 a n d to e'~/~.
Since / was an a r b i t r a r y function o r t h o g o n a l to ~ 0 , it follows t h a t e ~ 0 a n d e' ~ 0 are con- t a i n e d in ~ 0 .
I n the decomposition
~tl0 = eTt/0@e':~t/0, (35)
which is analogous to (25) in t h e proof of t h e W o l d decomposition, t h e s u m m a n d s are m u t u a l l y orthogonal a n d c o n t a i n e d in ~ 0 , a n d it is obvious t h a t each m u s t be closed.
The a r c h i m e d e a n p r o p e r t y of Rd has n o t been used to this point, b u t it is involved in the discussion which follows a n d in succeeding sections.
Suppose n o w t h a t the process {Z~} ill L~ depends continuously on 2, or w h a t is equi- valent, m e r e l y t h a t t h e inner p r o d u c t (Z0, Z~) is a continuous function of ~ in the o r d i n a r y sense. This is the kind of process t r e a t e d b y H a n n e r [8], K a r h u n e n [13], Wiener [19], a n d others, a n d complete results have been o b t a i n e d for the questions we are considering [7].
I n order t h a t the process be of this t y p e it is necessary a n d sufficient (as one proves w i t h o u t difficulty) t h a t / x be s u p p o r t e d b y B0, so t h a t in particular/~ is singular with respect to a.
If a I denotes linear measure on B0, t h e n ju has a decomposition
d~ = w d a 1 + d~ts, (36)
where n o w w is s u m m a b l e for da 1 a n d d / 4 is singular with respect to a 1 b u t carried on B 0.
The process is deterministic i/ and only i/
o o
(log w (x) d
J 1 + x 2 31 (x) = - oo. (37)
- - o O
I / t h e integral in (37) is/inite, the remote past is the subspace o/L~ consisting o/ /unctions which vanish almost everywhere/or da r
This result can be p r o v e d f r o m t h e corresponding t h e o r e m a b o u t discrete processes b y a change of variable ([1], p. 263), or it can be o b t a i n e d with a certain complication b y the m e t h o d s of [10]. The a n a l o g y with discrete processes is pressed further a n d more deeply b y Halmer. B u t we shall n o t be concerned with processes of this type, whose i n n o v a t i o n
13 - 61173060. Acts mathematica. 106. Imprim6 le 20 d6cembre 1961.
186 H E N R Y H E L S O I ~ A N D D A V I D L O W D E N S L A G E R
c o m p o n e n t s a l w a y s v a n i s h i n t h e W o l d d e c o m p o s i t i o n . I n s t e a d we consider t h e o t h e r e x t r e m e t y p e : processes w i t h a b s o l u t e l y c o n t i n u o u s s p e c t r a l m e a s u r e s , w h i c h c a n n o t p o s s i b l y b e c o n t i n u o u s .
5. C a u c h y m e a s u r e s
A c e r t a i n f a m i l y of m e a s u r e s on B is i n t i m a t e l y c o n n e c t e d w i t h t h e p r o b l e m s we a r e s t u d y i n g , as M a l l i a v i n p o i n t e d o u t t o us, a n d t h i s s e c t i o n sets d o w n t h e i r d e f i n i t i o n a n d r e l e v a n t p r o p e r t i e s .
F o r e a c h r (0 < r < 1) consider t h e m e a s u r e # r o n B whose F o u r i e r - S t i e l t j e s t r a n s f o r m is
hr(~) = rill- (38)
T h e f u n c t i o n on t h e r i g h t side is p o s i t i v e definite on Rd, so/~r e x i s t s on B a n d is n o n - n e g a t i v e , w i t h t o t a l m a s s one. B u t t h e t r a n s f o r m is c o n t i n u o u s in t h e o r d i n a r y t o p o l o g y of t h e line, a n d one p r o v e s e a s i l y t h a t t h i s is t h e case if a n d o n l y if t h e m e a s u r e is c a r r i e d on B 0.
T h e r e f o r e e a c h #r is c a r r i e d on B0, a n d hence is s i n g u l a r w i t h r e s p e c t t o o. These p r o p e r t i e s a r e also e a s y t o verify:
/~r-x-/~ =/~rs; l i m #r = a; l i m / ~ r = 8, (39)
r-->0 r-->l
w h e r e in t h e I a s t r e l a t i o n s t h e l i m i t refers t o t h e w e a k s t a r - t o p o l o g y of m e a s u r e s , a n d 5 is t h e u n i t m a s s a t t h e i d e n t i t y of B.
T h e e x p l i c i t f o r m of/~r is e a s y t o give, a l t h o u g h i t is n o t n e c e s s a r y t o o u r work; d/z~
is t h e m e a s u r e
y dt (r=e_y), (40)
z (t 2 + y2),
w h e r e t is t h e l i n e a r c o o r d i n a t e on B 0. This is t h e classical C a u c h y kernel, a n d t h e r e f o r e w e call t h e / ~ Cauchy measures.
E v e r y s e g m e n t of B o carries p a r t of t h e m a s s o f / ~ , a n d since B 0 is dense in B, i t follows t h a t / ~ r assigns p o s i t i v e m e a s u r e t o e v e r y n o n - e m p t y o p e n set in B. M u c h m o r e t h a n this s i m p l e s t a t e m e n t is t r u e , a n d t h e n e x t t h r e e l e m m a s give f u r t h e r e s s e n t i a l in-
f o r m a t i o n .
L EM~A 3. I / g is the characteristic ]unction o/ a set o/ positive measure in B, then
#r+g > 0 almost everywhere/or (~.
T h i s r e s u l t is s u r e l y n o t new, b u t we h a v e f o u n d i t w i t h M a l l i a v i n . I f i t were false, t h e n for a c e r t a i n v a l u e of r a n d a c e r t a i n f u n c t i o n g we s h o u l d have/~r44g = 0 on a set of p o s i t i v e m e a s u r e , whose c h a r a c t e r i s t i c f u n c t i o n we call h. L e t h' (x) = h ( - x); t h e n
P R E D I C T I O : N TiHEOI{Y A:ND FOURYEIr SEI:{IES I N S:EVEI~AL V A / ~ I A B L E S . I I 187 o = fh (x)~eeg (x) da (x) = ~ . g . h ' (0).
(41)
Now geeh' is a continuous non-negative function, and we know t h a t / ~ has positive mass i~
every non-empty open set. Therefore (41) is only possible if geeh' vanishes identically. Bub
fgeeh'd(r =fgda.fh'dcr 4=0. (42)
This contradiction proves that/~reeg is positive almost everywhere.
L]~MMA 4. There exists a set _F having positive measure in B with characteristic/unctiou g such that/~reeg is not essentially bounded/rom zero.
Since B 0 has measure zero, there is an open set G of small measure containing B 0- Moreover a translate Gx of G b y the element x of B still contains virtually all the mass of /~r provided x is close to the identity, because nearly all the mass of ~u, is carried on a com- pact segment of B 0 which remains in G under small translations. Therefore the characteristic function g o f / v , the negative of the complement of G, has the required property.
L]~MMA 5. There is a non-negative/unction u defined on B such that
f u d a = ~ ; p r e e n < ~ a.e. ( 0 < r < l ) . (43) Let t, r, and s satisfy the relations
0 < t < r < l ; t = r s .
(44)
We require the set F and the function g of Lemma 4, with the observation t h a t they were constructed independent of r. Since #teeg is not bounded from zero we can find a non- negative function u which is not summable but which satisfies
f~eeg (x) u ( - x) d(~ (x) =/~t~geeu (0) < ~ .
(45)
Using the associativity of the convolution operation a second time, this is equivalent t(>
f # t ~ u ( - x)d~(x) < ~ , (46}
F
so that/~teeu ( - x) is finite almost everywhere on F. We conclude that/~r%u is finite almosl~
everywhere on B from the representation
/~teeu = ~u~ee(#reeU), (47)
because b y Lemma 3 convolution with #~ would detect the set where/~r~u is infinite with probability one. Since r was arbitrary, the lemma is proved.
188 I=IE~I~Y H E L S 0 ~ A N D D A V I D L O W D E ~ S L A G E R
Actually the relation
~u~u < oo a.e. (48)
for a non-negative function u is true or false simultaneously for all values of r, and it is impossible for the convolution to converge on a set of positive measure unless it converges almost everywhere. These facts can be derived from (40), but they are contained in our later results, and we shall not prove them here.
The importance of the Cauchy measures lies in this property:
MALLIAVIN'S TIIEOREM. For any/unction / o] H 1 we have
log I/I
a.e. (49)Of course this is a generalization of the classical fact t h a t a subharmonic function in the u n i t circle is dominated b y the Poisson integral of its boundary values. I n our general context, however, the functions #r~-/do not need to be continuous, and they exist only as summable functions almost everywhere. The theorem of Malliavin is closely related to this result which we shall use as well, and which was found independently for the class of double power series b y M. l~osenblum:
TI~EOl~E~ 4. / / / is an outer /unction in H 1, then (49) is almost everywhere equality.
The proof of these theorems belongs to the function-theoretic part of this work, and is postponed until section seven. We proceed meanwhile to the solution of the prediction problem which is the main result of these sections.
6. The prediction problem
THEOREM 5. For every summable non-negative weight/unction w the process {)~) in L ~ is pure. It is o] type one i]
flog
w d ~ > - ~ ;(50)
o / t y p e two i / ( 5 0 ) / a i l s but
# r ~ l o g w > - ~ a.e. ( 0 < r < l ) ; (51) and o] type three i] (51)/ails, when necessarily
/ ~ r % l o g w = - c ~ a.e. ( 0 < r < l ) . (52) The proof will take the rest of this section, and for the sake of clarity we shall break i t into a number of simple lemmas.
Let d (T) denote the distance in L~ from Z 0 to ~ , the manifold spanned by linear
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 . IX 189
9 2
combinations of Z~ with 2 > r. F o r s > 0, d~(r) is the corresponding distance m L ~ , w h e r e w~ = m a x (w, e). These distances of course are zero for ~ < 0; ~nd we k n o w [10] t h a t d(0) > 0 if a n d only if (50) holds.
The condition d ( 0 ) > 0 e v i d e n t l y means t h a t t h e process has non-trivial i n n o v a t i o n component, b u t it is easy to go f u r t h e r a n d show t h a t the process is purely of t y p e one.
Consider @1, t h e i n v a r i a n t subspace s p a n n e d b y the i n n o v a t i o n elements y~ in L~. Since these elements are m u t u a l l y orthogonal, we h a v e f r o m (15)
fZ~lyol~wd,r
= 0 (2 # o), (53) so t h a t go has c o n s t a n t m o d u l u s different f r o m zero 9 Therefore in t h e decomposition (16) only e~ can be different f r o m the null function, a n d t h e process is pure as we asserted.N o w in the rest of the proof we m a y assume t h a t (50) fails, a n d t h e process h a s nc~
i n n o v a t i o n c o m p o n e n t .
L E M M A 6. I] d(v) = 0 / o r some ~ > O, then the same is t r u e / o r every 7:.
F o r if Z 0 belongs to ~ for some ~ > O, t h e n more generally Z~ belongs to ~/~+~ f o r every 2. I t follows t h a t all t h e subspaces ~l~ are identical, a n d so Z o is in ~/~ for e v e r y v.
LE~IMA 7. d(~) = lira d~@)/or each ~ > O.
e--~0
W e have for each s
d (~)~ =i~ffll +
z Pl wd i fl + z Pl w d ,
(54)where P ranges over all analytic trigonometric polynomials (6) with m e a n value zero.
T a k i n g t h e limit in e we find d (T) ~< lim d~ (z).
Conversely, for each fixed P
f] 1 + X, el~wd~=limfl l + X, P l ~ w ~ d ~ limd~(~)~.
(55)T a k i n g t h e i n f i m u m over P gives d (z) ~> lira d~ @), as we h a d to show.
e
F o r each e, let g~ be the outer function in H 2 such t h a t
= > 0 .
(This is t h e representation o b t a i n e d in T h e o r e m 3 of [10].) W e h a v e g ~ ( x ) " , S a ~ ( 2 ) Z ~ ( x ) ; 51a~(2)1 ~= f l g ~ ] ~ d ~ = f w ~ d c r .
2~0
(56)
(57)
190 t t E N I ~ u H E L S O N A N D D A V I D L O W D E : N S L A G E I ~
L E M M A 8. I n order ]or the process to be
o/type
three it is necessary and su//icient thatlim ~ l a~ (~)I ~ = 0 (58)
e-->0 0~<A~<~
]or every positive ~. This is the ease i / t h e condition holds/or a single value o/ 3.
I t is obvious t h a t the process is of t y p e three just if d (3) = O for all 3, or b y L e m m a 6, e v e n for a single T. W e shall show t h a t t h e sum in (58) is e x a c t l y d~ (3) 2. Therefore, using L e m m a 7, the limit is d(3), a n d (58) is necessary a~d sufficient for t h e process to be deter- Ininistie.
Since g~ is outer, functions of the form Pg~ where P is an analytic trigonometric poly- n o m i a l are dense in H e. We conclude easily t h a t functions Z~Pg~, where n o w P is analytic w i t h m e a n value zero a n d 3 is fixed, can a p p r o x i m a t e a n y function in H ~ whose coefficients v a n i s h for indices less than or equal to 3. I n particular, if we a p p r o x i m a t e t h e function
- Za~ (~)Z~ (59)
).>~:
t h e n in the expansion
t h e second s u m has n o r m as small as we please. Therefore
(60)
a s we h a d to prove.
P - 0~<.~<~
( 6 1 )
LEMi~IA 9. I n order/or (58) to hold it is necessary and su//iclent that i~r+g~ tend to zero i n the norm o / H 2 as s tends to O,/or each r (0 < r < 1). I t su//ices that this be true/or a single value o / r .
T h e l e m m a is obvious f r o m the equality
/J#~+g~ II ~ = 5 1 < (~)l er2~, (62)
t o g e t h e r with t h e fact t h a t 5 [ < ( 2 ) l e (63)
is u n i f o r m l y b o u n d e d as e tends to 0.
L E M ~ A 10. #r+g~ tends to zero in H 2 i / a n d only i] # r + l o g w = - ~ almost everywhere.
F r o m T h e o r e m 4 we h a v e a l m o s t everywhere
log [jur+g~ [~ =/~r%lOg We, (64)
so f I 'r+g:l:d = fe: p
(fir+log w~)da. (65)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 . I I 191 The left side is finite, so t h e i n t e g r a n d on t h e right is at a n y rate summable; since w~
decreases to w the l e m m a follows.
F r o m L e m m a s 8, 9 a n d 10 we see t h a t the process is deterministic i / a n d only i/ (52) holds; and (52) is t r u e / o r every r i / i t holds/or a single one. Therefore t h e process has non- trivial evanescent c o m p o n e n t just if (52) fails. W e assert t h a t in this case (51) holds. Indeed, suppose for some r there is a set of positive measure on which/~r%log w is finite. W e know, f r o m (47) as before, t h a t if this convolution is infinite on a set of positive measure, t h e n /~t-~log w = - o o almost everywhere for each t < r , a n d hence for all t, c o n t r a r y to hypo- thesis. Therefore (51) is t h e only alternative to (52).
I f (51) holds, the process has some evanescent c o m p o n e n t . W e complete the proof of t h e t h e o r e m b y showing t h a t t h e canonical process in L~ is pure, so t h a t it m u s t be p u r e l y evanescent.
LEMMA 11. I / W vanishes on a set o/positive measure, the process is deterministic.
Indeed, if w vanishes on a set of positive measure, t h e n b y L e m m a 3 (52) holds.
(This result was p r o v e d in a direct w a y in our conversation with Malliavin, before we knew the criterion (52) for determinacy.)
m Lw has b o t h r e m o t e p a s t a n d evanescent compo- N o w suppose t h e canonical process " ~
nent. B y the decomposition t h e o r e m we h a v e
w = w 2 + w 3 , w 2.w a ~ 0 , (66)
where t h e process {Za} in L~, is p u r e l y evanescent, a n d t h a t in L ~ is deterministic. B u t according to (66), w~ m u s t vanish on a set of positive measure. L e m m a l l asserts t h a t this is impossible. Therefore one or t h e other c o m p o n e n t in (66) m u s t h a v e been null. This shows t h a t t h e canonical process in L~ is pure, a n d the t h e o r e m is completely proved.
I t is i m p o r t a n t to remark, finally, t h a t conditions (50) a n d (51) really are different, or in other words t h a t t h e canonical process in L~ can be evanescent. I f log w is summable, so t h a t (50) holds, t h e n indeed (51) is true. To show t h a t t h e converse implication is false, let w = e -~ where u is t h e function of L e m m a 5. T h e n w is s u m m a b l e a n d satisfies (51) b u t n o t (50).
N o w we leave prediction t h e o r y a n d t u r n to t h e s t u d y of analytic functions on B.
7. Outer functions
I n order to prove Malliavin's T h e o r e m a n d T h e o r e m 4, the m o s t n a t u r a l m e t h o d (and t h e one followed b y Malliavin) is to refer the problem to the complex plane b y means of the canonical image B 0 of the line in B. This technique (which has been conspicuously
192 ] : I E ~ I ~ Y J=IELSON A N D D i V I D L O W D E ~ T S L A G ] g R
e x p l o i t e d b y Bochner) gives c o m p l e t e r e s u l t s a b o u t B a t l e a s t in p r i n c i p l e , b e c a u s e B c a n be c o n s t r u c t e d d i r e c t l y o u t of R so t h a t R b e c o m e s B 0. B u t t h e d e t a i l s of p r o o f b y t h i s m e t h o d are o f t e n f o r m i d a b l e , a n d we p r e f e r t o p r e s e n t a n i n t r i n s i c f u n c t i o n t h e o r y on B w h i c h f u r n i s h e s proofs w h i c h seem t o us closer t o t h e s u b j e c t m a t t e r . I n t h i s s e c t i o n we offer p r o o f s of M a l l i a v i n ' s T h e o r e m a n d T h e o r e m 4 in t h i s spirit. F i r s t we h a v e t o e x t e n d B e u r l i n g ' s n o t i o n of o u t e r / u n c t i o n f u r t h e r t h a n we d i d in [10], of w h i c h t h e r e s u l t s were s u m m a r i z e d in s e c t i o n two.
L e t u a n d v be r e a l f u n c t i o n s d e f i n e d on B. W e s a y t h a t v is conjugate t o u if u § i v is a n a l y t i c i n s o m e s u i t a b l y g e n e r a l sense, a n d v h a s m e a n v a l u e zero. I f u is i n L 2, t h e r e is e x a c t l y one f u n c t i o n v in L 2 h a v i n g m e a n v a l u e zero such t h a t u + i v belongs t o H2; b u t if u is m e r e l y s u m m a b l e , t h e r e m a y b e no s u m m a b l e f u n c t i o n v such t h a t u § i v is in H 1.
W e do h a v e t h i s w e a k e r r e s u l t [9]: for 0 < p < 1 t h e r e e x i s t s a c o n s t a n t K v s u c h t h a t
(f lvl do) <K fluld (67)
for e v e r y t r i g o n o m e t r i c p o l y n o m i a l u w i t h c o n j u g a t e v. I f a sequence (u~) of t r i g o n o m e t r i c p o l y n o m i a l s converges t o a f u n c t i o n u in t h e m e t r i c of L 1, t h e n t h e c o n j u g a t e f u n c t i o n s f o r m a f u n d a m e n t a l sequence in t h e m e t r i c space L p for e a c h p < 1 a n d so converge in m e a s u r e t o a u n i q u e l i m i t f u n c t i o n v. B y d e f i n i t i o n v is t h e f u n c t i o n c o n j u g a t e t o u. I t will b e i m p o r t a n t in t h e p r o o f s t o follow t h a t v is besides t h e pointwise l i m i t a l m o s t e v e r y w h e r e of a s u i t a b l y chosen s u b s e q u e n c e of (vn).
The n e x t t h e o r e m gives a c h a r a c t e r i z a t i o n of o u t e r f u n c t i o n s w h i c h is r e a l l y a n a b s t r a c t v e r s i o n of t h e i n t e g r a l r e p r e s e n t a t i o n f o r m u l a of B e u r l i n g .
TH~O~]~M 6. I / u is a real /unction such that u and e u are summable, then e u+i" (where v is conjugate to u) is an o u t e r / u n c t i o n in H 1. Conversely, i / a summable o u t e r / u n c t i o n / has the representation e ~+~" with u and v real, then u is summable with e ~ and v is equal to its conjugate modulo 2~, a s i d e / r o m an additive constant.
Proo/. S u p p o s e t h a t u a n d e u are s u m m a b l e a n d v is c o n j u g a t e t o u. W e h a v e t o show t h a t / = e u+~v b e l o n g s t o H I, a n d t h e n t h a t i t is o u t e r . F i r s t s u p p o s e t h a t u ( a n d so also v) a r e t r i g o n o m e t r i c p o l y n o m i a l s . The p o w e r series e x p a n s i o n for t h e e x p o n e n t i a l is u n i f o r m l y c o n v e r g e n t , a n d t h e a n a l y t i c i t y of / is t h e r e f o r e obvious. T h e p o w e r series also shows t h a t (7) holds:
More g e n e r a l l y , if b o t h u a n d v a r e b o u n d e d , u + i v c a n be a p p r o x i m a t e d boundedly b y t r i g o n o m e t r i c p o l y n o m i a l s , a n d t h e t w o p r o p e r t i e s of / p e r s i s t in t h e l i m i t .
P R E D I C T I O I ~ 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 . I I 193 Two m o r e l i m i t processes a r e r e q u i r e d t o t r e a t t h e g e n e r a l case, a n d t h e y use t h e r e s u l t a b o u t c o n j u g a t e f u n c t i o n s j u s t s t a t e d . S u p p o s e u is b o u n d e d , b u t n o t n e c e s s a r i l y v.
Choose t r i g o n o m e t r i c p o l y n o m i a l s u~ c o n v e r g i n g b o u n d e d l y t o u a n d such t h a t t h e c o n j u g a t e t r i g o n o m e t r i c p o l y n o m i a l s v n t e n d t o v a l m o s t e v e r y w h e r e . T h e n
(69) t e n d s t o zero b y t h e t h e o r e m on d o m i n a t e d convergence. A s t h e l i m i t in n o r m of e l e m e n t s of H 1, / itself belongs t o H 1. M o r e o v e r in t h e e q u a t i o n
fu,,
d a = logfe u-+~'- d a (70)w h i c h expresses t h e f a c t t h a t e a c h f u n c t i o n eUn+% is o u t e r , we can p a s s t o t h e l i m i t on b o t h sides t o o b t a i n t h e s a m e r e s u l t for ].
F i n a l l y , for a n u n b o u n d e d f u n c t i o n u define
u w h e r e - n < u < n
u ~ = n w h e r e u > n (71)
- n w h e r e u < - n .
Once m o r e we c a n a s s u m e t h a t t h e c o n j u g a t e f u n c t i o n s vn t e n d t o v p o i n t w i s e , a n d t h e t h e o r e m o n d o m i n a t e d c o n v e r g e n c e shows t h a t t h e n o r m in (69) t e n d s t o zero, a n d t h a t t h e e q u a t i o n (70) is v a l i d in t h e l i m i t .
Conversely, l e t / = e u+~v b e a n o u t e r f u n c t i o n in H 1. W e k n o w f r o m (7) t h a t u is s u m - m a b l e ; l e t v' be its c o n j u g a t e f u n c t i o n . T h e p a r t of our t h e o r e m a l r e a d y p r o v e d s t a t e s t h a t g = e u+~v" is a n a l y t i c a n d o u t e r . W e h a v e t o s h o w t h a t o u t e r f u n c t i o n s w i t h t h e s a m e m o d u l u s c a n differ o n l y b y a c o n s t a n t f a c t o r .
S e t w = I/[ = [g]" I n L ~ t h e t r i g o n o m e t r i c p o l y n o m i a l s 1 + P , w h e r e P is a n a l y t i c w i t h m e a n v a l u e zero, f o r m a c o n v e x s u b s e t in whose closure t h e r e is a u n i q u e e l e m e n t 1 + H of m i n i m a l n o r m . W e h a v e s h o w n [10] t h a t 1 + H is n o t t h e n u l l f u n c t i o n , a n d f u r t h e r t h a t [ 1 + H [ ~ w is a l m o s t e v e r y w h e r e e q u a l t o ~o = e x p (flog w d a ) . H e n c e (1 + H ) 2 / a n d (1 + H ) 2 g h a v e m o d u l u s c o n s t a n t a n d e q u a l t o o).
O n t h e o t h e r h a n d (1 + H ) 2 / i s t h e l i m i t i n H 1 of f u n c t i o n s of t h e f o r m (1 + P)~/, e a c h of w h i c h h a s m e a n v a l u e e q u a l t o t h e m e a n v a l u e o f / . Therefore, using (7), we h a v e t h e t w o r e l a t i o n s
1(1 + H ) ~ / l = o ~ ;
If(l+H)~/d(rl=lf/d,rl=~o.
(72)I t follows t h a t (1 +
H)2/is
a l m o s t e v e r y w h e r e e q u a l t o a c o n s t a n t .1 9 4 H E N R Y t t E L S O ~ AND DAVID LOWDENSLAGEI%
The same reasoning applies to g, a n d t h e n e w function 1 + H is t h e same as the old, because it depends only on w. Therefore (1 § H)2g is a l m o s t everywhere c o n s t a n t as well, a n d since 1 + H a l m o s t never vanishes, / a n d g differ at m o s t b y a c o n s t a n t factor. This completes t h e proof of the theorem.
Proo/ o/ Theorem 4. The C a u c h y measures/~r have t h e characteristic p r o p e r t y
/~r-)ee h = exp
(~r-)6h)
(73)if h is analytic a n d suitably restricted. F r o m this formula, (49) with equality follows in a formal w a y for the function ] = e~:
log I ~ e ~ I = Re (~r~h) = ~ r ~ Re (h) h = / ~ + l o g ] eh]. (74) U n f o r t u n a t e l y (73) is n o t easy to prove as generally as it is true, a n d it is n o t generally e n o u g h true to prove our t h e o r e m because the right side is n o t defined when h is n o t sum- mable, even if e h belongs to H i.
Suppose however t h a t h is a trigonometric polynomial. I f P a n d Q are a n y analytic trigonometric polynomials, t h e n
lu~(PQ) = (#~+P) (/~r~eQ), (75)
as one sees b y c o m p a r i n g coefficients in the Fourier expansions of left a n d right sides. I n particular, /ur~P ~= (kt~-)eP)~, with the same equation for higher exponents b y induction.
Hence
o n ! 0 n ~ ' (76)
a n d this is e x a c t l y (73). B y (74), then, T h e o r e m 4 holds for outer functions ] = e h where h is a n y analytic trigonometric polynomial.
According to t h e characterization of outer functions given b y T h e o r e m 6, we h a v e to prove t h a t
l~ I/~r ~+e=+~ I =/~,.~+log u (77)
for each real function u, s u m m a b l e together with e u, with c o n j u g a t e function v. W e h a v e just established this fact if u is a trigonometric polynomial. N o w we i m i t a t e t h e a p p r o x i m a - tion a r g u m e n t used in t h e proof of T h e o r e m 6 to establish (77) for larger classes of functions.
The details are neither novel n o r v e r y tedious, a n d so we shall n o t reproduce them.
T h e relation (77) is a k i n d of generalization of the defining p r o p e r t y (7) of outer func- tions. I f / = e =+~" is continuous a n d b o u n d e d f r o m zero, we can let r t e n d to 0 in (77) a n d o b t a i n
log
if Idol
= f l o gI/Id . (7s)
PI~EDICTIOI~I 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 i L V A R I A B L E S . I I 195 I n the function t h e o r y of the u n i t circle, w h e r e / ~ is s i m p l y t h e Poisson kernel, t h e general f o r m (77) can be o b t a i n e d f r o m its special case (78) b y a p p r o p r i a t e conformal t r a n s f o r m a t i o n s of t h e interior of the circle. B u t on B, #r is singular w i t h respect to t t a a r measure, a n d the limiting case r = 0 is distinguished f r o m the regular ones r > 0. Therefore some m o r e c o m p l i c a t e d device, such as the one used in our proof, m u s t be used to derive (77) f r o m (78).
P r o o / o / M a l l i a v i n ' s Theorem. L e t ] belong to H 1, a n d suppose its m e a n value is different f r o m zero. T h e n [10, p. 178] / = g'h, where g is outer a n d h is a n analytic function w i t h c o n s t a n t m o d u l u s equal to 1. T h e result to be p r o v e d t a k e s t h e f o r m
log [~trV~-(g'h)[ <~tr-~log [g[. (79)
NOW (75) is still true, b y a s t a n d a r d limit process, if P a n d Q are replaced b y g a n d h; a n d s i n c e / 4 has unit t o t a l m a s s we h a v e
< 1 a.e. (80)
Therefore using T h e o r e m 4 for t h e outer function g,
log
I#r-)/v/]
= l o g [/~r~eg] + log ]/~r~+h] ~< log [/Ur-)eg ] =/~r-)+log [g]. (81) T h u s (79) is proved, u n d e r t h e a s s u m p t i o n t h a t t h e m e a n value of / is n o t zero.I f t h e integral of / does vanish, we h a v e (49) a n y w a y for / + e in place o f / . F a t o u ' s L e m m a shows t h a t the i n e q u a l i t y is p r e s e r v e d in the limit.
8. Analytic functions
THEOREZ~ 7. I / a / u n c t i o n / in H 1 vanishes on a set o/ positive measure in B, or indeed i / m e r e l y
*log I/I = - a.e., (82)
then / vanishes identically. Otherwise under either condition
f / d ~ + 0 (83)
or / is continuous (84)
w e h a v e f l o g
I/Ida
> - c o . ( 8 5 )Nevertheless there are n o n - n u n / u n c t i o n s in H ~ /or which (85) i s / a l s e . More generally, i/ w is a non-negative summable /unction such that
ff~-)elog w > - c~ a.e. (86)
196 H E N R Y H E L S O N A N D D A V I D L O W D E N S L A G E R
then there is some ] in H 1 satis/ying
0 < [/[ < w a.e. (87)
This t h e o r e m contains m o s t of w h a t we k n o w a b o u t analytic functions on B. The proof m a y be clearer if we c o m m e n t on its various s t a t e m e n t s before we begin. On t h e circle g r o u p (85) holds for non-null a n a l y t i c functions subject to v e r y mild g r o w t h condi- tions inside t h e circle. The fact t h a t (85) follows f r o m (83) in general is an e l e m e n t a r y result f r o m [10]; t h e same conclusion from (84) was p r o v e d b y Arens [3] in two ways, b u t neither one is easy. W e shall give a new proof of Arens' T h e o r e m which relates it to t h e other s t a t e m e n t s of T h e o r e m 7.
I n spite of these positive results, (85) is n o t true in general even for analytic functions which are bounded. This fact follows f r o m t h e last assertion of the t h e o r e m if we t a k e w to be bounded, a n d satisfy (86) b u t n o t
flog w d a > - ~ . (88)
T h e existence of such weight functions is asserted b y L e m m a 5.
I f we ask o n l y t h a t t h e c o u n t e r e x a m p l e belong to H ~, t h e n t h e existence of functions violating (85) follows easily f r o m Theorem 5, a n d this simple proof is given before t h e more complicated discussion of (87).
P r o o / o / Theorem 7. Suppose / belongs to H 1 b u t satisfies (82); this will be the case in p a r t i c u l a r if / vanishes on a set of positive measure, b y L e m m a 3. F r o m Malliavin's Ine- q u a l i t y
- ~ =/~-~log ]/[ >~ log I/zr-x-[[. (89)
Therefore/~r-X-[ vanishes identically. B u t t h e Fourier-Stieltjes coefficients (38) of/~r vanish nowhere, so t h a t ] m u s t itself be t h e null function, as we h a d to prove.
To prove Arens' Theorem, let / be a continuous analytic function which is n o t every- where zero. Then/~r-~/is continuous a n d n o t identically zero. F i n d a n open set E on which / ~ + / i s b o u n d e d f r o m zero, a n d c o n s t r u c t a continuous f u n c t i o n h such t h a t h ( - x) is non- negative everywhere, positive somewhere on E, a n d zero outside E. T h e n we h a v e
h log I r /I (0) > - (90)
Using Malliavin's I n e q u a l i t y a n d t h e associativity of convolution,
(h~e#~)~log 111 (0)/> h ~ l o g [ttr-~[[ (0) > -- c~. (91) B u t h-)e#r is a continuous n o n - n e g a t i v e function, which m o r e o v e r can never vanish, because h is non-negative a n d ~u r has mass in every open set. H e n c e h-)e/zr > ~ > 0 on B, a n d (91)
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 . I I 197 implies
@og]llda
> - oo. (92)Our proof t o this p o i n t was f o u n d in conversation with Malliavin. T h e rest of the proof depends on T h e o r e m 5, a n d has a different character.
W e are to construct an analytic f u n c t i o n / n o t identically zero b u t such t h a t (85) is false. F o r this purpose find a weight f u n c t i o n w such t h a t (86) holds b u t n o t (88); w can m o r e o v e r be t a k e n bounded. B y T h e o r e m 5 there is a non-null element g o r t h o g o n a l to
9 2 .
'~0 in
Lw.
fX~ywda
= 0 (all 2 > O), (93) so t h a t / = ~w is s u m m a b l e a n d analytic. Definean element of L1; t h e n
(94)
I g l =
(W/w)%
I / I = Igl =(Ww)',,.
( 9 5 )F r o m (95) we see t h a t / is square-summable (because w is bounded), a n d so belongs t o H 2, a n d we have
1 ~'l ~ + ~ f l o g
wda.
flogl/I, , = )oo
1 (96)The first t e r m on t h e right side is finite or n e g a t i v e l y infinite, a n d b y t h e choice of w t h e second t e r m is - o o . H e n c e t h e left side diverges also, a n d / has the properties sought.
W e come to the final assertion of the theorem 9 Suppose t h a t w satisfies (86); once more b y T h e o r e m 5 we can find g in L~ n o t identically zero such t h a t (93) holds 9 T h e n
I glw '~
is in
L 2,
a n d soU = rain (1,
Igl-lw -'z*)
(97)satisfies flog
Uda
> - oo. (98)Using T h e o r e m 6, find h in H a so t h a t U = ]h I a l m o s t everywhere. (We do n o t need t h e fact t h a t h can be chosen to be outer.) N o w ~w belongs to H 1 a n d h to H ~, so t h e p r o d u c t
h~w
belongs to H 1, a n d e v i d e n t l y[h~w[
= r a i n (Iff[w, w '/') < w '/'. (99) Neither analytic factor ofh~w
vanishes on a set of positive measure, a n d so this function is a non-null element of H ~. Therefore / =(hyw) 2
belongs to H i a n d satisfies (87). All p a r t s of t h e t h e o r e m have n o w been proved 9198 I~IEI~I%Y ~IELS01W A N D D A V I D L O W D E N S L A G E I t
We do not know whether (87) can be improved to
I11=~
~.e.; (100)this question is related to interesting problems with prediction-theoretic import which lie deeper t h a n Theorem 5. A result we are publishing elsewhere [11] can be mentioned in this connection, although it does not directly extend our knowledge a b o u t analytic functions.
9. Moving averages
We come now to the second p a r t of the paper, having to do with multivariate stochastic processes and the related theory of matrix-valued functions. This section is devoted to the notion of moving average, and more generally to the prediction-theoretic point of view.
The results and proofs are n o t new, and we shall only sketch the development; a complete exposition has been given b y Masani and Wiener [16]. Beginning with the n e x t section it will be essential t h a t we s t u d y the group of integers, rather t h a n ordered groups of more general t y p e to which the m a t r i x theory of our first p a p e r applied.
First we set down the basic definitions. Functions are defined now on the unit circle, or more rarely inside it. The measure dx/2~ on (0, 2~) is denoted b y da. An integer N is given and fixed, and reference is usually made to complex Euclidean space of N dimensions.
Matrices with N columns operate on the column vectors of this space b y multiplication on the left. The trace function is normalized so t h a t I , the identity m a t r i x of a n y order, has trace 1. We s a y t h a t two matrices have the same shape if t h e y have the same n u m b e r of rows and the same n u m b e r of columns.
We consider functions F defined on the circle and taking matrices of various shapes a s values:
F(e ~x)
= (~'j~ (e~)). (101)The entries Fjk are always measurable complex scalar functions. I f F and G are constant matrices, their immr product is b y definition
(F, G) = tr (FG*), (102)
meaningful whenever F and G have the same shape. This definition is extended to m a t r i x functions b y integration:
(F, G) = ftr (FG*)d~. (103)
I f F and G have p rows, then evidently
(F, G) : p-1 ~ f Fjk Gjk d G. (104)
j.k
