(ii) is trivial and independent of the hypothesis that / b e zero-free in C + . Indeed, assuming, for example, that (i) holds with a = 1, we have ( — A) = ^(A) for all A 6IR, whence (ii) is immediate. It is the reverse implication that seems to be rather a striking result, since condition (ii) alone yields no kind of pointwise relation for . The proof of Theorem 1 makes use of the following formula for the Dirichlet integral of an entire function. THEOREM
2. Let gePW and a > 0 be such that h+(g) = 0 and
h~(g) = 2a.
(1-3)
Denote by {z^} the zero sequence of g, the zeros being counted with multiplicities. Then (1-4) (it is understood that z = x + iy and Zj = Xj + iy^). Remarks. (1) The convergence of the series in (1"4), as well as the convergence of the Blaschke products appearing in the proof below, is due to the fact that
(see [3], chap. III). (2) It might be interesting to compare (1-4) with Carleson's formula [1] for the Dirichlet integral of a generic analytic function on the disc. The rest of the paper is organized as follows. In Section 2 we cite (and partly prove) a few facts that will be employed later on. In Section 3 we prove Theorem 2 and, finally, in Section 4 we use it to derive Theorem 1.
2. A few lemmas We begin with the following elementary assertion.
Fourier transforms of entire functions
359
1. Let gePW and a > 0 be such that (1-3) holds true. Then the Dirichlet integral of g over C + , defined by LEMMA
is finite and satisfies @(g) = n f ° A|$(A)|8cW = 1 P° g'(t)gjf)dt.
(2-1)
Zt
Jo J-oo Proof. To prove the first equality in (2-1) (which in turn implies @i(g) < +oo), notice that g is supported on [0,2a] and Use Plancherel's formula to compute J ^ \g(x + iy)\2 dx and then integrate over y. The second equality in (2-1) follows again from the Plancherel formula. Remark. Alternatively, one can verify that £2>{g) = (2i)~l \g'g via Green's formula. The next lemma is also well known. Before stating it, we recall that the Hardy space H2 is denned by Hz = {feL*: f = 0
a.e. on ( — oo,0)}.
LEMMA 2. Let FePW, F =j= 0, and let {wn} be the zeros of F in C + . Then, for zeC+, we have F{z) = ce~iAz&(z)B(z),
(2-2)
where c is a constant with \c\ = 1, A = h+(F), ft f°° / 1 \.n J-oo \ 2 ~ ^
def
def
t \ < + 1/
] J
1—z/w
The integral and the infinite product involved are convergent and represent analytic functions in C + . Moreover, (9{z) extends to an entire function of class PW n H2 satisfying \(9(x)\ = \F(x)\,
xeU,
whereas B(z) is analytic on R and \B(x)\ = 1,
XER.
Finally, for the extended & and B, the factorization (2-2) holds true with zeR as well. Lemma 2 is but a special case of the canonical factorization theorem for H2 functions (see e.g. [2], chap. II). The version above, dealing with entire functions of exponential type, can also be found in [3], chap. III. The functions (9 and B, denned by (2-3) and (2-4), are the so-called outer factor and Blaschke product associated with the original function F. LEMMA 3. Under the hypothesis of Lemma 2, the Blaschke product B = B{Wn) defined by (2-4) satisfies "•
~
' -
^ ^ ,
xeR.
360
KONSTANTIN M. DYAKONOV
Proof. For an individual factor
we have |fen(a;)| = 1 on IR and xe as shown by a straightforward calculation. Hence
(x) n bn(x) + b'2(x) U & yi(x) + b2(x)b'2(x) + ... = 2i
as required. 3. Proof of Theorem 2 Let g, a and {z;} satisfy the hypotheses of the theorem. Set G(z) = g(z) exp (2iaz). As is readily seen, GePW; besides, h+(G) = 0 and hr(G) = 2a. Letting 6{x) denote exp (2iax), we obviously have G = gd on IR. In view of Lemma 1,
2i®(Q) = [G'G = Ug6)'gO = [w0 + W) $ = \tfg + \\gW6 = - 2»0(y) + 2ia [\g\\ (Here and throughout, / / stands for J" f(t)dt. The prime denotes differentiation along IR and is, therefore, interchangeable with the bar.) Thus,
-i:
•dt.
(3-1)
Now let {AB} be an enumeration of the set {zy. y^ > 0}. Using Lemma 2 and taking (1*3) into account, we get
g=c1®B1, where cx is a constant with |cj = 1, O = Cj9) and Bx = B{X ) (see (2-3) and (2-4) for the definitions). In particular, || = \g\ and \BX\ = 1 on 05. Hence
= jg'g =
J
n I1
A
n\
where the last equality relies on Lemma 3. Thus, \9(t)\2 S
= | -oo
uZTT^-
y-Vj>o I1
z
j\
(3-3)
Fourier transforms of entire functions
361 +
Further, let {/*„} be an enumeration of the set {z}: y} < 0}. Since h (G) = 0 and the /i n 's are precisely the zeros of G in C + , Lemma 2 yields G = c2$>B2,
where c2 is a constant with \c2\ = 1, is the same as above (note that \G\ = \g\ on U) {/n)
Rewriting (3-2) with g replaced by G, B1 by B2, and An by pun, we arrive at the following analogue of (3-3):
= -[
\g(t)\* S uzHi*-
(3 4)
'
Finally, the identity
when combined with (3-1), (3-3) and (3-4), yields (1-4). 4. Proof of Theorem, 1 Assuming that the hypotheses of the theorem are fulfilled, we prove the nontrivial implication (ii) => (i). In view of (ii), the supporting interval for equals [ — a, a] with a suitable a > 0. Set g(z) = /(z)exp(mz). Clearly, gePW. Further, the zero sequence of/ (denote it by {z^}) coincides with that of g. In particular, g has no zeros in C + , so that 2/, < 0
for all j
(4-1)
(here, as before, y} = Imz ; ). Besides, the relation a)
(4-2)
shows that g has supporting interval [0,2a]. This in turn means that g satisfies (1"3), and so Theorem 2 applies to g. Letting
R(g)d=2>(g)~
Z
P° \g(t)\*dt, J-00
we rewrite formula (l-4) from Theorem 2 as ^ j
.
(4-3)
\t~Zj\
On the other hand, using Lemma 1 and (4-2) we get = n By (1'2), the latter integral equals zero; thus R(g) = 0. Now a glance at (4-3) and (4-1) convinces us that y} = 0 for all j ; in other words, ZjeU
(.7=1,2,...).
(4-4)
362
K O N S T A N T I N M. D Y A K O N O V
The Hadamard factorization theorem (see e.g. [3], chap. Ill) says that, for suitable constants aeC, /?elR, yeU and meZ + , one has (4-5) where p{z) = zme^\ From (4-5) it follows that pePW, and from (4-4) we see thatp(IR) c U. Consequently, the supporting interval for p is symmetric with respect to the origin. Since = ap( • —y) also has this property, we conclude that y = 0. Now (4-5) reduces to f(z) = a.p(z), whence/(IR) c aU. REFERENCES [1] L. CABLESON. A representation formula for the Dirichlet integral. Math. Zeit. 73 (1960), 190-196. [2] J. B. GARNETT. Bounded analytic Junctions (Academic Press, 1981). [3] P. Koosis. The logarithmic integral (Cambridge University Press, 1988).
