Non-oscillating Paley-wiener functions

(1)

By

I. V. OSTROVSKII AND A. ULANOVSKII

A b s t r a c t . A non-oscillating Paley-Wiener function is a real entire function f of exponential type belonging to L2 (R) and such that each derivative f('~), n = 0, 1, 2 , . . . , has only a finite number of real zeros. It is established that the class o f such functions is non-empty and contains functions of arbitrarily fast decay on t t allowed by the convergence of the logarithmic integral. It is shown that the Fourier transform of a non-oscillating Paley-Wiener function must be infinitely differentiable outside the origin. We also give close to best possible asymptotic (as n ~ oo) estimates of the number of real zeros of the n-th derivative o f a function f of the class and the size of the smallest interval containing these zeros.

1 I n t r o d u c t i o n

A Paley-Wiener function (PW-function) is an entire function f of exponential type such that f E L2(R). By the Paley-Wiener theorem, the class of all PW- functions coincides with the class of functions f admitting the representation

/;

y ( z ) = e-iZSF(s)ds,

o o

where F (called the spectral function of f ) is an L2-function with bounded support. We say that a PW-function f is real if f ( R ) C R. Oscillatory properties of real PW-functions have been the subject of investigation of a number of works (see, for example, [1], [3] and [8]). J. R. Higgins [3] (p. 77) has constructed a sequence of real PW-functions s,~ with the following property. Each sn has a derivative of order 2n with infinitely many real zeros, but each derivative of order less than 2n has only a finite number of real zeros. W. J. Walker [8] (p. 1254) and J. R. Higgins [3] (p. 72) posed the question: Does a real PW-function f have for some order n a derivative f(n) which has infinitely many real zeros? For some classes of real PW-functions, a positive answer was obtained by W. J. Walker [8] and J. Clunie et al. [1]. In the latter work, it is mentioned (p. 167) that the answer to this question is still unknown.

211 JOURNAL D'ANALYSE MATH~MATIQUE, Vol. 92 (2004)

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212 I. V. O S T R O V S K I I A N D A. U L A N O V S K I I

In this paper, we answer this question in the negative. Let us call a PW-function f non-oscillating if it is real and each derivative f('~) has only a finite number of real zeros. Simple examples o f such functions are given in the following result.

T h e o r e m 1. (i) Suppose a real function p E C ~ ( R ) has bounded support and satisfies p(O) ~ O. Then the function

fO ~176

fl (x) := p(s) sin xs ds is a non-oscillating PW-function.

(ii) Suppose p E C~176 has a bounded support, p ( - s ) = p(s), f o r all real s, and f_~176 p(s) ds ~ O. Then the function

1/?

f2(x) := - (1 - ei=S)p(s)ds

X oo

is a non-oscillating PW-function.

P r o o f . Since p has compact support, the function fl is a real PW-function. C h o o s e any integer n _> 1. The statement (i) can be proved simply by integrating by parts 2n times:

5

/ ~ 2 n - 1 ) ( X ) = ( - - 1 ) n - 1 s2"-lp(s) cosxsds

[ c o s x s ( ) ] ~ 1 J~O oo

= L x-~i-~- ~

_82n-lp(8)_

(2n-1)

_o _{x ~n} _{(s2,_lp(s)) (2.) cos xs ds.} Since the function (s2n-lp(s))(2n) is continuous and has bounded support, the last integral tends to zero as Ixl ~ r We conclude that

(2n- 1 11p(~

(1 + o(1)) Ix I ~ oo, n = 1,2,.

~" x 2 n ' . . .

This shows that for any n > 1 the function f~2n-~) can have only a finite number o f real zeros. Hence, by Rolle's theorem, any derivative f~n), n = 1, 2 , . .. has only a finite n u m b e r o f real zeros.

(ii) It is clear that f~ is a real PW-function. Observe that

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Since the the function p is infinitely differentiable on the real line, for any natural number k, the second integral in the right hand side is

o(Izl -k)

as z -~ co. This shows that

] f ~ n ) ( x ) ] -

Ix~+ , / _ ~ p(s)ds

( 1 + o ( 1 ) ) , ]x]-+ co, n = 1 , 2 , . . . ,

so that every derivative f~") can have only a finite n u m b e r o f real zeros.

T h e functions f l and j'2 in T h e o r e m 1 tend to zero like [x1-1, and their n-th derivatives tend to zero like ]x] - n - 1 as Ix] -~ co, x E R. Since the n-th derivative o f a non-oscillating PW-function is also a non-oscillating PW-function, for any n = 1 , 2 , . . . there exist non-oscillating PW-functions f which satisfy ]f(x)] =

O(]xl -'~)

as ]x] -+ co. Below, we show that non-oscillating PW-functions f can have arbitrarily fast decay provided that the logarithmic integral converges (which is true for e v e r y PW-function):

~176

II~

< co.

1 + x 2

For instance, T h e o r e m 4 below implies that for any p E (0, 1) and any a > 0, there exists a real non-oscillating PW-function f o f type a such that

1 Ixl p log - -

l$(x)l

is bounded from below and above by positive constants for all sufficiently large

Ixl, x ~ It.

T h e r e remains an open question related to the

slowest

possible rate o f decay o f a real non-oscillating PW-function on R . For example, we do not know whether there exist non-oscillating PW-functions decreasing on R like

Ixl-~

with 1/2 < ~ < 1.

Observe that the spectral function o f the function f l in T h e o r e m 1 is

(i/2)p(Isl)signs,

s E R. This function is infinitely differentiable outside the origin. One can verify that this is also true for the spectral function o f f2. Our next result shows that the spectral function o f any non-oscillating PW-function has this property.

T h e o r e m

2. Let f be a non-oscillating PW-function. Then its spectral function

belongs to

CC~(R \ {0}).

Let us d e n o t e by

r(n, f)

the m a x i m u m o f the moduli o f real zeros o f the n-th derivative o f a non-oscillating PW-function f and by

p(n, f) the

n u m b e r o f its real zeros. Rolle's t h e o r e m implies that both

r(n, f)

and

p(n, f) are

strictly increasing functions o f n and, moreover,

p(n, f) > n.

It seems natural to ask how fast

r(n, f)

and

p(n, f)

may grow as n ~ co.

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214 I. v. OSTROVSKII AND A. ULANOVSKII

T h e o r e m 3. Let f be a non-oscillating PW-function. Then

(1) logr(n,f) >_ (1 + o ( 1 ) ) l o g n , n -+ co.

The inequality in (1) is sharp in the sense that there are functions f for which the inequality can be replaced by equality. We deduce Theorem 3 from the more general Theorem 5 below.

Observe that inequality (1) also remains valid for p(n, f). This follows from the trivial inequality p(n, f ) > n and Corollary 1 below. The problem o f finding more precise hounds and any upper bound for r(n, f ) and p(n, f ) remains open.

2 R a t e o f d e c a y a n d real z e r o s o f a n o n - o s c i l l a t i n g

P W - f u n c t i o n

We assume familiarity with the notion of a proximate order ([2], Ch. 2, w [4], Ch. 1, see. 12). Recall that a proximate order is a continuously differentiable positive function p(r) on [0, co) satisfying the conditions

Put

(i) 3 l i m , ~ p(r) =: p > 0; J

(ii) limr~oo(p

(r)lp(r))r

l o g r = O.

V ( r ) : = r p(r).

The simplest example o f a proximate order is p(r) - p > 0. In this case, V(r) = rP;

and the considerations below would be significantly simpler and independent o f results o f [2] and [4] related to proximate orders.

Condition (ii) implies that the function V is strictly increasing for sufficiently large r. Evidently, it is possible to change p(r) on a large interval [0, R] in such a way that V becomes strictly increasing on [t3, co) and V(0) = 0. We shall assume that this change has been made.

The following result relates to the rate of decay on R o f non-oscillating

Let p(r) be a proximate order such that

(2) lim p ( r ) = p > O , and d r < c o .

r - - ~

There exists a real non-oscillating PW-function f such that

1 1

(3)

c~ < v(--T~ log ~

< c~,

Ixl > to, x e R,

where ro, Ct and 02 are positive constants.

PW-functions.

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Let f be a non-oscillating PW-function and r(n, f ) the m a x i m u m of the moduli of real zeros o f f(n). Our next result estimates the rate o f the increase o f r(n, f).

T h e o r e m 5. (i) For any proximate order p(r) satisfying (2), there exists a non-oscillating PW-function f such that

(4) r ( n , f ) < v ( O n l o g n ) , n = 0 , 1 , 2 , . . . ,

where v is the inverse function f o r V and C is a positive constant.

(ii) For any non-oscillating PW-function f o f type 1,

_ 7 r 1

(5) r(n, f ) > -~en( + o(1)), n ~ oo.

We derive from Theorem 5 (i) the following fact, related to the possible growth of the number p(n, f ) of real zeros o f the n-th derivative o f a non-oscillating PW function f .

Corollary

1. For any proximate order p(r) satisfying (2), there exists a non- oscillating PW-function f such that

(6) p ( n , f ) <_ v ( C n l o g n ) , n = 0 , 1 , 2 , . . . ,

where v is the inverse function f o r V and C is a positive constant.

Evidently, T h e o r e m 5 (ii) implies inequality (1). Applying Theorem 5 (i) to the proximate order p(r) such that V ( r ) = r p(r) = r/log2(1 + r), we see that there exists a non-oscillating PW-function f for which r(n, f ) = O(n log 2 n), n ~ oo. For this function, the inequality in (1) becomes equality. Similarly, Corollary 1 implies that there exists a non-oscillating PW-function such that logp(n, f ) = (1 + o(1))logn, n -> oo.

The rest o f this paper is organized as follows. Some auxiliary results are proved in the next section. Section 4 is then devoted to the proof o f Theorem 4, and Sections 5 and 6 to the proof o f Theorem 5 and Corollary 1. An auxiliary result which is used in the proof o f Theorem 5 is proved in Section 7. Finally, Theorem 2 is proved in Section 8.

Theorems 4 and 5 were announced in [7] with a short description of the method of proof.

3 Auxiliary l e m m a s

Lemma

1. Let Pl ( r ) be a proximate order such that l i m r ~ pl (r) = Pl E (0, 1)

(6)

the solutions o f the equations

(7) Vl(r) = k, k = 1,2, 3,...,

a n d set

h i ( z ) = 1 - ~ .

k = l

Then the circles

C k ( e ) = { z : l z - - a k l < e a k / V l ( a k ) } , k = 1 , 2 , 3 , . . .

do n o t intersect each other provided that n u m b e r e is small enough. Further, the estimate

(8) log Ihl(re~)[ = ~r cos P l ( ~ - 7r) Vx(r) + o(V1 (r)), r -+ oo, 0 < ~ < 2~

sin 7rpl

holds outside the union o f the cicrles Ck (e).

Proof.

It follows immediately from the definition (7) and the properties o f proximate order that the circles Ck(e) do not intersect for all sufficiently small e > 0. This means that the roots ak o f the function hi form a so-called R-set (for the definition see [4], p. 95). The asymptotic formula (8) for re i~ f[ Uk Ck(e) now follows from a known result on entire functions o f completely regular growth

([4], p. 96, Theorem 5). []

Corollary

2. Set

h2(z) = h l ( - z ~) = IX 1 + . k = l

Then h2 (z) is positive on R , real on i R and

(i)

logh2(x) - ~ V ~ ( z 2) + o(Vl(z2)), Izl-~ ~ ;

sin 7rpl

(ii) on every interval ( v / ~ , av/-d-~-~), there is a p o i n t bk, k = 1,2,..., such that

loglh2(• = ~rcot~rp,. Vx(b~) + o(Vx(b~)), bk -~ c~.

L e m m a 2. Let p(r), l i m r ~ p(r) = p E (0, 1] be a proximate order. Then

there exists an entire function

(9) g(z) = 1 + , 0 < c l < c 2 < . . . , c-~ 2 < c~,

k = l k

(7)

(i)

71"

(10)

logg(x) - Y ( I x l ) + o ( Y ( l ~ l ) ) , Ixl -~ o~;

sin(Trp/2)

(ii) every interval (ok, Ck+l) contains a point dk, k = 1, 2, ..., such that

7rp . V(dk) + o(V(dk)), dk --+ oo.

(1 l) log Ig(+idk)l = 7r cot -~-

To establish this lemma, it suffices to apply Corollary 2 with pl = l p and set

g(z) = h2(z), Ck = v/-a~, dk = bk.

L e m m a 3. Suppose p(r) is a proximate order satisfying (2). Then there exists

an even real PW-function h with real zeros such that

(12) loglh(x)l < - V ( I x l ) + O(1), Ixl ~ o~, x C R.

P r o o f . This ]emma is a special case of the Beurling-Malliavin multiplier theorem. We give the p r o o f below for the reader's convenience.

Assume additionally that l i m , ~ p(r) = 1. Let {ak} be a sequence o f positive numbers defined by the equations

Set

V(ak) = k , k = 1 , 2 , . . . .

~ I sin(z/ak) h i ( z ) = Z/ak

k = l

It is easy to verify that (2) yields ~ k ( 1 / a k ) < Oo. Hence, this infinite product converges and represents an entire function o f exponential type.

Let n(r) be the n u m b e r o f points ak to be found in the disc {z : Izl < r}. We have, for e v e r y real x and integer n, that

1 n

Ihl(=)l < ~yz I I ak.

Now set n = n(Ixl) and observe that

{

}

Ihl(=)l < exp - dt .

- a o t

It is easily seen from (2) that

n(t)

is asymptotically equivalent to

t o(t).

Hence, by known properties of the proximate order ([4], p. 34), we conclude that

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218 I. V. O S T R O V S K I I A N D A. U L A N O V S K [ I

This gives

" 1

log

Ihl(x)J <

-W(Ixl)(1 + o(1)) <

-2Y(Ixl)

+ c .

Hence the function

h(z) = h~ (z 2)

satisfies the conclusions of the Lemma.

In the case limr-4oo

p(r)

= p < 1, one can construct hi (z) by taking a constant Pa E (p, 1) instead o f the function

p(r).

Then one gets

loglha(x)] <

- l l x l P l

(1 + o(1)) < -V(lxl)(1 + o(1)), and again one can set

h(z) = h21(z).

L e m m a 4.

Suppose p(r) is a proximate order, and q(z) is an entire function

of completely regular growth with respect to p(r). Assume that

(i)

q(z) does not vanish in

{z: I arg z I < ~ < rr}, (ii) 3 lim~--,+oo(log

Iq(x)])/V(x) ~ O.

Then the derivative q'(z) does not vanish in

{z : largzl <

a/3, Jzl > R} for

all

large R, and the asymptotic equation

(13)

~z

q(z)

= ( - 1 ) ' ~ , q ( z ) ]

- ~

1 + 0

~

,

z ---r cx~, largzl <

a/3,

holds for all n = 1, 2,...

P r o o f . We require the following fact (see [2], p. 99, Theorem 5.1). Let

p(r),

limr--,~

p(r) = p

be a proximate order. Then there exists a function

L(z)

analytic in {z : l a r g z I < 7r} such that

(14) V(r)(= r p(~)) =

rPL(r)(1 +

o(1)), r -+ co,

(15)

L(re i~)

= L(r)(1 + o(1)), r ----r c~,

where (15) holds uniformly in ~o E (-Tr + 6, 7r - 6) for every 6 E (0, r). Let us verify that (15) gives

(9)

uniformly with respect to qo E [-7r/2, 7r/2]. Indeed, since ([2], p. 73, [4], p. 32) lira

L(kr) _ 1

~-r~ L(r)

for every k > 0, then, in view of (15),

L(~)

=

L(r) + o(L(r)),

r --+ 0%

uniformly with respect to ~ in the sector {~ : [argO] <

2rr/3, r/2

< ]~1 < 3r/2}. It now follows from the Cauchy integral formula that

,

9 1 f r

L(~)

d~

L (re '~)

= ~ -rei~'l=r/2 (r

__--~e/~)2

1 fl i

L ( r

(L__~)

= 2ri --re"l=~/2 - ~ : ~ e / - - ~ ) 2 de = o , r --+ 0o, which establishes (16).

By known results in the theory of .functions of completely regular growth ([4], pp. 94-95, Theorem 4), we have

logq(z) =

AzPL(r) +o(V(r)),

r --+ oo, z = re i~, Iqot < a/2,

where A is the limit in assumption (ii) of the lemma. This and (15) give logq(z) =

AzVL(z) + o(Y(r)),

r ~ oo, z = re i~, ]qo] <_ a/2.

Let us now differentiate the formula. Using (16) and estimating the derivative of the remainder term with the help of the Cauchy integral, we get

(17)

q'(Z) - P A z P - l L ( z ) + ~

, r-+oo, z = r e i~,

,qo,<c~/3,_

*

~, q(z) ] = p ( p - 1 ) A z p - 2 L ( z ) + ~

r2 ] ,

r -+ oo, z = re '~, kol < a/3,

!

(18)

\

q(z) ] \q'(z)/

v - ~ ' r + ~ , z = ,'e'~, I~1 < c~/3.

It is clear from (14), (15) and (17) that the derivative

q'(z)

does not vanish in {z: largzl __ ~/3,]z I > R} forall large R.

To verify (13), we use induction. When n = 1, (13) is evident. Assume that (13) holds for some n. Then we have

( d ) n+l 1 d [

(q'(z"~n

1 (

(V----~))]

(10)

220

I. v. OSTROVSKII A N D A, U L A N O V S K I I = ( _ l ) n [

( q ' ( z ) ) n - l / q ' ( z ) ) '

1

{ q ' ( z ) ) n + l 1 ] ( ( 1 ) )

n \ q(z) ]

~ q(z) ] q ~ - \ q(z) ]

q~z)

1 + 0

~ - ~

(q'(z)'~n 1

(

1 ) =

( _ 1 ) . + 1

{q'(z) ~n+l

1 + ( - 1 ) n

\ q(z) ] - ~ " 0 . ~ .

\ q(z) ]

q(-z)

{[

- 1 1

))

n

_{\ q(z) ] \q'(z)] J}

- ~, q(z) ]

"

This, together with (18) and (17), establishes (13).

4 P r o o f o f T h e o r e m 4

Suppose that 9 is a function whose existence is established in Lemma 2. Let us show that there exists a real even entire function G of exponential type with real zeros such that

(19) lim

x'~G(x)g(x) = O, n = 1,2,....

]zl-*cr zO:t

Let h be a function from Lemma 3 and let m > 2 + 27r/sinQrp/2) be an even number. Then, by (10) and (12), there is a constant D > 0 such that for all real x,

log

Ihr"(z)g(x)l < -V(Ixl) + D.

Set

G(z) = hm(z).

We see that an even stronger statement than (19) holds:

IG(x)a(x)l <_ exp{-U(Ixl) + D}.

Since h is even,

h(iy)

is real, so that G is positive on iR. Moreover, G is bounded on R and all its zeros are real. A well-known result on entire functions of exponential type (see, e.g., [4], p. 240, Theorem 5) then implies the asymptotic equality

logG(iy) = Aly I +

o(]y[), lY[ -~ oc,

where A is some positive constant. In view of (10) and (11), this gives

[G(+idk)g(+idk)] ~ cr

k --> ~ ,

sign

(G('4-idk)g(+idk))

= ( - 1 ) k, k = 1 , 2 , . . . .

We see that there exists a small number c > 0 that every interval (c2~-1 ,c2k) contains at least two points, say P2k-1 and P2k, which are roots of the equation

(11)

Set and define

( zD

q(z) = H

1 +

,

k----1

(20)

f ( z ) - C ( z ) g ( z ) + c q ( z )

We claim that this function satisfies the conclusion of Theorem 4. Since f is entire and is a ratio of two functions of exponential type, it is itself of exponential type.

Let us show that f satisfies condition (3). Using (19), we get

c + o(1) 1

loglf(x)l = log ~ : log q--~ + 0(1)

1 ig(x/i

= ] o g ~ + l o g q - ~ +O(1), Since C2k-1 < P 2 k - I < P2k < C2k, w e see that

IX] --')" 0 0 . log

g(x)

= ~ I o g

q(x)

k=l

1

+

x2/•k_1

~

1

+ z21c~k

1

+

x2/p'~k_,

+

log

1 +

x2/p~k

k = l

1 + x2/c'.22k_l

< log 1

+

x2/plk_l

k = l

(Xc-~Z'zl)

~

1+x2/c2k+ 1

- log 1 + + Z log 1 +

12/p2/2k-1

k = l < log (1 + ~.~2z) - One can get a similar estimate of log

Ig(x)/q(x)l

from below. This gives

1

log

lf(x)t

= log ~ + O (log Ix[)), Ix I -+ oo. Now (3) follows from (10).

It remains to verify that f is non-oscillating, that is every derivative of f has only a finite number of real zeros. Set

F(z) = G(z)9(z).

It follows from (20) that

(21)

( d ) n

_"~z

_{f ( z ) = c "~z}

( d ) n

_{" ~ +}

1 ( d ) nF(z)

_{q ( z ) "}

By construction, function q is an entire function with purely imaginary roots at

+ipk.

Denote by nff(r) and nff(r) the number of roots ofq and 9 in {z : Izl _< r, +()z > 0}, respectively. It follows from (9) and the construction of q that

I n , C,') - n ~ ( , ' ) l = o ( 1 ) , ,- ~ oo.

Since

n~(r)

= V(r)(1 + o(1)),

n~(r)

has the same asymptotics. It follows that q is an entire function of completely regular growth with respect to the proximate

(12)

222 I. V. OSTROVSKII AND A. ULANOVSKII

order

p(r).

Observe that

Iq(z)l <

q(-i-[z[), so that q satisfies the assumption (ii) o f L e m m a 4. Thus, formula (13) holds. By L e m m a 4, q' has only a finite number of real zeros; therefore, the first term on the right-hand side of (21) has only a finite number of real zeros. B y (13) and (20), to finish the proof it now suffices to establish that (22)

-~z

q(x) - o

q - ~

q ~

,

Ixl -+ c~.

We use L e m m a 4 to get = _j=O \ dx,/ q(z) j=0

\ q(x) ,l

q(x) 1 + 0

= fq'(x)'~ '~ 1

y~C~F(J)(x ) ( _ l ) . _ j ( q ( x ) ~ J

1 + 0 1 \ q(x) ] q--~ j=,

\ q ' ( x ) ]

~

'

It follows from (17) and (14) that

q(x)

q'(z)

kv(Izl)]

Since F is an entire function of exponential type and

F(x)

= o(Izl -m) for any natural number m,

F(J)(x) = O(Ixl-m),

Izl ~ 00

for any natural numbers j and m. This establishes (22) and completes the proof of T h e o r e m 4.

5 P r o o f o f T h e o r e m 5 (i)

Let h, h(0) = 1, be a real entire function of exponential type whose existence is established by L e m m a 3. We shall need the estimate.

(23)

Ih(x + iy)l <

cexp[alyl -

bV(Ixl)],

lyl < Ix],

where a, b, c are positive constants.

Let a be any number strictly greater than the type o f h, and L be a function analytic in {z : [argz I < rr} satisfying (14) and (15). Set

(13)

Clearly, hi is analytic in the quadrant Q = {z : 0 < argz < 7r/2}. Formulas (14) and (12) show that hi is bounded on the positive half-axis. Further, it follows from (14), (15) and (2) that

(24) ~ ( z P L ( z ) ) = ~ cos(p~) . V(~)(1 + o(1)) = o(~), ~ -~ ~ .

We see that hi is of order _< 1 in Q and bounded on the positive imaginary half-axis. By the Phragmtn-LindelSf principle, hi is bounded in Q. Hence

Ih(z)l _< cl

e x p [ - i a z - ( 1 / 2 ) z P L ( z ) ] l , z 9 Q ,

c being a positive constant. Recalling that (by (2)) p < 1 and using (24), we obtain (23) for the angle { z = x + iy : 0 < y < x } . For the other three angles of the form {z 9 0 < +y < +x}, the proof of (23) is similar.

Set

/ ( z ) - 1 - h ( z )

z

We show that (4) holds for this function (and hence it is a non-oscillating PW- function). Clearly,

fr

~

+

We estimate the second term in the right-hand side for large x. Let

M x ( r ) = max Ih(z)l.

I z - ~ l - - r

Inequality (23) shows that for x > r,

(25)

M z ( r ) < c e x p { a r - b V ( x - r)}. The Cauchy integral formula gives

< n!(x - r)r n" It is now clear that fCn) (x) does not vanish, provided that

x n § Hence, f(n)(x) does not vanish if

nW M x ( r ) < n!

(14)

For r < x / 2 , this can be rewritten as

M,(,-)<~ ~

.

By (25), the last inequality holds if

1 / r \

(26)

Observe that there is a constant q > 0 such that

V ( x - r) > V (x/2) > q V ( x ) , 0 < x < r/2.

The first inequality holds because V is increasing; the second follows from a well- known property of proximate order ([2], p. 73; [4], p. 33). Therefore, (26) holds if

ar - bqV (x) < n log(r / x ) + cl,

where cl is a positive constant. Set

r = b~qv(x). z a

Then we see that f(n)(x) cannot vanish provided that

bq _{V ( x ) < _ n l o g}

(bq V(x)) -tr

~a" x o r

(27)

_{V ( x ) > -~q}_log _"V-(x) _{bq "}

Since V ( x ) > 6x p/2 for all x >_ 1 if 6 is sufficiently small, v(u) < (u/6) 2/p for all sufficiently large u. Taking this into account, one easily checks that x = v ( C n log n) satisfies (27) for n = 1, 2 , . . . , provided that C is large enough. Hence (4) holds.

6 P r o o f o f T h e o r e m 5 (ii) a n d C o r o l l a r y 1

For any natural number n, we denote by un(t) the number of zeros of f(n) in the disc {z : ]z I _< t} (counting multiplicities). If f(n)(O) # O, then by Jensen's formula,

(28) i r Un(t) dt = _t _~ _{loglf(n)(rei~')ldr}_{p -} _{logl/(")(0)l,} _{r > 0.}

7r

We choose r = "er(n, f) and estimate the left-hand (right-hand) side of (28) from below (above) for this value of r.

(15)

Observe that, by the definition o f

r(n, f)

and

p(n,

f ) , we have

Vn(r(n, f)) >_ p(n, f).

Since we get (29)

/o

v~t) dt >

dt > u,~(r),

fo ~(n's) v'(t) dt > p(n,f).

t

To estimate the first term o f the right-hand side of (28), we observe that f admits the representation

f_

l

f(z) =

ei~tr

1 where ~b E L 2 ( - 1 , 1); therefore,

If(n)(z)l :

f_l 1

eizt(it)nr

/

/_l

<_

e-Y'ltlnlr

<

e-Utlr

<_

11r

lul,

1 1 Hence

xf

27r ~r and 7r

loglf(n)(rei~O)ldqo <( - - f log(l]~b]12e r[sin~~ dqo

: ~rd--/ogll~bll2

- 27r J _ , 7r

To estimate the second term o f the right-hand side o f (28), we need the following lemma, whose proof will be given in the next section of the paper.

L e m m a 5. (31)

where # is a complex-valued Borel measure on

[-1, 1]

of finite total variation such

that

{ - 1 , 1} C supp #.

Then there exists an increasing sequence {nj } ~=1 of natural

numbers such that

(i) lira

n~+x/n~

= 1; j-~oo

(ii)

loglf(n)(0)l =o(n),

n = n ~ ~ o o .

Let f be an entire function admitting the representation

f l

f(z)

= e/Ztdu(t), 1

n = 0 , 1 , 2 , . . . .

1 f_-~

If('~)(er(n,f)e'~)ld~ <

2-er(n,f)

+1og11r

(30)

~

log

(16)

226 I. V. OSTROVSKII A N D A. U L A N O V S K I I

B y L e m m a 5, we have

log If~n)(0)l = o(n), n = n~ ~ ~ . Substituting this, (29) and (30) into (28), we obtain

(32) p(n, f ) < 2er(n, f) + o(n), n = nj ~ oo. B y Rolle's theorem, p(n, f) > n; therefore, (32) yields

71"

r ( n , / ) >_ Ve n + o(n), n =

Taking into account that r(n, f) increases in n and the sequence {nj}~~ satisfies L e m m a 5(i), we get (5).

Let f be the function whose existence has been established in Theorem 5 (i). Evidently, (32) is applicable to f. Using the inequality (4), we get

p ( n , f ) < 2 e v ( C n l o g n ) + o(n), n = n: ~ to.

Since p(n, f) increases in n and the sequence {nj }~1 satisfies L e m m a 5(i), we can increase the constant C in such a way that (6) will hold.

7 P r o o f o f L e m m a 5

Note that the equality

(33) lim sup 1 log If ('~) (0)l = 0

rl---q, O ~ / t .

is trivially true because f is o f exponential type 1. Therefore, the existence o f some sequence {n~}j~ t satisfying (ii) is evident. We show that such a sequence can be chosen to be rather dense, namely that (i) is fulfilled. First we prove this under the additional assumption that the measure ~ is even, that is the function f admits the representation

(34) f ( z ) = cos(zt)dp(t),

where 1 E supp p. In this case,

1

f(2k)(0) = ( - 1 ) k f0 t2kdl~(t)' f(~k+l)(0) = 0, k = 0 , 1 , . . . .

We prove the lemma by contradiction. Let us assume that L e m m a 5 is wrong. Then there exist numbers q > 1, ~ > 0 and a sequence o f disjoint intervals [a:, b:] such that aj and b: are natural numbers satisfying the condition

(17)

(36) where

Consider the function

lim sup 1

k~oo, keA k log [f(2k) (0)1 < - e ,

O O

a = O[aj,b ].

j = l

f0

1

F ( z ) = t2Z+2d#(t).

This function is analytic and bounded in the closed half-plane {z : Rz > 0}. Since

F ( k - 1) = (--1)kf(2k)(0), k = 0, 1 , . . . , (36) implies

lim sup i log IF(k)l -- 0. k ~ k

Hence, since F is bounded on R, we have

(37) lim sup I log IF(x) I = 0.

x---*+oo X On the other hand, (36) implies

(38) _{k--,oo, keA -k log IF(k)l < - ~ .}lim sup 1

By a well-known result ([5], p. 104, Theorem 3),

(39) f ~o _dr(s) log I F ( z ) l = - z__

j _

zr

o o x 2 + ( y - s ) 2

=:

- u l ( z ) + u 2 ( z ) - c z , I + ' ~ , l o g z ~--~at I I Z "t- at I

where v is a non-negative Borel measure on R such that

f ) dv(s)

(40)

oo 1 + s ~ < o o ,

the at's are points in the half-plane {z : ~ z > 0} satisfying the condition c o s ( a r g at)

(41) ~ ] [atl < ~

l and c is a non-negative constant.

Observe that (37) yields c = 0, and (40) implies ul (x) = o(x), x ~ +oo.

Therefore, by (38) we see that

(42) lim sup u2(k)

(18)

228 1. V. O S T R O V S K I I A N D A. U L A N O V S K / I

To prove the lemma, we show that this inequality cannot hold.

B y the H a y m a n - A z a r i n theorem (see, e.g., [5], p. 109, Theorem 1),

(43)

us(z) >

-(1/2)~lz 1, Y~z > 0,

outside some exceptional set o f discs C,~ = {z : Iz - z,~l < 6m} o f finite view, that is, such that

~,(6m/IZ~l) < co.

Since

(6,n/lZmD ~ 0 as R ---r +c~, Iz,.l>R

we can choose R and the rays A+ := {z : arg z = +0}, 0 < 0 < 7r/4, in such a way that

(n• n {z: Izl > R}) n U c,,, -- o.

The H a y m a n - A z a r i n theorem ([5], p. 109) also implies that there is a sequence o f segments

{[dp, dp +

~Tp]}p=l, 0 < dp < dp + y~, < dp+l 1" +oo, satisfying the condition

~-'~(,1,/d,) < oo

p = l

and such that (43) holds outside the half-annuli

{z : dp <

Izl

< dp + ~p, ~ z > 0}. Let us consider the system o f sectors

gp:= {z:dp <lzl<dp+rlp, largzt<O},

P>_P0,

where P0 is so large that the circumference {z : Iz[ = R} does not intersect Kp for p > P0. Evidently, the sectors Kp are pairwise disjoint and their union covers the set A n {z :

Izl > R).

Moreover, we have

us(z) > -(E/2)lzl,

z e o g ~ , p > po.

We split the system o f sectors

{Kp}~~

into two groups. The first (second) one consists o f those, for which ~/r > 1 (r/p < 1). We denote the sectors o f the first (second) group by K'p (K~).

Setting

A = A N P

(19)

we obtain 1 < kEA' 1 + 1 rlr,>l dp<k<dp+rlr~ rlp>l dp _ _ < _ E 2r/P < 2 ~ r/p r/p_>l dp - p=l dPP < oo. Since (33) implies

using the notation

we have

(44)

1 Z ~ = o ~ , kEA " @ K ' /

A = A n

,

P 1 kEA" Ale tt ii

Each point k E belongs at least to one sector Kp. Moreover, each sector

Kp

contains at most one point from

A"

because ~7~ < 1. Thus, to each k E A", there corresponds a unique sector K~, which will be denoted by K;(k).

Now observe that each sector

K;(k)

must contain at least one point

at

from the representation (39). Indeed, otherwise, the function u2 would be harmonic in

K;(k),

and so

u2(k) >

min _II

uz(z) >

-(e/2)(k+rlp(k)) > - ( ~ / 2 ) ( k + 1).

z~DK,(~)

This contradicts condition (42) for sufficiently large k.

Thus, to each sufficiently large k E A", there corresponds at least one point K t l .

al = al(k) E p(k)' and to different values of k, there correspond different values K "

at(k).

Since both k and at(k) belong to the same p(k), Ik - ]at(k)]] < 7)p(k) < 1. Therefore, (44) implies

On the other hand,

1

Z

]at(k)l

kEA"

- - - - 0 0 .

1 1 cos(argal(k)) < 1 ~ cos(argat)

la,(k)l -<

cosQr/4) Z

la,(k)l

- Cos(,q4) --7-

I~1

kEA" kEA"

Clearly, this inequality and (41) contradict each other, which proves Lemma 5 under the additional assumption that the function f admits the representation (34) and 1 E supp#.

(20)

If f admits the representation

f ( z ) = sin(tz)d#(t), 1 9 supp#, then the proof is similar. In general, when f satisfies (31), we set

where

fo

fl

(z) = cos(tz)d#x (t), =

fo

f2(z) = sin(tz)d#2(t),

i#2(E)

=

#(E)-#(-E)

and observe that 1 E supp/q [.J supp/ze. Therefore, we can apply what we have already proved to either fl or f2. Noting that

f(2k)(0) = f~k)(0), f(2k+l)(O):f~2k+l)(O), k : 0 , 1 , 2 , . . . , we obtain the desired assertion.

8 P r o o f o f T h e o r e m 2

We begin this section with the following theorem, which we think has independent interest.

T h e o r e m 6. Suppose k > 0 is an integer, f is a real PW-function, and F is its spectral function. Assume that the derivative f(k+3)(x) has only a finite number of changes o f sign on the real line. Then F 9 C k ( R \ {0}).

Theorem 2 follows immediately from this result and the definition of a non- oscillating PW-function.

Another immediate corollary of Theorem 6 is the following result, which for any k > 0 gives a description of a wide class of real PW-functions f for which the n-th derivative of f, n > 3, must have infinitely many real zeros.

Corollary

3. Let f be a real PW-function with spectral function F. Assume F ~ C k ( R \ {0})for some integer k. Then f(k+3) has infinitely many changes o f

sign.

We deduce Theorem 6 from

Lemma

6. Let g 9 Lt (R) and g(z) > 0 f o r all large Izl, and let G be the spectral function of g. Suppose there is an even integer n > 2 such that G is n times differentiable at the origin. Then G E Cn(R).

(21)

P r o o f . W h e n g is non-negative on R , this lemma is contained in the well- known L6vy theorem (see Theorem 2.1.1 and its Corollary 1 in [6], p. 21).

In the general case, there exists a > 0 such that g(x) _> 0 for Ixl _> a. Write

g l ( x ) : = X ( - a , a ) ( X ) g ( x ) , g~(x) := g ( x ) -- gl (x),

where

X(-a,a)

is the characteristic function of the interval

(-a,a).

Then G = G1 + G2, where Gj is the spectral function o f

gj,j

= 1,2. Clearly, GI can be continued to the complex plane as an entire function, so G1 is infinitely differentiable. The l e m m a now follows from Corollary 1 in [6], according to which

G~ n) exists and is continuous on R . []

P r o o f o f T h e o r e m 6. Let f be a real PW-function with spectral function F. We begin with the observation that if f ' has only a finite number o f real zeros, then f ' E L1 (R). Indeed, there is a number a > 0 such that f ' does not change the sign (i.e., is either non-positive or non-negative) in (a, oo) and in ( - o o , - a ) . This gives

/_~[f'(x),dx= /):f'(s)ds +/_i 'f'(s)'ds+ fa ~176

<_ [f(-a)l +

Vz~[]/'[IL~ + I/(a)l < oc.

The spectral function o f f ' is

isF(s).

Since it is the inverse Fourier transform o f f ' , which belongs to L1 (R), it follows that

sF(s)

is continuous on the real line. Hence, for any n = 1 , 2 , . . . , the function

s'~F(s)

is n - 1 times differentiable at the origin.

In what follows, we assume that

f(x) >_ 0

for all large negative x (otherwise, consider the function - f ( x ) ) . Then f satisfies one o f the conditions

(i)

f(x) > 0

for all large positive x,

(ii)

f(x) <_ 0

for all large positive x.

(i) By Rolle's theorem, for any integer s, 0 < s < k + 2, the derivative

f(s)(x)

has only a finite number o f real zeros. It is then clear that for any even number s,

f(~)(x) >_ 0

for all large [x[.

Assume that k is even. Then

f(k+2)(x) >_ 0

for all large [x[. The spectral function of f(k+2) is

(is)k+2F(s).

It is k + 1 times differentiable at the origin. Hence, by L e m m a 6 with g = f(k+2), we conclude that

((iS)k+2F(s))(k)

exists and is continuous on R. It follows that F E C k ( R \ {0}).

Assume now that k is odd. Then f(k+s)(x) _> 0 for all large x. The same argument shows that the derivative F E C k+l ( R \ {0}).

(22)

232 1. V. OSTROVSKII AND A. ULANOVSKII

(ii) Clearly, for any odd integer s, 1 < s < k + 3, we have f ( ' ) ( z ) > 0 for all large Iz[. The same argument as in (i) establishes that F E C'k(R \ {0}) if k is odd, and F E C k+t (R \ {0}) if k is even.

R e m a r k . The observation at the beginning o f the proof implies that, for any non-oscillating PW-function f , one has f(k) E L1 (R), k = 1, 2 , . . . . For k = 0, this is not always true, as the example o f functions fl, f2 in Theorem 1 shows. It can be shown that f(k) E L1 (R), k = 1, 2 , . . . for any real PW-function f such that

f'

has only finitely many real zeros.

A c k n o w l e d g e m e n t . A substantial part o f this research was done during a visit o f the second-named author to Bilkent University at Ankara. This visit was supported by the Scientific and Technical Research Council o f Turkey (TUBITAK).

REFERENCES

[1] J. Clunie, Q I. Rahman and W. J. Walker, On entire functions of exponential type boundedon the real axis J. London Math. Soc. (2) 61 (2000), 163-176.

[2] A.A. Goldberg and I. V. Ostrovskii, Value Distribution of Meromorphic Functions, Nauka, Moscow,

1970 (Russian).

[3] J. R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, Clarendon Press,

Oxford, 1996.

[4] B. Ya. Levin, Distribution of Zeros of Entire Functions, Translations of Mathematical Monographs,

Vol. 5, Amer. Math. Soc., Providence, RI, 1980.

[5] B. Ya. Levin, Lectures on Entire Functions, Translations of Mathematical Monographs, Vol. 150.

Amer. Math. Soc., Providence, RI, 1996.

[6] Ju. V. Linnik and I. V. Ostrovskii, Decomposition of Random Variables and Vectors, Translations

of Mathematical Monographs, Vol. 48, Amer. Math. Soc., Providence, RI, 1977.

[7] I. V. Ostrovskii and A. Ulanovskii, Non-oscillating Paley-Wienerfunctions, C. R. Acad. Sci. Paris,

S6rie 1 3 3 3 (2001), 735-740.

[8] W. J. Walker, Oscillatory properties of Paley-Wiener functions, Indian J. Pure Appl. Math. 25

(1994), 1253-1258.

L V Ostrovskii

DEPARTMENT OF MATHEMATICS BILKENT UNIVERSITY

06533 BILKENT, ANKARA, TURKEY

VERKIN INSTITUTE FOR LOW TEMPERATURE PHYSICS AND ENGINEERING 61103 KHARKOV, UKRAINE

email: [email protected], [email protected]

A. Ulanovskii

STAVANGER UNIVERSITY COLLEGE P.O. BOX 2557 ULLANDHAUG

4091 STAVANGER, NORWAY

email: Alexa nder.Ola [email protected]