417
I nternat. J. Math. Math. Sci.
Vol. 5 No. 3 (1982) 417-439
ZEROS OF SMALLEST MODULUS OF FUNCTIONS
RESEMBLING exp(z)
KENNETH B. STOLARSKY
Department of Mathematics, University of Illinois
Urbana, Illinois 61801 U.S.A.
(Received July 17, 1981)
ABSTRACT.
To determine (in various senses) the zeros of the Laplace transform of a
signed mass distribution is of great importance for many problems in classical
analysis and number theory.
For example, if the mass consists of finitely many atoms,
the transform is an exponential polynomial.
This survey studies what is known when
the distribution is a probability density function of small variance, and examines in
what sense the zeros must have large moduli.
Bessel function zeros, of
In particular, classical results on
Szeg8 on zeros of partial sums of the exponential, of
I. J. Schoenberg on k-times positive functions, and a result stemming from Graeffe’s
method, are all presented from
a unified probabilistic point of view.
KEY WORDS AND PHRASES. Bessel function, characteristic function, exponential function, exponential polynomial, Graeffe’s method, Laplace transform, multiply positive
function, normal density, probab density function, vance.
A.M.S. CLASSIFICATION CODE.
i.
INTRODUCTION.
number
z.
property.
30C15.
It is fundamental that
exp(az+b)
In fact, it is the only entire function of exponential type with this
Now one could say that the zeros of the exponential lie "unborn" in the
essential singularity at
.
This leads us to ask how the zeros
as the exponential is "slightly perturbed" in the space
exponential type.
Write
is never zero for any complex
E
of
"stream out"
ntire
of
fnctions of
418
K.B. STOLARSKY
f(z)
where
g(t)
o
the
0
exp z.
z
tZg (t)dt
(1.1)
is a probability density function of mean
g(t)
becomes the Dirac "delta function" at
Thus, when all the mass is concentrated at
mass spreads out from
to
e
i,
the zeros move in from
i
I
.
and variance
I,
and
f(z)
When
is then
there are no zeros; when the
and may come closer and closer
although they cannot attain it (nor any other point on the real axis).
0,
In probability theory
[LU 1,2].
f(iz)
is known as the characteristic function of
g(t)
We note that if
g(t)
consists of a finite number of atoms, then (i.i)
becomes an exponential polynomial.
For an example, say
and
g(t)
I + a respectively.
and as the parameter
a
ga(t)
has two atoms of mass
f(z)
e
1/2
located at
i
a
Then
Z
(1.2)
cosh az,
increases the zeros become more and more dense along the
imaginary axis, with the one of smallest modulus approaching
For another example, note that as the parameter b
0.
increases from
0
I,
to
the modulus of the zeros of
f(z)
b
nearest to the origin tends to
;
Here we have two atoms of mass
(l-b)/2
the mean is
0
+ (l-b)cosh z
(1.3)
in fact, it can be shown to do so monotonically.
at
+i
and one atom of mass
b
at
0,
so
in this case.
The following informal assertion seems reasonable.
The -Hypothesis.
The distance
from the origin to the nearest zero tends
to vary inversely with the variance
Our goal in this paper is to determine to what extent this hypothesis is true.
To do this we assemble results scattered throughout the literature.
theorems of 3 and 4 may be new.
However, the
ZEROS OF
SMALLEST
MODULUS OF FUNCTIONS
419
To achieve an appropriate level of generality, we shall study the hypothesis in
the class of all functions of the form
I
f(z)
where
(w)
eWZd(w)
(1.4)
is a probability measure on the complex plane, not necessarily of corn-
"mean i"
pact support, with
o,"
and "variance
i.e.,
2
(w)
For example, when
fined to
Re(w)
is purely atomic, symmetric about the real axis, and con-
then
i,
f(z)
%j,
for appropriate real
of radius
r
about
z
j
i,
r"
z
2
J(z;g)
z
(1.6)
ajcos
If the mass is distributed on
g(8)
with density
e__
J(z)
e
1,2,3,
I
where
o2
at
i
I
i8
then
exp[z
reiO ]g(O)dO eZJ(zr;g)
-"
neither an exponential nor a constant, let
f(z)
re
exp[z(l+reiO]g(O)d( rO.
is an entire function such that
Then the 6G-hypothesis for
+
a circle
z
O
J(O)
I.
If
J(z)
is
be its zero of smallest modulus.
has in this case the very simple form
O
since
G
r.
For (1.7) it is obvious that increasing
r
brings the nearest zero
closer; for (1.6) it is mearly plausible that increasing one of the
%j
may have
this effect.
REMARK i.
For a non-trivlal characterization of
in a function class much larger than
REMARK 2.
zeros of
J(z).
E,
exp z,
via its lack of zeros,
see [HA, pp. 66-67].
It may be of interest to investigate the angular distribution of the
If
g(8)
is considerably larger for
lel
< /2
than for
K.B. STOLARSKY
420
181
> /2
-
(say
<_
<_ ),
8
J
will
have more zeros in the left half plane than
The results of P61ya [PO 4] on Mittag-Leffler’s function suggest that
in the right?
g(8)
a nearly uniform
will have its zeros distributed almost uniformly with respect
to angle.
COUNTEREXAMPLES TO THE O-HYPOTHESIS.
2.
When" (w)
is chosen so that
f(z)
has
the form (1.6), one can sometimes create a larger zero free region about the origin
by dispersing some of the mass arbitrarily far away from w
i.
We exhibit such an
example.
Let
D
0
be the closed disc about
f(z;e,%) --k[cos z +
where
k
n.
Fix a positive integer
D,
D
one near
O
cos z
to
(2.1)
cos %z],
z
O
For
I%1
<_ 5n,
e
0.
Hence for small
as
near
(2.2)
1.
/
the functions of (2.1) converge equlunl-
w/2 and the other near -w/2.
slder the zero
z
and set
is chosen so
dl(w)
formly on
n,
of radius
w/2.
%
For
i
z
is even, we need only con-
f
Since
this
they have only two zeros in
O
is
w/2.
For
%
4n write
/z’" + so
cos 4ne
is small)
and (since
24n2-i 3 +...,
3
+
Thus for
(2.3)
sin
e(n)
24n2-i E3 +.-3
or
sufficiently small, the zeros of
are more distant than the nearest zero of
f(z;e,4n)
(2.4)
nearest the origin
f(z;e,l).
For further examples of a similar nature (involving cosine sums on the real
line) see [NU].
counterexamples.
One might doubt, however, the
For
t
>_
0
let
compact support, continuous in
t(w)
t,
existence of far more extreme
B(w;t)
be a family of mass distributions
such that for
t
o
< t
I
the distribution
ZEROS OF SMALLEST MODULUS OF FUNCTIONS
(w;t I)
away from
the
U
in the radial direction.
i
Let
i.
R
t
f(z)
D
further
We shall even make the assumption (A):
and
i,
Dt
about
0
such that
ezwdt(w)
ft(z)
all
has no mass in some neighborhood
0
be the radius of the largest disc
has no zeros interior to
(w;t 0)
by moving some of the mass of
are centrally summetric about
t
of
(w;t 0)
is obtained from
421
(2.5)
t.
Is the quantity
PROBLEM.
R
lim sup R
(2.6)
t
finite?
If
0(w)
is the mas distribution corresponding to
to conjecture that
Roo
Rt
corresponding to a nearly maximal
smallest mass.
cos z,
it seems safe
and that any mass distribution
will have a large proportion of its mass very
For mass distributions along the real axis, the answer can be nega-
rive, no matter how large
REMARK 2.
z
i + i.
near the points
REMARK i.
3n/2,
is rather less than
e
U
See 7.
is!
In the example (2.1), the highest frequency was attached to the
The distribution of the roots of
f(z)
b
etZg(t)dt
a
where
when
g(t)
g(a)
is continuous is sometimes more peculiar or more difficult to analyze
g(b)
0;
see [CART, TI].
g(t)
When
is monotone, or
M
f(z)
[
k=N
a
k
cos kz,
a
N
<
aN+1
<--.<
aM,
(2.8)
the situation is far simpler; see [PO 2].
The simplest quantitative formulation of the 6-hypothesis is that
o
>> I.
(2.9)
422
K.B. STOLARSKY
More precisely, this means that given a family
duct
6o
G
of probability measures the pro-
exceeds a positive constant depending only on
is not true unless
G
eZ[ee -az
+ ee az +
We now show that this
Let
is somehow restricted.
f(z)
G.
2],
i
a,e > 0.
(2.10)
Here
(2.11)
By solving a quadratic equation, we find there are roots as close as
a
-I
ln[-e +
0(e
2)]I,
and
a
-I
ln[-(4e) -i +
0(i)]
(2.12)
Since
el/2n
e
0
the formulation (2.9) cannot be correct.
e
as
O,
(2.13)
In fact, if
P(z)
is any function that
assumes positive values, we cannot have a universal rule of the form
6 >> PCc)
(2.14)
Co/2,]"
(2.15)
since for
a
it would imply that
6
2 n e -I
c
o
a contradiction for an appropriate
3.
c
o
P(c0),
>>
(2.16)
by (2.13).
THE 6O-HYPOTHESIS FOR SYMMETRIC PEAKED DISTRIBUTIONS.
for functions
symmetric about
is large if
f(z)
I
of the form (1.1) when
and
g(t)
"strongly" peaked near
is very small.
The 6c-hypothesis is true
is a probability density function
1.
The result (3.3) asserts that
6
ZEROS OF SMALLEST MODULUS OF FUNCTIONS
423
Note that here and elsewhere we use "peaked" as an informal adjective rather
than as a concept with a precise logical definition.
THEOREM.
If
g(l-t)
g(l+t)
(3,1)
and
-t
g(l+t) < Ke
2
t > 0,
(3.2)
then
6 > min[F(O) ,/F(O)
where
PROOF.
Let
z
0
6
201
(3.4)
2k+l
be a zero of minimal modulus of
e
thus
.
(3.3)
-zf (z)
zolIf(o)
e
eZt g(l+t)dt;
_oo
zf(Zo)
z0t
(3.5)
(3.6)
and by expanding the hyperbolic cosine into an infinite series we easily see that
this is
T
< 2
t
2g (l+t)
0
<
Now choose
202
cosh t-i
dt
2
+ 2
t
cosh
T
6T-I + 2K
e
6t-t2 dr.
T
g(l+t)[cosh 6t-l]dt
(3.7)
424
K.B. STOLARSKY
T
1
+ 6 + 4K
+
(i
(4K+1)6-i)6
(3.8)
Then the last integral on the right of (3.7) is bounded by
{exp [-M(M-I) 6 2
/ (M-l) 6.
(3.9)
Hence
1 <
202 exp[62
+ (4K+I)6] + 1/2
(3.10)
+ (4K+I)6]
(3.11)
and
exp[62
1/4o 2 <
exp(4K+2)
62
6 > i
<
6 < i.
exp(4K+2)6,
The result follows.
Note that if
were replaced by
g
GRAEFFE’S METHOD.
pulse of width
C
g(t)
(2.9),
in (i.i) is very close to a smooth unimodal dis-
"somewhat" peaked at
2e
(3.2), the proof would make no essen-
The first naive formulation of the 6o-hypothesis,
does seem to hold when the
tribution that is
in
g.
tial use of the nonnegativity of
4.
gl
centered at
i
1.
For example, when
g(t)
is a rectangular
(this can be approximated arbitrarily well by a
function) we have
O
where
z
0
e/r,
and
z
0
denotes zeros of minimal modulus.
60
_+izle
Hence
/’.
We now show that something like (2.9) holds when
trated near
i
and
6
is small.
(4,1)
(4.2)
g(t)
is suitably concen-
In this case the old root-squaring method
(Graeffe’s method) that has been popular in the past for determining minimum modulus
zeros of polynomials is quite usable.
In fact, this method has been used previously
425
ZEROS OF SMALLEST MODULUS OF FUNCTIONS
on certain transcendental entire functions; see [POI, 5] and
g(t)
that the power moments of
[DIR].
We shall require
do not grow too rapidly; our condition (4.3) seems
quite natural since it is satisfied for
exp(-It-I I)
g(t)
exp[-(t-l)2],
g(t)
and
and commonly occurs in the study of characteristic functions [LUI, pp. 19ff.
THEOREM 4.1.
(t)
If the
,27ff].
of (i.I) satisfies
(4.3)
A
for positive constants
and
K,
and
z
is any zero of
0
f(z)
less than
I/K
modulus, then
l_zO21
PROOF.
<
A
IKz014
(5+3K21 z01 2)
(I_K21 z01 2) 2
2
(4,4)
Define
G(z)
f eiZtg(t)dt.
(4.5)
Since
g(t)dt
tg(t)dt
i,
(t-1)2g(t)dt
(4.6)
we find for the even function below on the left that
G(z) G(-z)
e
iz(t-s)
T.[ (t-l)
[i
Iwhere
R(z)
(s-l)
12 + m=2[
(-i)
mz2m t-s) 2m__] g( t
(2m)
2
[.[2-2" 0+2
z
(4.7)
g(t) g(s) dtds
+ R(z)
is an infinite sum of double integrals.
Since
g(s) dtds
in
K.B. STOLARSKY
426
(s-l) ]2mg(t)g(s)dtds
(t-l)
<[
3
<
(2m)flt_ll2m-j g(t)
j
m
A2K2m
f Is-ll
dt
(2m)
(2m-j) ’j’
J
j=0
(48)
j
g(s)ds
A2K2m(2m+l)
<
we have
< A2
IR(z)
[Kzl2m(2m+l)-
[
(4,9)
m=2
G(z 0)
Since
the result follows by summing the series on the right of (4.9).
O,
For example, if
A
K
g(t)
(4.4) shows that the variance of
g(t)
If
f(z)
and
1
has a zero in
must exceed
zl
< 1/4,
inequality
15.
has compact support, we can prove much more.
g(t)
If the support of
THEOREM 4.2.
lies in the interval
[a,b],
then
> I/(b-a).
PROOF.
Modify the proof of Theorem 4.1 by estimating
IR(z)
z (b-a)
Y.
<
say that contrary to (4.10) there is a zero
.
[Zo(b-a)
Since the double integral on the right of
JR(z0)
Now the variance
and
b
<
0
2
4(b-a)4(
+
R(z)
g(t)g(s)dtds.
(2m)
m=2
a
(4.10)
z
0
as follows:
(4,11)
with
(4,12)
< I.
-
(4.11) equals one,
+.-.) < 1
64 (b-a) 4 < 2-I
(4.13)
is maximal when the mass consists of two equal atoms located at
respectively, in which case
0
From (4.7) and (4.13) with
z
z
0
(b-a)/2.
we obtain
(4.14)
427
ZEROS OF SMALLEST MODULUS OF FUNCTIONS
6202 <_ 1123
I-
(4,15)
and so
22/23
< (b-a)/2.
(4,16)
But this contradicts (4.12).
Note that the proofs of the above theorems use (4.6), but not the nonnegativity
of
g.
Since the function
(eaZ+e bz)/2
f(z)
corresponding to two equal atoms at
4.2 is within a factor of
n
a
and
has zeros at
b
of being best possible.
[a,b],
If the mass is in
-+in/(b-a),
Theorem
For this function
T/2.
60
CONJECTURE.
(4,17)
(4,18)
then
6 > w/(b-a).
(4.19)
[a,b],
We add that if the mass is not supported by any inteval smaller than
(b-a)/W
then
5.
is the "linear density" of the zeros of
PARTIAL SUMS OF THE EXPONENTIAL SERIES.
see [TI].
f(z)
By Hurwitz’s Theorem, the partial sums
N
SN(Z)
have no zeros in a disc of radius
r
N
about
statement has been made very precise by
of
s
N(z)
the subset
are divided by
H
of the curve
N,
Re(z) < i.
(5.1)
0,
Szeg [SZ]
where
r
/
N
C
In fact,
as
oo.
N
This
who has shown that if the zeros
these normalized zeros, for large
N,
lie very near
defined by
ze
for which
Z zk/k!
k=0
H
1-z
1
is a loop that encircles the origin.
(5.2)
The curve
428
K.B. STOLARSKY
"a".
(5.2) roughly resembles the Greek letter
at
z
i,
and to the right of
I
z
It intersects itself at right angles
it consists of two curves, mirror images in
the z-axis of each other, with the one in the first quadrant being convex, and having
an ordinate that grows exponentially with its abscissa.
(5.2) to the left and right of
respectively of
z
We speak of the parts of
as being the "head
i
H"
and the "tail
T",
C.
Curiously enough,
Szeg
also proved that the normalized zeros of the "tail" of
the exponential series
k
TN(Z)
lie very near
T!
(5.3)
k=N+l k
In particular, all the zeros are at least
and this is asymptotically best possible.
N
in absolute value,
We now show how this bears on the present
investigations.
Consider the probability density given by
(N+l)t
N
0
_<
t
_< 1
g(t)
(5.4)
0
Then
g(t)
densities.
can be approximated arbitrarily well by sharply peaked
tg(t)dt
0
N
C
unimodal
A simple calculation yields
i
so for
otherwise.
N+I
N--$’
C
N+I
(N+2) 2 (N+3)
2
g(t)
large it is very nearly the sort of
rN(z)
we considered in (i.i).
ezlz e-SsNds
(5.5)
Since
(5.6)
0
and (by change of variable) we have
f(z)
e
Ztg (t)dt
0
it follows from
N+I
(-z) N+I
e-SsNds,
(5.7)
0
(5.6) and (5.7) that
f(z)
(N+I)!e
(_z) N+I
z
TN(-Z).
(5.8)
ZEROS OF SMALLEST MODULUS OF FUNCTIONS
Thus
>> i,
and
o
i
as
429
.
N
(5.9)
This provides another example where (2.9) is valid, and with a smaller constant than
in (4.2) or (4.18).
REMARK.
Szeg
also determined the asymptotic angular distribution of the zeros.
His results (particularly in a neighborhood of
z
I)
have been both greatly
refined and also extended to entries of the Pad table for
exp(z).
See [BU, NEW,
S-3].
6.
SMALL ZEROS OF BESSEL FUNCTIONS.
in the absence of the sort of "side
The rule (2.9) seems to hold quite generally
lobes" that played a role
in constructing the
"over-
counterexamples of 2.
This leads us to suspect it may be satisfied, or even
satisfied," when g(t)
is convex aside from some small neighborhood of its maximum.
For an example of this we turn to the book of B. C. Carlson, Special Functions of
Applied Mathematics
[CAR], that emphasizes the utility of the fact that
most of the
commonly occurring special functions are expectations of elementary functions (sometimes exponentials) with respect to common probability measures.
Bessel functions
Jr(X)
have the form
Jr (x)
where
J
B(x,y)
tribution
(6.1)
exp[wz I +
(6.2)
(l-w)z2]d
I
0
exp[uzl +
(w)
(l-u)z 2]
(w)
u
bl-I (l-u) b2-1
du.
B(bl’b2)
For a random variable
is the usual beta-function.
X
with the dis-
one has
b
E(X)
so
(x/2) r
F(r+l) S(r+i/2,r+I/2;ix,-ix)
[CAR, pp. 93-96]
S(bl,b2;Zl,Z 2)
Here
For example, the
I
bl+b 2
E(X
2)
bl(bl+I)
(bl+b2) (bl+b2+l)
(6.3)
430
K.B. STOLARSKY
blb 2
O2=
(6,4)
2
(bl+b2) (bl+b2+l)
Hence
r(_r+l)
S (r+i/2, r+i/2; z/2 ,-z/2)
f(z)
is a function
(z/41) r
J (z/2f)
r
(6 5)
of the form (i.I) with
o
8(r+l) ]-i/2.
(6 6)
Thus the rule (2.9) yields
6 >> r I/2
for
Jr(Z).
est zero of
(6.7)
But Watson’s treatise on Bessel functions [WA] tells us that the smallJ (z)
is
r
r
+ 0(r I/3)
-
so in this case
6 >> i
7.
SOME IDEAS FROM PROBABILITY THEORY.
(6.8)
The example of 6 leads us to ask if there
are some further hypotheses that can be made on the shape of a unitary spike so that
the nearest zero would have to be even further away for
o
small, with perhaps a
rule such as
6Om
m
where
>> 1
(7.1)
is some positive integer.
If we think of
g(t)
as the distribution function of a random variable
X,
it
is natural to also consider the distribution function for the average
X
x
where
f(z;Y)
O(Y)
XI,... ,Xn
I +’’’+
Xn
(7 2)
n
are random variables with the same distribution as
X
denote the moment generating function for a given random variable
denote its variance, etc.
f(z;X--)
itself.
Y,
Let
let
Then
[f(;X) in,
O(X--)
O(X)/n,
(7.3)
ZEROS OF SMALLEST MODULUS OF FUNCTIONS
431
and
(X)
n(X).
Thus if we begin with a class of functions
g(t)
replace each
(7.4)
g(t)
for which the rule (2.9) holds, and
by the density for the corresponding
"much more room to spare."
X,
the rule holds with
What (7.3) and (7.4) suggest, of course, is that for
"nicely shaped" pulses, the appropriate rule to investigate is
o2
>> 1,
(7,5)
a rule we already stumbled upon in 6.
g(t)
We next observe that for
large, and
f(z)
is zero free.
with infinite support, it may be that
g(t)
For example, let
1
g(t)
exp[-
0. 2/"
is
0.
be the normal density
(t-l)2]
2
(7.6)
20"
Since
exp(-z+z20.2/2)
f(z)
there are no zeros!
nearly as good.
(7.7)
We can now construct densities of compact upport that are
Simply multiply
g(t)
fine it to be zero outside of some large interval
again has total mass one.
1
by a number
+ e
where
e > 0
and rede-
[-L,L], so the resulting function
Various densities other than the
g(t)
of (7.6) are also
available; in fact, when an infinitely divisible distribution has an entire charac-
f(z),
teristic function
then
f(z)
is zero free
The graph of the normal density is
derivative changes signs
n
times.
[LUI, p. 187].
bell-shaped; this means that its
If this holds merely for
n < M
nth
we can say it
It is conceivable that some such subtle measure of the shape of
is M-bell shaped.
peakedness
may determine the extent to which (2.9) or
(7.5) may be replaced by a sharper rule.
The concept of bell-shaped will also occur
the spike and not merely its
below in 8.
We now return to the problem of
bution" with
1
and
0.
2.
Let
g(t)
be a "normal density distri-
extremely large, but modified as above to be of compact
432
K. B. STOLARSKY
support.
e > 0
Fix
[I-B,I+]
and support
i
r(t;)
and let
be a rectangular pulse distribution with
where
i+n
go(t) dt
1-rl
/2
First shift
r(t;N)
of the mass of
and then shift the remaining mass of
bution of
go(t).
By choice of
< e"
to each of the points
r(t;)
PROBLEM.
i
N
and
,
+
i
outwards so it becomes the mass distri-
we can insure that before the second mass shift
/ while after that shift
the nearest zero is at a distance of approximately
is far away as desired.
(7.8)
it
Thus the problem of 2 has a negative answer in this case.
Can the mass of the rectangular pulse be moved out to infinity con-
tinuously with respect to time so that the distance of the nearest zero from the
origin is monotonically increasing?
8.
MULTIPLY POSITIVE FUNCTIONS.
function
g(x)
In [SCH03] Schoenberg defines a real measurable
such that
g(x)dx <
0 <
(8.1)
to be k-times positive if for
Xl
<
x2
<’’’<
Xn; Yl
<
Y2
<’" "<
(8,2)
Yn
we have
det[g(xi_Yj)]ni,j=l --> 0
for
If
i < n < k.
tally positive.
g(x)
is k-times positive for every
are known to be totally positive.
function, then
k,
it is said to be to-
For example,
exp(-x 2)
THEOREM.
(8.3)
If
g(x)
exp(-x-e -x)
(cosh x) -I
(8.4)
Schoenberg [SCH02] has proved the
is totally positive, but not the exponentl,al of a linear
ZEROS OF SMALLEST MODULUS OF FUNCTIONS
o
e
zt
433
i
g(t) dt
E(-z)
where
E(z)
Ce
-Tz2+az --l-(l+amZ)e -amz
(8.6)
m=l
Here the parameters
C,y,a,a
> 0
are real, with
and
(8.7)
Moreover, it is
Thus the Laplace transform of such a function is zero free.
known that the totally positive functions are bell-shaped [SCHOI, HIRI-2]!
But of
greatest interest in connection with the Go-hypothesis is Schoenberg’s theorem
[SCH03]
on k-times positive functions of compact support.
THEOREM.
Let
g(x)
[0,hi.
be k-times positive and identically zero outside of
Then
f(z)
e
Ztg (t)dt
(8.8)
< k,
(8.9)
0
has no zeros in the strip
lIm(z)
and this is best possible.
Schoenberg’s example that shows the result is best possible is particularly
illuminating.
He defines the function
sln
g(t)
t)
(x
0
<__
x
<_
T,
(8.10)
g(t)
otherwise,
and observes that
e
-zw/2
f[ eZt
g(t)dt
0
for
> -i
(note that
g(t)
has mean
r(a+)
2r (1/2 (+2+iz)) r (1/2 (a+2-1z))
w/2
rather than
1).
For
> k
2
he
434
K. B. STOLARSKY
g(t)
shows that
Now by (8.11) the zeros of its transform are
is k-times positive.
+(e+2+m) i,
and (set
Scale
0)
m
m
0,i
it follows that the strict inequality in
g(t)
so that it becomes a probability density.
variance tends to
(8.9) cannot be relaxed.
Then as
and all the mass becomes concentrated at its mean
0,
terms of variance we have, just as for the Bessel functions of
6 >>
REMARK i.
a-
its
n/2.
6, that
-2.
0
In
(8.13)
I know of no paper or monograph that discusses infinitely divisible
distributions and totally positive functions together, although they seem to be similar in some ways.
REMARK 2.
exp(-t 2)
Note that
The property of being
belongs to both of these function classes.
k
times positive is very delicate and can be
g(t).
destroyed by a small perturbation of
However, the parameters
at most slightly changed by such perturbations, so a law such as
9.
SEVERAL COMPLEX VARIABLES.
6
and
o
are
(8.13) is "robust."
The nonvanishing of the exponential has an analogue
in several complex variables, namely that
[
exp z. =0
cannot occur if the
z.
3
(9.1)
3
j--i
exhibit only "small deviations from the
this precise, define the distance
A
(a
I
d(A,P)
n)
a
mean." To make
between any points
P
(z
I
z
n)
(9.2)
in complex n-space by
d2(A, P)
r..la.l-Z.i 12
(9,3)
The diagonal of complex n-space is the set of complex numbers having all components
identical.
ZEROS OF SMALLEST MODULUS OF FUNCTIONS
If (9.1) holds and
THEOREM.
n > 2,
435
P
then the point
(z I,
z
n)
has
distance at least
d
n
from the diagonal.
(l+n -l)n n
P
On the other hand, there is a
[l+0(n n)
D
n
-i
(9.4)
with distance at most
]n n
(9.5)
from the diagonal for which (9.1) holds.
The above result again comfirms the "60-hypothesis," though in a rather differ-
ent setting.
For the proof, and further comments on the relation of this result to
various statistical estimates, see
i0.
OSCILLATION THEORY.
[ST].
It is curious that the 6o-hypothesis does not seem to be
However, if
directly addressed by any previous literature.
meter
t
we think of the para-
in
h(z)
ht(z)
as being expressed by square roots of
or by fourth roots of
p
q
e
tz
+
e
-tz
(i0.i)
in the differential equation
h"(z)
qh(z)
0
(10.2)
(4)(z)
ph(z)
O,
(10.3)
in
h
then the oscillation theory of ordinary differential equations, starting with the
comparison theorems of Sturm (see [HI2, pp. 373-384; pp. 576-644]), is possibly
relevant.
The literature in this direction is large; we mention merely [HII-2, LE,
NEI-2].
To illustrate its bearing on the 6o-hypothesis, consider
y"
where
q
q(z)
qy
(10.4)
0
is analytic in a neighborhood of
z
0
and real for real
z.
Let
436
f(z)
K.B. STOLARSKY
be the solution with
f(0)
i, f’(O)
d(w)
If
0.
is a measure on the
real axis satisfying (1.4), then a simple calculation (note that
f"(0)
q(0))
shows that (1.5) is valid with
02
q(O) + 1.
(i05)
Now a theorem of Hille [HI2] asserts that for
and satisfying
lq(z)
<M
q
q(z)
analytic in
zl
< R
there, we have
6 > min(R,W/2).
(10.6)
Thus, in some loose sense, Hille’s oscillation theorem gives the same lower bound as
the
6
hypothesis for slowly varying
q.
Both are quantitative versions of the
classical principle that the larger the potential the more rapidly the solution
oscillates.
ii.
REMARKS. ’The location of minimum modulus zeros has been extensively investi-
gated for polynomials [MA], but the literature for analytic functions does not seem
terribly large, despite the importance of the related problem of location of minimal
eigenvalues.
The literature on zero-free half planes, at least, for exponential
polynomials, is perhaps larger owing to its importance for stability theory.
See
[PON] and also [BE] for many further references.
For the asymptotic distribution of zeros of exponential polynomials and closely
related functions see [BE, CART, DII-2, LA, ME, MO, P02-4, POO2, SCHW, TI, TU].
Also relevant here, as well as to 5, is [GAl.
For the local distribution of zeros
of exponential polynomials in the complex plane see [POOl,
ACKNOWLEDGEMENT.
TIJ, VOI],
I thank the referee for a number of valuable suggestions, and also
for removing many misprints and obscurities.
ZEROS OF SMALLEST MODULUS OF FUNCTIONS
[BE]
[BER]
437
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Academic Press, New York, 1963.
BERENSTEIN, CARLOS A. and TAYLOR, B. A., Mean Periodic Functions, Int. J. of
3(1980), 199-235.
Math. and Math. Sci.
[BU]
[CAR]
[CART]
[DII]
BUCKOLTZ, J. D., A characterization of the exponential series, Amer. Math.
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CARLSON, B. C., Special Functions of Applied Mathematics, Academic Press,
New York, 1977.
CARTWRIGHT, M. L., The zeros of certain integral functions, Quart. J. Math.
1(1930), 38-59; II, ibid 2(1931), 113-129.
DICKSON, D. G., The asymptotic distribution of zeros of exponential sums,
Publ. Math. Debrecen 11(1964), 295-300.
[DI2]
Zeros of exponential sums, Proc. Amer. Math. Soc. 73(1965),
84-89.
[DIR]
DIRSCHMID, H., Bemerkungen zu einer Arbeit yon G. P61ya zur Bestimmung der
Nullstellen ganzer Funktionen, Numer. Math. 13(1969), 344-348.
[GA]
GANELIUS, T., Sequences of analytic functions and their zeros, Arkiv. Mat.
3(1953), 1-50.
[HA]
}lAYMAN, W. K., Meromorphic Functions, Clarendon Press, Oxford, 1964.
[HII]
HILLE, E., Nonoscillation theorems, Trans. Amer. Math. Soc. 64(1948),
234-252.
[HI2]
Lectures on Ordinary Differential Equations, Addison-Wesley,
Reading, 1969.
[HIRI]
HIRSCHMAN, I. I. and WIDDER, D. V., The inversion of a general class of
convolution transforms, Trans. Amer. Math. Soc. 66(1949), 135-201.
[HIR2]
Proof of a conjecture of I. J. Schoenberg, Proc. Amer. Math.
Soc. 1(1950), 63-65.
[LA]
LANGER, R. E., On the zeros of exponential sums and integrals, Bull. Amer.
Math. Soc. 37(1931), 213-239.
[LE]
LEIGHTON, W. and NEHARI, Z., On the oscillation of solutions of self-adjoint
linear differential equations of the 4t__h order, Trans. Amer. Math. Soc.
89(1958), 325-377.
[LUI]
LU2
[MA]
LUKACS, E., Characteristic Functions, Hafner, New York, 1960.
Characteristic Functions, 2nd. end., Griffin, London, 1970.
MARDEN, M. Geometry of Polynomials, 2nd. ed., Providence, Amer. Math. Soc.,
1966.
438
K.B. STOLARSKY
[ME]
MEYER, G. P., Uber die Nulstellen von Exponentialsummen lngs einer logarithmischen Wechsellinie, Archly der Math. 32(1979), 479-486.
[MO]
MORENO, C. J., The zeros of exponential polynomials (I), Compositio Math.
26(1973), 69-78.
[NEI]
NEHARI, Z., Non-oscillation criteria for n-th
equations, Duke Math. J. 32(1965), 607-616.
[NE2]
Disconjugacy criteria for linear differential equations,
J. Differential Equations 4(1968), 604-611.
[NEW]
NEWMAN, D. J. and RIVLIN, T. J., The zeros of partial sums of the exponential
function, J. Approximation Theor 5(1972), 405-412.
[NU]
[POI]
order linear differential
NULTON, J. D. and STOLARSKY, K. B., The first sign change of a cosine polynomial, Proc. Amer. Math. Soc., to appear.
P6LYA,
G., Sur la mthode de Graeffe, C. R. 156(1913), 1145-1147.
ber
[PO2]
die Nullstellen gewisser ganzer Funktionen, Math. Zeit.
2(1918), 352-383.
[PO3]
Geometrisches ber die Verteilung der Nullstellen gewlsser
ganzer transzendenter Funktionen, S. B. Bayer Akad. Wiss. (1920), 285-290.
Bemerkung 5ber die Mittag-Lefflerschen Funktlonen
[P04]
E(z),
TShoku Math. J. 19(1921), 241-248.
Graeffe’s method for eigenvalues, Numer. Math. 11(1968),
[P05]
315-319.
[PON]
[POOl]
PONTRYAGIN, L. S., On the zeros of some elementary transcendental functions,
Izv. Akad. Nauk SSSR, Ser. Mat. 6(1942), 115-134 (Russian); Amer. Math. Soc.
Translations l(Ser. 2), Providence (1955), 95-110.
VAN DER POORTEN, A. J., On the number of zeros of functions, L’Enseigenement
Math. 23( 1977), 19-38.
[PO02]
A note on the zeros of exponential polynomials, Comp. Math.
31 (1975)
i09-i13.
[SAI]
SAFF, E. B. and VARGA, R. S., Zero-free parabolic regions for sequences of
polynomials, SlAM J. Math. Anal. 7(1976), 344-357.
[SA2]
The behavior of the Pad table for the exponential, in
Approximation Theory, Vol. II (G. G. Lorentz, C. K. Chui, L. L. Shumaker,
eds.), New York, Academic Press, 1976, pp. 195-213.
[SA3]
On the zeros and poles of
Numer. Math. 30(1978), 241-266.
[SCHOI]
SCH02
Pad approxiamnts
e
Z
III,
SCHOENBERG, I. J., On smoothing operations and their generating functions,
Bull. Amer. Math. Soc. 59(1953), 199-230.
On
P61ya frequency
functions I:
tions and their Laplace transforms, Journal
[scHo3]
to
The totally positive func-
d’Analyse Math. 1(1951), 331-374.
On the zeros of the generating functions of multiply positive
sequences and functions, Ann. Math. 62(1955), 447-471.
ZEROS OF SMALLEST MODULUS OF FUNCTIONS
[SCHW]
439
SCHWENGELER, E., Goemetrisches 5her die Verteilung der Nullstellen spezleller
ganzer Funktionen, Thesis, Zrich, 1925.
[ST]
STOLARSKY, K. B., Zero-free regions for exponential sums, Proc. Amer. Math.
Soc., to appear.
[SZ]
SZEG,
G., Uber
Sitzungsber
[TI]
eine Eigenschaft der Exponentlalrelhe, Berlin Math. Ges.
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[TIJ]
[TU]
[VOI]
TIJDEMAN, R., On the number of zeros of general exponential polynomials,
Proc. Nederl. Akad. Wetensch. Ser. A 74(1971), 1-7.
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P., Eine Neue Methode
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VOORHOEVE, M., On the oscillation of exponential polynomials, Math. Zeit.
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[vo2]
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[WA]
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