Internat. J. Math. & Math. Sci.
3 (1987) 443-452
443
Vol. i0, No
FOURIER COEFFICIENTS AND GROWTH OF HARMONIC FUNCTIONS
A. FRYANT
Department of Mathematics
Utica College of Syracuse University
13502
Utica, New York
and
H. SHANKAR
Department of Mathematics
Ohio University
Athens, Ohio
4570|
(Received October 7, 1986)
ABSTRACT.
We consider Harmonic Functions,
H
of several variables.
We obtain
H
necessary and sufficient conditions on its Fourier coefficients so that
entire harmonic (that is, has no finite
is an
singularities) function; the radius of har-
monicity in terms of its Fourier coefficients in case
H
is not entire.
Further, we
obtain, in terms of its Fourier coefficients, the Order and Type growth measures,
both in case
H
is entire or non-entire.
KEY WORDS AND PHRASES.
Harmonic function, Spherical harmonics, Entire harmonic func-
tion, Radius of harmonicity, Fourier coefficients, Order and Type growth measures,
Liouville’s theorem.
1980 AMS SUBJECT CLASSIFICATION CODES.
|.
3|B05,
42A16.
INTRODUCTION.
Let
iables
H
x
k
denote the set of all homogeneous, harmonic polynomials in the
x
2
x
having real coefficients.
n
H
k
n
var-
is a vector space of
dimension
d
k
(n+k-3)!
(n + 2k- 2)
k! (n- 2)!
(see [, p. 237], or [2, p. 145]).
<f
g>
With respect to the inner product
f(x) g(x) do
2
and
n-l
do
is the element of surface area on this sphere, let
{YImdk|__
n
be an ortho-
A. FRYANT AND H. SHANKAR
444
H
normal basis for
Such orthonormal, homogeneous, harmonic polynomials
k
m
Yk
are
It is well known that spherical harmonics
called spherical harmonics of degree
k
of different degrees are orthogonal.
That is,
(see [2, p. 141]).
H
Let
be harmonic in a neighborhood
of the origin in
n
l
H
That is,
satisfies Laplace’s equation
2H + 2H
x21 -22
If
throughout
then on this sphere
0
>
0
2H
+
0
x n2
xl
is chosen so that the sphere
is contained in
0
has the Fourier (spherical harmonic) expansion
H
d
k
H(x)
k=O m=l
akm r
r
where
02k+n-I (’On- Ix I=0
akm
Here
do
series
0
n-1
(1 1)
do
H(x)
Y:(x)
do
xl
is the element of surface area on the sphere
2
L
sense on the sphere
0
Ixl
converges in the
absolutely and uniformly on compact subsets of the interior ball
The
0
and thus converges
Ix]
<
The series
0
(l.l)
might, of course, converge absolutely and uniformly within larger balls centered
at the
origin, depending on the location of the singularities of
H
The objective of this paper is to characterize growth properties of
H
explicit-
Specifically, necessary and sufficient
akm
are found so that H is entire (that is, has no
ly in terms of its Fourier coefficients
conditions on the coefficients
akm
finite singularities), the radius of harmonicity of the spherical harmonic expansion
H
is computed in the case where
is not entire, and the order and type of
H
found in both the entire and non-entire case.
2.
CONVERGENCE OF SPHERICAL HARMONIC EXPANSIONS.
The following result
THEOREM.
If
{Ykm}dk
m=l
[2, p. 144] is
.
d
k
m=l
for all
x
Since
on the unit sphere
d
(n + 2k
k
d
k
m=l
[y(x)]2
of fundamental importance in our work:
is an orthonormal basis for
2)
lYe(x) ]2
d
and
k
then
k
n-!
Ixl
(n + k
3)!
2)
kW (n
H
2n/2
n-I
2)(n + k
r(n/m)(n + 2k
n/2 k! (n
2)!
2
r(n/2)
3)!
we have
Ixl
This result is a natural generalization of the Pythagorean identity
are
FOURIER COEFFICIENTS AND GROWTH OF HARMONIC FUNCTIONS
sin2@
For, in
k
R2
_"
we have
d
2
k
and
-2 cos kO
sin kO
are
(akl, ak2
k
+ cos
2
0
The two spherical harmonics of degree
and the result of the preceding theorem becomes
(_I__ sin k@) 2
Let a
k
terms of degree
445
akd k)
in the expansion
akl (a2kl
2
(I
,-cos k@)
+
2
2
be the vector of Fourier coefficients of the
(l.l)
+
We define
2
+
ak2
+
akd k )
2
Our first result gives the radius of harmonicity of the series
these
(l.l)
in terms of
]akl.
THEOREM I.
The series
d
[
k=O
k
[ akm r kY(x/r)
m=l
xl
r
Ixl
converges absolutely and uniformly on compact subsets of the disk
< R
where
is given by
R
-I
lim
suplakl I/k
Further, such convergence obtains within no larger ball centered at the origin.
PROOF.
d
d
k
Z Z
k=O m=l
akm k (x/=)l
k
<
I =k m=!Z akmY(x/r)[
k=O
..
d
<
k=O m=!
[
k=O
d
k
a
2
km
) (Ik
m=!
[Y(x)]2)
[ak[(dklmn_l ) r k
Z
=n-I k=O
I"kl=
Now,
lim
(dk) llk k-K:,lim
lira
k-o
Further, since
r(k + a)/r(k + 8)
[(n + 2k- 2)
(!k-2)!3)!
+
l12k
+ k- 3)
[(n (."] I/2k
1)!
ka-8
the above limit is
n-2)l12k
lim --(k
Ikl rk has radius of convergence
R-I li_sup( dvk lakl I/k li_suplak[ l/k
Thus the power series
’k=O
R
where
R
A. FRYANT AND H. SHANKAR
446
d_k
’k=O m-I
Since this series dominates the spherical harmonic series
akm rky(x/r
the spherical harmonic series converges absolutely and uniformly for all
Ixl
< R
>
E
0
Ixl
and thus on compact subsets of the ball
such that
x
< R
To see that such convergence cannot obtain within any larger disk centered at the
origin, consider
d
M2(r)
oo
n-I r
I:!:
n-I
k
m=O k=l
do)J
akm
Ixl
If the spherical harmonic series converges uniformly for
then term by term
r
integration yields
d
[Mm(r)]
2
k
2
I
m=l
. ,.
k--O
akm r
d
r2k
k=O
k
r2k
to
n-!
2
k
[Y
rn-1 [x[--r
akm
ton- f ixi--1
m=
d
2k
(x/r)]2
do
eYe(x) ]2 di
2
akm
k=0
m=
[
lak 12 r2k
This series then must be convergent, which implies
r
Thus
r
< R
-I
and the proof of our theorem is complete.
dk
k
If H(x)
r
is harmonic in a neighborhood of
r
k=0R m=O akm Y(x/
COROLLARY.
the origin, then the distance
given by
R
lisuplaki I/k__ R
>
-I
H
from the origin to the nearest singularity of
is
ll/k.,
suplak
k-o
lim
By Harnack’s theorem, the uniform limit of a sequence of harmonic functions is harmonic. Thus our previous theorem ensures that H is harmonic in the open
-I
ball ixl < R
where R
Further, H cannot be harmonic throughPROOF.
li_supiak il/k
out any larger open ball centered at the origin.
would exist a
0
> R
H
such that
For if this were the case, there
would be harmonic on the closed ball
(that is, harmonic throughout an open set. containing the closed ball
ix[
Ixl
0
0 ).
Since the spherical harmonics are complete in the space of square integrable func-
tions on any sphere
Ixl
(see [2, chap. 4, sec. 2]),
O
H
would then have the
Fourier series expansion
d
k
H(x)
k=Om=O
where the series was at least
L
2
akm r kY(x/r)
convergent on the sphere
convergent on compact subsets of the interior ball
the result of our previous theorem.
where on the sphere
Ixi
R
Thus
H
Ixl
<
0
ixi
0
and uniformly
But this contradicts
must have at least one singularity some-
FOURIER COEFFICIENTS AND GROWTH OF HARMONIC FUNCTIONS
In view of the result of this corollary, it is natural to call
H
As an aid in investigating the growth of harmonic functions
analytic functions of
the radius of
r
we next obtain
which provide good upper and lower bounds on
M(r)
max
THEOREM 2.
If
"(x)
[:l- akmrk (x/r)
[-0
M2(r)
for all
R
H
harmonicity of
M(r, H)
44?
r
< M(r) <
R
less than the radius of harmonicity
H
of
where
lakl 2 2k)
([
2(r
then
and
PROOF.
Theorem I.
M(r) < M(r)
Derivation of the upper bound
is contained in the proof of
For the lower bound, we have
[
s2(r)
k=O
lakl 2 r2k) %
.F
H2(x)
do] i
dO) %
max H(x)
n-I
< (,
n-l
M(r)
3.
ENTIRE HARMONIC FUNCTIONS
The results of the previous section allow for the characterization of entire hat-
monic functions in terms of their spherical harmonic coefficients.
That is, a har-
monic function having the series expansion (l.l) is entire if and only if R
-l
where R
In this section we characterize the growth of entire
lisuplakll/k
harmonic functions explicitly in terms of the functions’ spherical harmonic coefficients.
As measures of growth we use the classical concepts of order and type.
p of an entire harmonic function H is defined by
The order
p
where
M(r)
M(r, H)
lira sup
loz log M(r)
log r
is as given in the previous section.
zero order, then the type
H
of
z
If
H
is of finite, non-
is defined by
M(r)
lim sup ,l..,og p
r-K=
r
THEOREM 3.
is given by
If
H(x)
k=O [’dkm=l akmrkYkm(x/r)
is entire, then the order of
H
448
A. FRYANT AND H. SHANKAR
k log k
lira sup
PROOF.
M2(r)
Let
lisuplak ll/k
0
t4(r)
and
I. 1
be as defined in Theorem 2.
.
[M2(r)]
Thus the analytic continuation of
[M2(z)]2
H
Since
2
is entire,
that is
lak 12 z2k
is an entire analytic function of the complex variable
2
where r
Also,
2
z
Further,
’’IM2(z)l 2
Izl
[M2(r)]
log log
lim sup
<
[M2(z)]
2
lira sup
r-o
log log
lim sup
log r
r-o
Thus the order of
[M2(r)]
log log M(r)
log r
M2(r)
log r
r-m
O(H)
is less than or equal to the order
f(z)
the order of an entire analytic function
o(f)
=0
k log k
lim sup
Ck
O(H)
of
H
But
is given by
z
[2, p. 9]
We conclude that
2k log 2k
lim sup
k-
M(r)
og
lim sup
k
lak I-z
is also an analytic function of
of its variable.
k log k
og
< O(H)
lakl-’
and thus can be continued to complex values
r
We have
n-I
Now,
lira sup
k-=
as was shown in the proof of Theorem
entire function.
since
?4(r) <
the order of
max
?4(z)
p. 9] the order of
I.
d/k
=
Thus since
H
is entire,
Appealing to the result of Theorem 2, we have
[t4(z)
it follows that the order of
H
t4(z)
is also an
M(r) < ?4(r)
and
is less than or equal to
But according to the classical function theoretic result [2,
4(z)
is
lira sup
k log k
lira sup
Therefore,
P(H) < lira sup
k’
k log k
log
lak[-I
k log k
FOURIER COEFFICIENTS AND GROWTH OF HARMONIC FUNCTIONS
449
and the proof of our theorem is complete.
Note that in the preceding argument we actually show that
p[M22(z)]
That is, the analytic functions of
lower bounds on
H
M(r)
p[M(z)]
p(H)
given in Theorem 2 which provide upper and
r
have the same order as
H
They also have the same type as
as is shown in the proof of our next theorem.
THEOREM 4.
dk
[k=O
r Y (x/r) is an entire harmonic function
m=l akm
of H is given by
then the type
H(x)
If
of finite, non-zero order
0
(e0) -I
PROOF.
Since
[M2(z)]
2
lisup
has order
log[
by definition its type is
0
max
iim sup
IM2(z)12
r0
r-o
log[ max
2 lim sup
r0
r
M(r)
lim sup log
r_o
rO
< 2
where as before
result
[3, p. ]
order
0
M(r)
the
max
Izl=r
type of
IH(x)
Further, by the classical function theoretic
[k=O
f(z)
an entire analytic function
Ck z
k
having
is
Thus the type of
(e0) -I
[M2(z)]
2
li_sup klCk lO/k
(eO) -I
(f)
is
2(e0) -I
2k(lakl2) 0/k
li_sup
li_sup
klak lO/k
and the previous inequality implies
(e0) -I
M(z)
Also, since the order of
z(H)
S
is
klak 10/k
and
0
(eP)-I li_sup
(e0)
4.
li_sup
-I
lim sup
k-o
(H)
M(r) < H(r) <
max
IM(z)l
we have
k(%li/k lakl) 0/k
klakl0/k
GROWTH OF HARMONIC FUNCTIONS IN SPHERES
..
We next consider the growth of non-entire harmonic functions interior to the
largest sphere centered at the origin which omits the function’s singularities. If
dk
H(x)
k=O m=O
akm r
k
Y(x/r)
and
A. FRYANT AND H. SHANKAR
450
-l
R
lisup
then we define the order
lakl ]/k
and type
0
%
R
0,
H
of
interior to the open ball
Ixl
< R
by
lim sup
0
r/R
lg+lg+M(r)
r)]
log[R/(R
and
log+M(r)
r)]0
[R/(R
lim sup
%
r-R
where
M(r)
of non-entire
lira sup
k_o
IH(x)
max
Ixl=r
analytic
ICk l|/k
0
<
The definitions agree with those for the order and type
functions (see [4]).
I/R
<
0
then the order and type of
have been computed in terms of the coefficients
c
f
of
k
,.=
k
z
k
interior to the disk
f(z)
Further, if
[4, 5, 6]
f
c
where
Izl
< R
These results
are
0
log+log+Ick
lim sup
k-o
o+l
R
k
log k
and
iO
T
0
(0 + I) 0+I
lira sup
k
(log+lekl Rk)O+
k
p
Arguing as in the proofs of Theorems 3 and 4, we obtain such results for harmonic
funct ions
k
R
II
0
then the order
0
T
H
of
H
l/R,
is given by
+
+
k
R
log log
lakl
lim sup
0+1
and the type
of
0
i=,<i’ /k
g(x
log k
is
0
(0 + I) 0+I
limk_osup
(log+lakl Rk)0+
k0
The proof of this theorem is obtained by again appealing to the analytic bounds
on
M(r)
5.
LIOUVILLE’S THEOREM
provided by Theorem 2.
A very simple proof of the analog of Liouville’s theorem for entire harmonic
functions can be given by arguing on the function
M2(r)
defined in Theorem 2.
We
give this proof below.
THEOREM 6.
all
x E R
n
PROOF.
then
If
H
If
H
H
is an entire harmonic function in
R
n
and
iH(x) <
is constant
is entire, then it has the spherical harmonic expansion
dk
H(x)
k=0 m=
akm r
k
Ykm(x/r
and the series converges uniformly on compact subsets of
Rn
Thus
B
for
FOURIER COEFFICIENTS AND GROWTH OF HARMONIC FUNCTIONS
2
[M2(r)
[H(x) 2 do
f
on-| r
451
k--O
Now, suppose
IH(x)[ <
B
for all
ik=O lakl2 r 2k < 0B 2 forfor kall I,r 2,> 0
implies
k
>
[a
k12
x
Rn
Then since
Thus
,ak ,2
But
a
r
M2(r)
2k < B 2
dk-- 2
for all
Im=l akin
This yeilds H(x)
]H(x)
< max
r
Hence
>
we have
which
0
akin
0
for all
=-
and all corresponding values of m
ao0
For proofs of stronger analogs of Liouville’s theorem for entire harmonic func-
tions, see [7, chap. 2].
Note that the argument given here could also be used to prove the classical LiouWe believe this apz
ville theorem for entire analytic functions f(z)
=0 Ck
to its wider
owing
presentations,
most
found
in
that
over
proach is to be preferred
applicability. In particular, it can be easily used to obtain analogs of Liouville’s
theorem for solutions of many partial differential equations. Nevertheless, to the
best of our knowledge this approach to Liouville’s theorem has not yet found a place
in the literature.
Finally, we note that all the results of this paper hold as stated for complex
k m(x/r
n
r
that is for H(x)
valued harmonic functions in R
m= akm
are complex numbers. For this, only minor changes are
where the coefficients
’k--O [dk
Yk
akm
required in the corresponding development.
6.
RELATED RESULTS
Work on the growth of harmonic functions in dimensions greater than 2 has a
history going back at least fifty years. In [8] and [9] the growth of harmonic func3
is considered, and the Bergman B 3 integral operator is used as a
tions in R
principal tool in the investigation. Paper [I0] contains results similar to those
3
The growth of harmonic functions
presented here, but for harmonic functions in R
n
at the origin in [II, 12
functions
the
of
to
derivatives
is related
n > 3
in R
equations is
differential
13]. Growth of entire solutions of certain elliptic partial
considered in [14,
15], and special cases of
these equations yield axisymmetric har-
n
monic functions in R
3
In [16] the growth of entire harmonic functions in R
is expressed in terms of
n
in terms of the maximum value attained by the
the function’s coefficients, and in R
h
on spheres centered at the origin.
harmonic
degree spherical
function’s kt
ACKNOWLEDGEMENT.
This paper is dedicated to Professor Morris Marden.
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