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For harmonic and separately harmonic functions, we give results similar to the Carlson-Boas theorem.. Introduction The well known classical theorem of Carlson see [2, p.153] states that

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Uniqueness Theorems for Harmonic and Separately Harmonic Entire Functions on CN

Bachir Djebbar

Department of Computer Sciences, University of Sciences and Technology “M B” of Oran,

B.P 1505, El M’naouer Oran 31000, Algeria

Received May 24, 2004

Abstract. For harmonic and separately harmonic functions, we give results similar to the Carlson-Boas theorem We give also harmonic analogous of the Polya and Guelfond theorems

1 Introduction

The well known classical theorem of Carlson (see [2, p.153]) states that an entire

holomorphic function of exponential type< π (i.e f satisfies an inequality of

the form|f(z)| ≤ A exp(τ |z|) with τ < π ) must vanish identically if it vanishes

onN.

In [3] Boas extended Carlson’s theorem to harmonic functions and proved the following theorem:

Theorem 1.1 (Boas theorem) Let h be an entire harmonic function on C of

exponential type < π.

If

h(z) = 0 for z = 0, ±1, ±2, , i, i ± 1, i ± 2, (1)

Then h ≡ 0.

Similarly, Ching in [5] showed that the same conclusion holds under the con-ditions

i) h is of exponential type < π.

ii) h(z) = 0 for z = 0, ±1, ±2, , ±i, ±2i,

iii) h(z) = −h(−z) for all complex z.

In [1] Armitage gives a similar result for harmonic entire function in RN.

Let us recall the classical:

Theorem 1.2 (Polya Theorem [2]) Let f be an entire function on C, of

exponential type < log 2 If {f(n), n ∈ N} ⊂ Z, then f is a polynomial.

Guelfond gives in [7] a similar result for an entire function that takes integers

values on a sequence (β n) under some growth condition near infinity

Theorem 1.3 (Guelfond Theorem [7]) Let g be an entire function on C, β an

integer greater than one If g(β n ) are integers for n = 1, 2, and g satisfies

the inequality:

log|g(z)|  log2|z|

4 log β −1

2log|z| − ω|z|, where ω :R+→ R satisfies lim

r−→∞ ω (r) = ∞ then g is a polynomial.

In this paper we give a result similar to Boas theorem but under different conditions Our proof is based on the properties of a polynomial basis estab-lished in [6] We extend this result to separately harmonic functions We give also a Guelfond and Polya type theorem in the case of harmonic function

2 Notations and Results

For all z = re iθ ∈ C and n ∈ N we put:

e1(z) ≡ 1, e n (z) =

r k

cos kθ, if n = 2k k ≥ 1,

r k sin kθ, if n = 2k + 1 k ≥ 1. (2)

The sequence (e j)j≥1 of harmonic polynomials with deg(e j ) = [j/2] ([ ] des-ignates the entire part ) is a basis for the space H(C) of all entire harmonic functions Moreover for all function h ∈ H(C), we have the following relation

between the growth of h and its coefficients in the basis (e j)

Theorem 2.1 [6] Let h be an entire harmonic function, and let h(z) =

∞



j=1 a j e j (z) be an expansion according to the basis (e j)j≥1 Then the growth order

ρ of h is given as follows

ρ = lim sup

j→∞

[j/2] log[j/2]

− log |a j | . (3) When ρ ∈]0, +∞[, the growth type τ of h is given by

τ = lim sup

j→∞

[j/2]

eρ (|a j |)

ρ

We will prove the following results

Theorem 2.2 Let h be an entire harmonic function on C of exponential type

< π If h(z) = 0 for z = 0, 1, 2, and h(z) = h(z) then h ≡ 0 on C.

Theorem 2.3 Let h be an entire separately harmonic function on C N of exponential type < π with respect to the norm |z| = sup

j |z j | (i.e : |h(z)| ≤

A exp(τ |z|) with τ < π).

For m ∈ {0, 1, , N} ⊂ N let:

E m=

(z1, , z N)∈ C N : z m+1=· · · = z N = 0

and L m=

(z1, z N)∈

E m : z

j ∈ N for j = 1, m.

If h ≡ 0 on L m and h

z 1, z j−1 , z j , z j+1, z N

= h(z1, , z j , z N ); j = 1 , m, then h ≡ 0 on E m .

Corollary 2.4 Let h be an entire separately harmonic function on C N of expo-nential type < π If h(z1, , z N ) = 0 for z j =0, 1 , and h

z1, z j−1 , z j , j+1 , ,

z N

= h

z1, z j−1 , z j , z j+1, , z N

; j = 1, , N then h ≡ 0 on C N

Corollary 2.4 is a direct consequence of Theorem 2.3

Theorem 2.5 [The harmonic analogous of Guelfond theorem] Let h be an

entire harmonic function onR2≈ C and q ∈ Z such that |q| > 1 Suppose that

i)



h(q n , 0) ∂h

∂y (q n , 0), n ∈ N ⊂ Z,

ii) There is a function ω :R∗

+−→ R+ such that: lim

r−→∞ r2ω(r) = 0 and M(h, r)  ω(r) √ r exp log

4 log|q|

, ∀r > 0,

where M (h, r) = sup

|z|=r |h(z)|.

Then h is a polynomial.

Theorem 2.6 (The harmonic analogous of Polya Theorem) Let h be an entire

harmonic function on R2 If h satisfies:

i)



h(n, 0), ∂h

∂y (n, 0), n ∈ N ⊂ Z,

ii) M (h, r)  A exp(Cr), C < log 2,

then h is a polynomial.

3 Proofs

Proof of Theorem 2.2 Let h be an entire harmonic function onC of exponential

type τ < π, and let h(z) = ∞

j=1

a j e j (z) be its expansion in (e j)j∈N One can write

h(z) = ∞

j=1

a 2j e 2j (z) +

∞

j=1

a 2j+1 e 2j+1 (z).

The condition h(z) = h(z), ∀z ∈ C implies that a 2j+1 = 0 ∀j ≥ 0 However,

for all m ∈ N, we get h(m) = ∞

j=1 a j e j (m) = ∞

j=1 a 2j m j = 0 Consider the

function f (z) = ∞

j=1 a 2j z j (z ∈ C) which is entire on C and of exponential type

β < τ < π We have: f(m) = ∞

j=1

a 2j m j = h(m) = 0 for all m ∈ N, and hence

f ≡ 0 by Carlson Theorem, so a 2j = 0 for j = 1, 2, , which finally implies

Proof of Theorem 2.3 We prove Theorem 2.3 by induction on m The case

m = 1 is an immediate consequence of Theorem 2.2 applied to the function v(z) = h(z, 0 , 0), z ∈ C Suppose the theorem is true for m such that 1

≤ m < N Assume that h satisfies the hypotheses of the theorem for m + 1.

Hence h is an entire separately harmonic function of exponential type σ < π

and satisfies the condition:

if h ≡ 0 on L m+1 and h(z

1, ., z j , , z N ) = h(z1, , z j , z N ), j = 1, ., m + 1, then h(z1, , z m , 0, 0) = 0, ∀(z1, , z m ∈ N),

since h ≡ 0 on L m then h ≡ 0 on E m So h(z1, , z m, 0, , 0) = 0, ∀(z1, ,

z m)∈ C m Let k ∈ N and consider the translation:

T k :CN → C N

(z1, , z N)→ (z1, , z m , z m+1 + k, z m+2 , , z N)

h ◦ T k (z 1, , z m , 0, , 0) = h(z 1, z2, z m , k, 0, 0) then h ◦ T k ≡ 0 on

L m h ◦ T k is a entire separately harmonic function of exponential type < π

which satisfies:

h ◦ T k (z1, , z j , , z N ) = h(z 1, , z j , z m , z m+1 + k, , z N)

= h(z1, , z j , , z m , z m+1 + k, z N ), j = 1, , m

then h ◦ T k ≡ 0 on E m,

i.e h ◦ T k (z1, ., z m , 0, ., 0)=h(z1, ., z m , k, 0, ., 0)=0, ∀z j ∈ C; j=1, ., m

and, k ∈ N For z 1, , z m fixed in C, we consider the function: g(z) =

h(z1, , z m , z, 0, , 0) z ∈ C g is an entire separately harmonic function

of exponential type≤ σ < π, and satisfies:

g(z) = h(z

1, z m , z, 0, , 0) = h(z1, , z m , z, 0, , 0) = g(z) ∀z ∈ C g(k) = h(z1, , z m , k, 0, , 0) = h ◦ T k (z1, z m , 0 0) = 0, ∀k ∈ N.

By Theorem 2.2 we deduce that g(z) = 0, ∀z ∈ C Since (z1, , z m) is arbitrarily fixed inCm then

h(z1, , z m, z, 0, , 0) = 0, ∀(z1, , z m)∈ C m and ∀z ∈ C.

Consequently h(z1, , z m , z m+1 , 0, , 0) = 0 ∀(z1, , z m+1)∈ C m+1 So

h ≡ 0 on E m+1 The induction is complete.  Proof of Theorem 2.5 Let h be an entire harmonic function and let f (z) =

∞



k=0 (a k + ib k ) z

k be its Taylor series expansion.

We consider the function F (z) = 1

2 f(z)+f(z)

Then F is an holomorphic entire function and F (z) =∞

k=0 a k z k , F (q n ) = Re f (q n ) = h(q n , 0) ∈ Z By

the following Carath´eodory’s inequality [2]

M(f, r) ≤ f(0)+ 2r

R − r M(Re f, R) − Re f(0)



, 0 < r < R,

we deduce that F satisfies conditions of the theorem of Gurelfond in the holo-morphic case, so F is a polynomial.

There is an integer N such that a k = 0, ∀k > N Consider now the

holomorphic entire function H defined by:

H(z) =1

2 if  (z) + if  (z)

=

∞

k=1

−2kb k z k−1

Then

H(q n) =1

2 if  (q n ) + if  (q n)

=−2 ∂h ∂y (q n , 0) ∈ Z, ∀ n ∈ N.

The classical result

⎧

⎨

⎩

if g is holomorphic in |z| < R + ε then we have :

|g 

(z) | ≤ R

(R − r)2M(g, R) for |z|  r < R,

gives

M(f  , r)  (r + 1)M(f, r + 1), ∀r > 0.

H satisfies the Gurelfond’s Theorem conditions in the holomorphic case, so H

is a polynomial; there exist N  such b k = 0, ∀k > N  Then f is a polynomial,

Proof of Theorem 2.6 Very similar to the proof of Theorem 2.5.

Remark. It would certainly be interesting to give Gelfond and Polya type theorems in the general case of harmonic entire functions inRN

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Hence h is an entire separately harmonic function of exponential type σ < π

and satisfies the condition:

if h ≡ on L m+1 and. .. interpolation formula for harmonic functions, J Approximation

Theory 15 (1975) 50–53.

6 B Djebbar,Approximation Polynomiale et Croissance des Fonctions N ... Armitage, Uniqueness theorems for harmonic functions which vanish at lattice points,J Approximation Theory 26 (1979) 259–268.

2 R Boas, Entire functions, Academic Press.,