A. El-Sayed Ahmed∗
Extended results of the Hadamard product of simple sets of polynomials in hypersphere
Abstract. In this paper we study some extended results of the Hadamard product set for several simple monic sets of polynomials of several complex variables in hy- perspherical regions, then we obtain the effectiveness conditions for this Hadamard product in hyperspherical regions.
2000 Mathematics Subject Classification: 32A17, 32A99.
Key words and phrases: Basic sets of polynomials, Hadamard product, several com- plex variables.
1. Introduction. The idea of studying effectiveness properties of the Hadamard product set of sets of polynomials in a single complex variable was introduced by Melek and El-Said in [7]. In [12] Nassif and Rizk introduced an extension of this product in the case of two complex variables using spherical regions. It should be mentioned here the recent study of this Hadamard product set in Clifford Analysis (see [1]). In the present paper, we aim to investigate the extent of a generalization of this Hadamard product set in Cn using hyperspherical regions.
We will let C represent the field of complex variables. In the space C2 of the two complex variables z and w, the successive monomial 1, z, w, z2, zw, w2, ...
are arranged so that the enumeration number of the monomial zjwk in the above sequence is 12(j + k)(j + k) + k, (j, k ≥ 0). The enumeration number of the last monomial of a polynomial P (z, w) in two complex variables is called the degree of the polynomial. A sequence {Pi(z; w)}∞0 of polynomials in two complex variables in which the order of each polynomial is equal to its degree is called a simple set ( see [3], [5] and [13]). Such a set is conveniently denoted {Pi(z; w)}, where the last
∗The author thanks the referees for their useful suggestions.
monomial in Pm,n(z, w) is zmwn.
If further, the coefficient of this last monomial is 1, the simple set is termed monic.
Thus, in the simple monic set {Pm,n(z; w)} the polynomial Pm,n(z, w) is represented as follows.
Pm,n(z, w) =
m+n
X
k=0 k
X
j=0
Pk−j,jm,n zk−jwj (Pm,nm,n= 1; Pm+n−j,jm,n = 0, j > n).
Let z = (z1, z2, ..., zn) be an element of Cn; the space of several complex variables.
The following definition was introduced in [9] and [10].
Definition 1.1 A set of polynomials {Pm[z]} =P0, P1, P2, ..., Pn, ... is said to be basic when every polynomial in the complex variables zs; s ∈ I = {1, 2, 3, ..., n can be uniquely expressed as a finite linear combination of the elements of the basic set {Pm[z]}.
Thus according to [10], the set {Pm[z]} will be basic if and only if there exists a unique row-finite matrix P such that P P = P P = I, where P = Pm,h
is the matrix of coefficients, P is the matrix of operators of the set {Pm[z]} and I is the infinite unit matrix.
Similar definition for a simple monic set can be extended to the case of sev- eral complex variables by replacing m, n by (m) = (m1, m2, m3, ..., mn) , j, k by (h) = (h1, h2, h3, ..., hn) and z, w by z, where each of (m) and (h) be multi-indices of non-negative integers.
The fact that the simple monic set {Pm[z]} of several complex variables is necessar- ily basic follows from the observation that the matrix Pm,h of coefficients of the polynomials of the set is a lower triangular matrix with non-zero diagonal elements.
( These elements are each equal to 1 for monic sets).
Definition 1.2 ([6]) The basic set {Pm[z]} is said to be algebraic of degree ` when its matrix of coefficients P satisfies the usual identity
α0P`+ α1P`−1+ ... + α`I = 0.
Thus, we have a relation of the form
Pm,h = δm,hγ0+
`−1
X
s1=1
γs1Pm,h(s1),
where Pm,h(s1)are the elements of the power matrix Ps1and γs1, s1= 0, 1, 2, ..., ` − 1 are constant numbers.
In the space Cn, an open hypersphere is defined by:
Sr=
z ∈ Cn:
n X
s=1
|zs|2
12
< r
,
the closed hypersphere is defined by:
Sr=
z ∈ Cn:
n X
s=1
|zs|2
12
≤ r
.
Thus the function f (z) of the complex variables zs; s ∈ I which is regular in Srcan be represented by the power series
f (z) =
∞
X
(m)=0
amzm=
∞
X
(m1,m2,...,mn)=0
am1,m2,...,mnzm11zm22...znmn
For the function f (z), we have from [2] that Mf ; r = sup
Sr
f (z) ,
then it follows that
lim
hmi→∞sup |am| σm
hmi1
= 1 r, where hmi = m1+ m2+ ... + mn and
σm= inf
|t|=1
1
tm = hmi hmi2 Qn
s=1msms2
(see [2] and [11]),
1 ≤ σm≤ (√
n)hmion the assumption that m
ms
s2 = 1, whenever ms= 0; s ∈ I.
For the basic set {Pm[z]} and its inversePm[z] , we have Pm[z] =X
h
Pm,hzh,
Pm[z] =X
h
Pm,hzh,
zm=X
h
Pm,hPh[z] =X
h
Pm,hPh[z].
For any function f (z) of several complex variables, there is formally an associated basic seriesP∞
(h)=0Ph[z]. When this associated series converges uniformally to f (z) in some domain it is said to represent f (z) in that domain; in other words, as in the classical terminology of Whittaker for a single complex variable (see [14]), the basic set Pm[z] will be effective in that domain. For more information about basic sets of polynomials we refer to [1], [2],[4] , [5] , [13] and [14].
The convergence properties of basic sets of polynomials are classified according to the classes of functions represented by their associated basic series and also to the domain in which are represented.
Let,
G Pm; r=X
h
Pm;h
M [Ph, r,
then the Cannon sum of the set {Pm[z]} for Sr will be Ω Pm; r = σmG Pm; r and the Cannon function for the same set is
Ω P ; r = lim
hmi→∞Ω Pm; r hmi1 .
Concerning the effectiveness of the basic set of polynomials of several complex vari- ables in hypersphere, we have from [9, 10] , the following results.
Theorem 1.3 The necessary and sufficient condition for the basic set {Pm[z]} of polynomials of several complex variables to be effective in the closed hypersphere Sr is that
(1) Ω P ; r = r.
If r → 0+ , then we will obtain the effectiveness at the origin as in the following corollary:
Corollary 1.4 The necessary and sufficient condition for the basic set {Pm[z]} of polynomials of several complex variables to be effective at the origin is that
Ω P ; 0+ = lim
r→0+Ω P ; r = 0.
In [7], the Hadamard product of simple monic sets of polynomials of a single complex variable was introduced and its effectiveness properties were studied. Also, in [12]
Nassif and Rizk extended this study in the case of two complex variables and they introduced the following definition:
Definition 1.5 Let {Pm,n(z, w)} and {qm,n(z, w)} be two simple monic sets of polynomials , where
Pm,n(z, w) =
(m,n)
X
(i,j)=0
Pi,jm,nziwj;
qm,n(z, w) =
(m,n)
X
(i,j)=0
qi,jm,nziwj.
Then the Hadamard product of the sets {Pm,n(z, w)} and {qm,n(z, w)} is the simple monic set {Um,n(z, w)} given by
Um,n(z, w) =
(m,n)
X
(i,j)=0
Ui,jm,nziwj,
where
Ui,jm,n=σm,n
σi,j
Pi,jm,nqm,ni,j ; ((i, j) ≤ (m, n)).
In this paper we will give an inevitable modification in the definition of Hadamard product of basic sets of polynomials of two complex variables as to yield favorable results in the case of several complex variables in hyperspherical regions in Cn, by using k basic sets of polynomials instead of two sets.
Now, we will extend the above product by using k basic sets of polynomials of several complex variables, so we will denote these polynomials by {P1,m[z]}, {P2,m[z]}, ..., {Pk,m[z]} and in general write {Ps2,m[z]}; s2= 1, 2, 3, ..., k.
Definition 1.6 Let {Ps2,m[z]}; s2 = 1, 2, 3, ..., k be simple monic sets of polyno- mials of several complex variables, where
(2) Ps2,m[z] =
(m)
X
(h)=0
Ps2,m,hzh.
Then the Hadamard product of the sets {Ps2,m[z]} is the simple monic set {Hm[z]}
given by
(3) Hm[z] =
(m)
X
(h)=0
Hm,hzh,
where
(4) Hm,h = σm
σh
k−1 k Y
s2=1
Ps2,m,h
.
If, we substitute by k = 2 and consider polynomials of two complex variables in- stead of several complex variables, then we will obtain Definition 1.5 . It should be remarked here that Definition 1.6 is different from that used in [8].
2. Main results. In this section, we will study the effectiveness of the extended Hadamard product of simple monic sets of polynomials of several complex variables defined by (3) and (4) in closed hyperspherical regions and at the origin.
Let {Ps2,m[z]} be simple monic sets of polynomials of several complex variables zs; s ∈ I, so that we can write
(5) Ps2,m[z] =
(m)
X
(h)=0
Ps2,m,hzh,
where
Psm1,m2,...,mn
2,m1,m2,...,mn= 1; s2= 1, 2, ..., k The normalizing functions of the sets {Ps2,m[z]} are defined by
(6) µ Ps2; r = lim
hmi→∞sup
σmMPs2,m; r
hmi1
(see [12]),
where MPs2,m; r are defined as follows:
MPs2,m; r = sup
Sr
Ps2,m[z]
We observe that, since the sets {Ps2,m[z]} are monic, then by applying Cauchy’s inequality in (2), we have
|Ps2,m,h| ≤ σm
rhmisup
Sr
Ps2,m[z]
which implies that,
MPs2,m; r ≥ rhmi σm . So that (6) gives
(7) µ Ps2; r ≥ r.
Next, we show if ρ are positive numbers greater than r, then
(8) µ Ps2; ρ ≤ ρ
rµ Ps2; r, ρ > r.
In fact, this relation follows by applying (6) to the inequality
MPs2,m; ρ ≤ K ρ r
hmi
MPs2,m; r
which in its turn, is derivable from (6), Cauchy’s inequality and the supremum of zm, where K = O(hmi + 1).
Now, let {Ps2,m[z]}; s2 = 1, 2, 3, ..., k be simple monic sets of polynomials of several complex variables, and that {Hm∗[z]} is the set defined as follows
(9) Hm∗[z] =
k
Y
s2=1
Ps2,m[z].
The following fundamental lemma is proved.
Lemma 2.1 If, for any r > 0
(10) µ Ps2; r = r.
Then
(11) µ H∗; r = r.
Proof We first observe that, if ρ be any finite number greater than r , then by (6), (7) and (9), we obtain that
(12) µ Ps2; ρ = ρ,
Now, given r∗> r, we choose finite number r0 such that
(13) r < r0 < r∗.
Then by (6) and (10), we obtain that (14) M Ps2,h; r < η
σh(r0)hhi where η > 1, where hhi = h1+ h2+ h3+ . . . hn. Also from (9), we can write
Hm∗[z] =
(m)
X
(h)=0 k
Y
s2=1
Ps2,m,hPs2,h[z].
Hence (13) and (14) lead to
MHm∗; r ≤ ηK
1 − r0
r∗
n−n
MPs2,m; r∗, Making hmi → ∞ and applying (11), we get
µ H∗; r = lim
hmi→∞sup
σmMHm∗; r
hmi1
≤ µ Ps2; r∗ = r∗,
which leads to the equality (10), by the choice of r∗ near to r, and our lemma is therefore proved.
From Lemma 2.1 if we consider the simple monic sets {Ps2,m[z]} accord to con- dition (10), then it is not hard to prove by induction for the j-power sets {Ps(j)2,m[z]}
that
(15) µ Ps(j)2 ; r = r.
Before getting the results for the effectiveness in the closed hypersphere Sr and at the origin, we need the following technical lemmas.
Lemma 2.2 Let {Ps2,m[z]}; s2= 1, 2, 3, ..., k be simple monic algebraic sets of poly- nomials of several complex variables, which accord to condition (10). Then the set will be effective in the closed hypersphere Sr.
Proof Suppose that the monomial zm admit the representation zm=X
h
Pm,hPh[z].
Since, the set {P1,m[z]} is algebraic, then there exists a relation of the form
(16) P1,m,h=
k
X
j=1
αjP1,m,h(j) ; (h) ≤ (m),
where k is a finite positive integer which together with the coefficients (αj)kj=1, is independent of the indices (m), (h). The coefficients P1,m,h(j) are defined by
P1,m(j)[z] =
(m)
X
(h)=1
P1,m,h(j) zh; 1 ≤ j ≤ k.
So that, (17)
P1,m,h(j)
rhmi≤ σhMP1,m(j); r.
According to (15) for given r∗> r and from the definition corresponding to µ P1(j); r, we may deduce that
(18) MP1,h(j); r < K
σh(r∗)hhi. Applying (17) and (18) in (16), it follows that
(19)
P(j)1,m,h
< ζKβσh
σm
(r∗)hmi (r)hmi , where
(20) β = max|αj|; 0 ≤ j ≤ k
and ζ be a constant.
In view of the representation
zm=X
h
Pm,hPh[z],
the Cannon sum of the set {P1,m(j)[z]} will be
(21) Ω P1,m(j); r = σm (m)
X
(h)=0
|P(j)m,h|MP1,h(j); r
where,
(22) MP1,h(j); r = sup
Sr
|P1,m(j)[z]|.
Therefore (18), (19) and (21) (for r∗> r) give
(23) Ω P1,m(j); r < ζKβ(r∗)hmi,
Hence the Cannon function of the set {P1,m(j)[z]} turns out to be
Ω P1(j); r = lim
hmi→∞
Ω P1,m(j); r
hmi1
= r∗
which, by the choice of r∗, implies that
Ω P1(j); r = r.
As very similar, we can obtain that the sets {Pν,m(j)[z]}; ν = 2, 3, 4, ..., k will be effective in the hypersphere Sr. Our lemma is therefore proved.
As a consequence of Lemma 2.1 , we obtain the following lemma.
Lemma 2.3 Let {Ps2,m[z]}; s2= 1, 2, 3, ..., k be simple monic sets of polynomials of several complex variables, which accord to the following condition:
(24) µ Ps2; 0+ = 0 , s2= 1, 2, 3, ..., k.
Then, for the set {Hm∗[z]} of (9), we have
(25) µ H∗; 0+ = 0,
for
(26) µ Ps2; 0+ = lim
r→0+µ Ps2; r.
Proof The proof of Lemma 2.3 is much akin to that of lemma 3 in [12].
We can again, apply Lemma 2.3, to prove by induction that, if the simple monic sets {Ps2,m[z]}; s2 = 1, 2, 3, ..., k accord to condition (24), then for the j-power set {Ps(j)2,m[z]}, we can obtain that
(27) µ Ps(j)
2 ; 0+ = 0.
The next lemma is the analogue of Lemma 2.2.
Lemma 2.4 Let {Ps2,m[z]}; s2 = 1, 2, 3, ..., k are algebraic sets of polynomials of several complex variables and accord to the following condition
(28) µ Ps2; 0+ = 0.
Then the sets will be effective at the origin.
Proof The proof of Lemma 2.4 is similar to that of lemma 4 in [12], so it will be omitted.
Remark 2.5 It should be remarked here that special cases from Lemmas 2.1, 2.2, 2.3 and 2.4 were obtained in [12] in the case of two complex variables.
The following theorem will give us the effectiveness of the Hadamard product set in the closed hypersphere Sr.
Theorem 2.6 Let {Ps2,m[z]}; s2= 1, 2, 3, ..., k be simple monic sets of polynomials of several complex variables, which are effective in the hypersphere Sr for all r ≥ ρs2,> 0. Then the extended Hadamard product set {Hm[z]} defined as in (3), (4) and is algebraic, will be effective in the hypersphere Sr for all
r ≥
k
Y
s2=1
ρs2
and this result is best possible.
Proof Since the set {P1,m[z]} is effective in the hypersphere Sr for r ≥ ρ1, then for the Cannon function of this set, we have
Ω P1; r = r.
Hence, if ρ∗1 be any positive number greater than ρ1, it follows that (29) Ω P1; ρ1 = ρ1< ρ∗1.
Therefore, for the Cannon sum of the set {P1,m[z]}, we get (30) Ω P1,m; ρ ≤ ζK(ρ∗1)hmi.
Since the simple set {P1,m[z]} is monic, then in view of (21), we infer that
(31)
P1,m,h
< σh
(ρ1)hmiM P1,m,h; ρ < Kσh
σm
(ρ∗1)hmi (ρ1)hhi. Similarly, we can obtain that
(32)
Pν,m,h
< Kσh
σm
(ρ∗ν)hmi (ρν)hhi
ν = 2, 3, 4, ..., k. Introducing (31) and (32) in (4), we can obtain Hm,h
< Kkσh
σm k
Y
s2=1
(ρ∗s2)hmi (ρs2)hhi. It follows therefore, that
sup
Sρ
Hm[z]
= MHm; [ρ] =
(m)
X
(h)=0
Hm,h
MPh; ρ
< λ1Kkσh
σm
k Y
s2=1
(ρ∗s
2)hmi (ρs2)hhi
1 σh
k
Y
s2=1
ρhhis
2
,
where λ1 be a constant. Taking the limit as hmi → ∞, we obtain the normalizing function of the set {Hm[z]}, as follows
µ H; ρ ≤
k
Y
s2=1
ρ∗s
2.
Since ρ∗s2 be arbitrary chosen near to ρs2, we conclude that
µ H; ρ ≤
k
Y
s2=1
ρs2.
Since the set {Hm[z]} is algebraic, then according to Lemma 2.2 above, we conclude that the simple set {Hm[z]} is effective in the hypersphere Sρ and consequently, in the hypersphere Sr for all
k
Y
s2=1
ρs2 ≤ r
and the first assertion of Theorem 2.6 is proved.
To complete the proof of Theorem 2.6, we give the following example.
Example 2.7 Let the set
P0,0(z, w) = 1; Pm,n(z, w) = zmwn+ 1 σm,n
(ρ∗)m+n,
q0,0(z, w) = 1; qm,n(z, w) = zmwn+ 1 σm,n
(r)m+n, where, m, n are not both zero, be effective in Sr for all ρ∗≤ r.
Now, according to Definition 1.6 of the Hadamard product, we have H0,0(z, w) = 1; Hm,n(z, w) = zmwn+ 1
σm,n(ρ∗r)m+n
It is easy to see that this set is not effective in Sρ, where 0 < ρ < r ρ∗ as required.
Theorem 2.6 , is therefore established.
Our last result in this section concerns the effectiveness of the Hadamard product as in Definition 1.6 at the origin.
Corollary 2.8 Let {Ps2,m[z]}; s2 = 1, 2, 3, ..., k be simple monic sets of polyno- mials of several complex variables, which are effective at the origin for all entire functions. Then, the Hadamard product set {Hm[z]} defined as in (3), (4) and is algebraic will be effective at the origin.
Proof Since the set {P1,m[z]} is effective at the origin, then for the Cannon function of this set, we have
Ω P1; 0+ = 0.
Hence, there exist finite positive numbers R∗ and R, where R∗> R such that
(33) Ω P1; R = R < R∗.
Thus, we can proceed in a similar way that given in the proof of Theorem 2.6 to obtain that
Hm,h
< Kkσh
σm
(R∗s)hmi (R)hhi . It follows therefore, that
sup
SR
Hm[z]
= MHm; R < λ2
Kk σm
(R∗)hmi (R)hhi , where λ2 be a constant.
Taking the limit as hmi → ∞, we obtain the normalizing function of the set {Hm[z]}, as follows
µ H; R ≤ R∗. By the choice of R∗, we conclude that
µ H; 0+ = 0.
Since the set {Hm[z]} is algebraic, then according to Lemma 2.4 above, we conclude that the simple set {Hm[z]} is effective at the origin.
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A. El-Sayed Ahmed
Faculty of Science, Mathematics Department, Sohag University 82524 – Sohag, Egypt
E-mail: ahsayed80@hotmail.com
(Received: 25.01.2006)
