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Similar Transposed Sets of PolynomialsM.M. Makky a

aMathematics Depart ment , Facult y of Science (Qena), South Vall ey

Universit y, Egypt

Available online: 15 Sep 2010

To cite this article: M.M. Makky (2002): Similar Transposed Set s of Polynomials, Complex Vari ables,Theory and Application: An International Journal: An International Journal, 47:11, 1003-1014

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Complex Variables , 2002, Vol. 47, No. 11, pp. 1003–1014

SimilarTransposed Sets of PolynomialsM.M. MAKKY *

Mathematics Department,Faculty of Science (Qena), SouthValley University, Egypt

Communicatedby H. Begehr

(Received 4 August1999; Revised 22 December 2000; In final form18 March 2002)

This paper studies the effectiveness of another kind of transposed sets of polynomials of one complex variablein closed regions, open discs, at the origin and for all entire functions. In addition, an upper bound for theorder of these sets in this case is obtained.

Keywords: Similar; Transposed; Effectiveness; Polynomials; Cannon sum; Cannon function

AMS Subject Classifications: 30A99

1. INTRODUCTION

A set f pnðzÞgof polynomials of the complex variable z is said to form a basic set, if everypolynomial zn , admits a unique finite linear representation in the form:

zn ¼ Xk

P n, kP kðzÞ ð1:1Þ

P nðzÞ ¼Xk

P n, kzk , ð1:2Þ

the matrix p ¼ ð pn, kÞ and p ¼ ð pn, kÞ are called the matrix of coefficients and the matrixof operators of the basic set f pnðzÞg, respectively each of which is row finite. Thus, thenecessary and sufficient condition for the set f pnðzÞgto be basic such that

pp ¼ pp ¼ I

where I is the infinite unit matrix.

*E-mail: [email protected]

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The basic set f pnðzÞgwill be called simple set if the polynomials pnðzÞ are of ordern, i.e.,

P nðzÞ ¼Xn

k¼0P n, kzk ð1:3Þ

and it is a monic set if pn, n ¼ 1 for all n.The basic set f pnðzÞg will be called Cannon set if the number N n of non-zero

coefficients in (1.2) is such that N 1=nn ! 1 and n ! 1 , otherwise it is called General set .

The basic set f pnðzÞgis said to represent the function f ðzÞin a closed circle, if the basicseries

f ðzÞ ¼

Xk

ck P ðzÞ,

where

ck ¼ XnP k, nan ¼ Xn

P k, n1n!

@n f ð0Þ@zn ,

converge uniformly to f ðzÞ in the closed circle and the set called effective in theopen circle, if it represents every function which is regular in the closed circle.

To justify effectiveness of sets of polynomials certain notations are used namelythe Cannon sum nð1=rÞ and the Cannon function ð1=rÞ, (c.f. [4, p. 449])

n1r ¼ Xk

jP n, k jAk1r ; r > 0

where

Ak1r ¼ max

jzj¼1=rj pkðzÞj

and

1r ¼ lim

n!1n

1r

1=n:

The order of the basic set f pnðzÞgof polynomials is given by

¼ limn!1

limn!1

log nðrÞn log n

:

Suppose that f pði Þn ðzÞg; i ¼ 1,2 are the basic sets of polynomials of the single variable z,

where (c.f. [4])

P ði Þn ðzÞ ¼Xk

pði Þn, kzk ; i ¼ 1, 2 :

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Let f ~ p pði Þn ðzÞg be the transposed sets of polynomials of the sets f pði Þ

n ðzÞg, i ¼ 1, 2, where(c.f. [4, p. 455])

~ PP

ði Þn ðzÞ ¼Xk

Pði Þk, nz

k;

and let f ^ p pði Þn ðzÞgbe the inverse transposed sets of polynomials of the sets f pði Þ

n ðzÞg, i ¼ 1,2,where

P̂P ði Þn ðzÞ ¼Xk

Pði Þk, nzk :

The similar set of polynomials of a single complex variable is defined in [6] when theconstituent sets are basic sets.

Here we define another kind of similar sets of polynomials of a single complexvariable, whenever the constituent sets are transposed basic sets as follows:

Definition Let f pði Þn ðzÞg; i ¼ 1, 2 be two transposed basic sets of polynomials of the single

complex variable z. Suppose that the set funðzÞgof polynomials is given by (c.f. [1,7])

funðzÞ g ¼ f~ p pð1Þn ðzÞgf~ p pð2Þ

n ðzÞgf̂ p pði Þn ðzÞg ð1:4Þ

where

funðzÞg ¼Xkun, kzk

and

un, k ¼ Xi , j ~ p pð1Þn,i

~ p pð2Þi , j

^ p pð1Þ j , k : ð1:5Þ

The set funðzÞgis called the similar transposed set of polynomials of a single complexvariable.

To construct the basic property for the similar set, we suppose that f ~ p pði Þn ðzÞg; i ¼ 1,2,are basic. In fact, suppose that ~ p pði Þ; i ¼ 1, 2 are the matrices of coefficients of the sets

f ~ p pði Þn ðzÞg, ~ p pði Þ ¼ ð~ p pði Þ

n, kÞ, and U ¼ ~ p pð1Þ~ p pð2Þ^ p pð1Þ is the matrix of coefficients of the similar trans-posed set funðzÞg. Thus if the matrix of coefficients of the similar transposed set funðzÞgand the matrix U ¼ ~ p pð1Þ^ p pð2Þ^ p pð1Þ then

UU ¼ ~ p pð1Þ~ p pð2Þ^ p pð1Þ ~ p pð1Þ^ p pð2Þ^ p pð1Þ ¼ I

and

UU ¼ ~ p pð1Þ

^ p pð2Þ

^ p pð1Þ

~ p pð1Þ

~ p pð2Þ

^ p pð1Þ

¼ I

where I is the unit infinite matrix.

SIMILAR TRANSPOSED SETS OF POLYNOMIALS 1005

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Hence the matrix U of coefficients of the set funðzÞg has a unique inverse U .Therefore the set funðzÞgis basic.

2. EFFECTIVENESS OF SIMILAR TRANSPOSED SET IN CLOSED REGIONS

The effectiveness of the similar transposed set funðzÞgof polynomials in closed regionswill be obtained whenever the general sets f pði Þ

n ðzÞg, i ¼ 1, 2, are effective in jzj < Rand satisfy the following conditions (c.f. [4, Theorem 19.2, p. 460])

i ðrÞ< R r < R

i ðRÀÞ R; i ¼ 1, 2ð2:1Þ

where

i ðrÞ ¼ limn!1

sup fAði Þn ðrÞg1=n

i ðrÀÞ ¼ limn!1

inf fAði Þn ðrÞg1=n

and

Aði Þ

n ðrÞ ¼ maxjzj¼r j pði Þ

n ðzÞ j, i ¼ 1, 2 :

On this concern, we have the following:

THEOREM 2.1 Suppose that each of the two general sets f pði Þn ðzÞg; i ¼ 1, 2 is effective in

jzj < R and satisfies conditions (2.1) . Then the similar transposed set funðzÞgis effectivein jzj 1=R.

Proof First, we suppose that f pði Þn ðzÞg; i ¼ 1, 2 are the inverse sets of the sets f pði Þ

n ðzÞg.Since each of the sets f pði Þ

n ðzÞgis effective in jzj < R and satisfies conditions (2.1), theneach of the inverse sets f pði Þ

n ðzÞgis effective in jzj < R and satisfy the conditions (c.f. [5])

i ðrÞ< R r < R

i ðRÀÞ R; i ¼ 1, 2 :ð2:2Þ

Also, construct numbers rm such that r < rm < R; m ¼ 1,2, . . . ,11 and

Að2Þn ðr7Þ< k1rn

8


10 ; n 0:( ð2:3Þ

Here the number k1 denotes positive finite numbers independent of the index n, whichdo not retain the same values at different occurrences.

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According to (2.3) and Cauchy’s inequality we see that

C n1

r11 ¼ maxjzj¼1=r11 junðzÞj Xi , j , kj ~ p p

ð1Þn, i jj ~ p p

ð2Þi , j jj ^ p p

ð1Þ j , k j

1r11

k

Xi , j , k

j pð1Þi , njj pð2Þ

j , i jj pð1Þk, j j

1r11

k

Xi , j , k

j pð1Þi , nj

j pð2Þ j , i j

r j 9

Að1Þk ðr9Þrk

11

< k1 Xi , j , k

j pð1Þi , nj

j pð2Þ j , i j

r j 9

r10

r11 k

< k1 Xi , j , k

j pð1Þi , n j

r i 7

Að2Þ j ðr7Þ

r j 9

r10

r11 k

< k1~ AAð1Þ

n1

r11 :

ð2:4Þ

Again, since the sets f pði Þn ðzÞg and f p

ði Þn ðzÞg; i ¼ 1,2 satisfy (2.1) and (2.2), respectively

we get


2


4:( ð2:5Þ

Since the set f pði Þn ðzÞgis effective in jzj < R, then according to [3,4], its inverse is also

effective in the same circles, from which we obtain

ð1Þ j ðr5Þ< k1r j 6; j 0: ð2:6Þ

From the relation (2.6) and using Cauchy’s inequality we have the Cannon sum of theset f ~ p pði Þ

n ðzÞgas follows:

~ ð1Þ j

1r7 ¼ Xk

j pð1Þk, nj ~ AAð1Þ

k1r7 1

rn5 Xk

j pð1Þk, n jrn

5 X j

j pð1Þ j , k j

1r7

j

¼ 1r5

n

Xk, j

j pð1Þ j , k j A

ð1Þk ðr5Þ 1

r7 j

¼1r5

n

X j

ð1Þ j ðr5Þ

r j 7

k11rn

5 X j

r6

r7 j

k11r5

n

ð2:7Þ

since

~ AAð1Þk 1

r7 ¼ maxjzj¼1=r11

j ~ p pð1Þk ðzÞj X j

j pð1Þ j , k j 1

r7 j


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Inserting (2.4), (2.5), (2.7) and using Cauchy’s inequality in the Cannon sum of thesimilar transposed set funðzÞgwe have

n 1r11 ¼ Xk

jun, k jC k 1r11 Xi , j , k


j , i jj pð1Þk, j jC k 1

r11 < k1 Xi , j , k



~ AA1k

1r7

k1 Xi , j j pð1Þi , njj pð2Þ

j , i j~ ð1Þ

j 1r7 k1 Xi , j j pð1Þ

i , njj pð2Þ j , i j

1

r j 5

k1

Xi , j

j pð1Þi , nj

Að2Þ j ðr3Þ

ri 3

1

r j 5

k11

rn1

Xi , j

Að1Þi ðr1Þ

ri 3

r4

r5

j

< k11rn

1 Xi

r2

r3 i

k11r1

n

:

Since r1 and r11 are arbitrarily chosen near to r and R; respectively we see that

n1

r11 k11r1

n

and

1R 1

r8r < R,

i.e., the similar transposed set funðzÞgis effective in jzj 1=R; as required.

3. EFFECTIVENESS OF SIMILAR TRANSPOSED SET IN OPEN DISCS

Now, we study the effectiveness of the similar transposed set funðzÞgof polynomials inopen discs, whenever the general sets f pði Þ

n ðzÞg; i ¼ 1, 2 are effective in jzj < r and satisfythe following conditions (c.f. [4])

i ðrþÞ < ri ðRÞ> r; 8R > r; i ¼ 1, 2 :

ð3:1Þ

For this purpose, we have the following:

THEOREM 3.1 Let f pði Þn ðzÞg; i ¼ 1, 2 be general sets of polynomials which are effectivein jzj < r and satisfy conditions (3.1). Then the similar transposed set funðzÞgis effectivein jzj < 1=r :

1008 M.M. MAKKY

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Proof Suppose that f pði Þn ðzÞg; i ¼ 1, 2 are the inverse sets of the sets f pði Þ

n ðzÞg; satisfyingthe following conditions:

i ðrþÞ r

i ðRÞ r; 8R > r; i ¼ 1, 2 : ð3:2Þ

Since the sets f pði Þn ðzÞgare effective in jzj r and satisfy (3.1), then we have


7


9


11 n 0:

8><>:ð3:3Þ

Inserting (3.3) and using Cauchy’s inequality in the maximum modulus C nð1=r12Þ of

the similar transposed set funðzÞg, we get

C n1

r12 ¼ maxjzj¼1=r12

junðzÞj Xi , j , k

j ~ p pð1Þn, i jj ~ p pð2Þ

i , j jj ^ p pð1Þ j , k j

1r12

k

Xi , j , k


j , i j j pð1Þk, j j

1r12

k

Xi , j , k

j pð1Þi , nj

j pð2Þ j , i j

r j 10

Að1Þk ðr10Þrk

12

k1 Xi , j , k

j pð1Þi , nj

j pð2Þ j , i j

r j 10

r11

r12 k

k1 Xi , j , k

j pð1Þi , nj

r j 8

Að2Þ j ðr8Þ

r j 10

r11

r12 k

< k11r7

n

: ð3:4Þ

Again, since the sets f pði Þn ðzÞgand f pði Þ

n ðzÞg; i ¼ 1, 2 satisfy (3.1) and (3.2), respectively wehave

Að1Þn ðrÞ< k1rn

1


5


3; n 0:

8>><>>:ð3:5Þ

A combination between (3.4), (3.5) and Cauchy’s inequality in the Cannon sum of the

similar transposed set funðzÞg, we get

n1

r12 ¼ Xk

jun, k jC k1

r12 Xi , j , k


j , i jj pð1Þk, j jC k

1r12 < k1 Xi , j , k



1r6

k

k1 Xi , j , k


j , i jA

ð1Þk ðr4Þ

r j 4

1r6

k

< k1 Xi , j j pð1Þi , njj pð2Þ

j , i j1

r j 4

< k1 Xi , j j pð1Þi , nj A

ð2Þ j ðr2Þ

r i 2

1r j

4

< k11rn Xi

r1

r2 i < k1

1r

n

:


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Since r12 are arbitrarily chosen near to R we see that

n1

r12

< k1

1r

n

and

1R <

1r

8r < R,

i.e., the similar transposed set funðzÞgis effective in jzj < 1=r as required.

4. EFFECTIVENESS OF SIMILAR TRANSPOSED SET FOR ALL ENTIRE FUNCTIONS

Now, we consider the following normalizing condition:

i ð0þÞ ¼ 0 ð4:1Þ

where

i ðrÞ ¼ limn!1

sup fAði Þn ðrÞg1=n

and

Aði Þn ðrÞ ¼ max

jzj¼rj pði Þ

n ðzÞj, i ¼ 1, 2 : ð4:2Þ

Suppose that f pði Þn ðzÞg, i ¼ 1, 2 are algebraic sets. We recall the algebraic property of these sets as given in [2], given by the relation

pð1Þn, k ¼ n, k

ð1Þ0 þ X

s1

t¼1

ð1Þt pð1Þ

n, k

pð2Þn, k ¼ n, k

ð2Þ0 þ X

s2

t¼1

ð2Þt pð2Þ

n, k ,

8>>>><>>>>:ð4:3Þ

where s1 , s2 are any positive intergers, ð ð1Þt Þs1

0 , ð ð2Þt Þs2

0 are constants and ðtÞf pði Þn ðzÞg, t 1

are the power of the sets f pði Þn ðzÞg.

First, we apply the condition (4.1). We see that


7

Að2Þn ðr8Þ< k1r9; nn 0:( ð4:4Þ

It follows from Lemma 2 of [4] that

ði ÞAði Þn ðrÞ< k1rn

11 ; n 0: ð4:5Þ

Inserting (4.5) in (4.3) and using Cauchy’s inequality, we have

j pð1Þn, k j < k1 1ðs1 þ 1Þ

rn11

rk10

; n ¼ k, 1 ¼ sup0 t s1

j ð1Þt j: ð4:6Þ

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By (4.4), (4.6) and using Cauchy’s inequality we obtain

C n1

r12

¼ max

jzj¼1=r12junðzÞj

Xi , j , k



1r12

k

< k1 1ðs1 þ 1ÞXi , j , k


j , i jrk

11

r j 10

1r12

k

< k1 1ðs1 þ 1ÞXi , j , k

j pð1Þi , nj

r i 8

Að2Þ j ðr8Þ

r j 10

r11

r12 k

< k1 1ðs1 þ 1Þ1r6

n

Xi , j , k

Að1Þi ðr6Þr i

8

r9

r10 j r11

r12 k

< k1 1ðs1 þ 1Þ1r6

n

Xi , j , k

r7

r8 i r9

r10 j r11

r12 k

< k1 1ðs1 þ 1Þ1r6

n

: ð4:7Þ

Again, from (4.1) and (4.3) we get

j pð1Þn, k j < k1 1ðs1 þ 1Þðrn

5=rk4Þ

j pð2Þn, k j < k1 2ðs2 þ 1Þðrn

3=rk2Þ, n ¼ k

Að1Þn ðrÞ< k1rn

1 , n 0

8>><>>:ð4:8Þ

where

2 ¼ sup0 t s2 jð2Þ

t j:

Inserting (4.7), (4.8) and Cauchy’s inequality in the Cannon sum nð1=r12Þof the similartransposed set funðzÞg, we get

n1

r12 ¼ Xk

jun, k jC k1

r12 Xi , j , k

j ~ p pð1Þn, i jj ^ p pð2Þ

i , j jj ^ p pð1Þ j , k jC k

1r12

< k1 1ðs1 þ 1ÞXi , j , k

j pð1Þi , n jj pð2Þ


1r6

k

< k1 21ðs1 þ 1Þ2

Xi , j , kj pð1Þi , njj pð2Þ j , i j r

k

5r j 4

1r6 k

< k121 2ðs1 þ 1Þ2ðs2 þ 1ÞXi , j , k

j pð1Þi , nj

r i 2

r3

r4 j r5

r6 k

< k121 2ðs1 þ 1Þ2ðs2 þ 1Þ

1r

n

Xi , j , k

Að1Þi ðrÞr i

2

r3

r4 j r5

r6 k

< k121 2ðs1 þ 1Þ2ðs2 þ 1Þ

1r

n

Xi , j , k

r1

r2 i r3

r4 j r5

r6 k

< k121 2ðs1 þ 1Þ2ðs2 þ 1Þ

1r

n


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where

Aði Þn ðrÞ ¼ max

jzj¼1r

j pði Þn ðzÞj:

Using Cauchy’s inequality with relation (5.4) we get

C n1r ¼ max

jzj¼1=rjunðzÞj Xi , j , k



1r

k

Xi , j , k



1r

k

Xi , j , k


j , i jð1Þk ðrÞr j

1r

k

k11r

n

Xi , j , k

jAð1Þ

i ðrÞj

r i Að2Þ j ðrÞ

ð1Þ

kðrÞ

r j

1r

k

k11r

n

nnð2 1þ 2Þþ 3: ð5:5Þ

Inserting (5.4), (5.5) and (5.6) in the Cannon sum nð1=rÞ of the similar transposedset funðzÞgwe have

n1r

¼

Xk

jun, k jC k1r

Xi , j , k



1r

Xi , j , k



1r

Xi , j , k

j pð1Þi , nj

ð2Þ j ðrÞr i

ð1Þk ðrÞr j C k

1r

< k1nnð4 1þ 2 2Þþ 6

:

Since i , are chosen arbitrarily near to i (i ¼ 1, 2); respectively then we have

¼ limr!1

limn!1

log nðrÞn log n

4 1 þ 2 2

as required.

References

[1] R.H. Makar and S. Awad (5 April, 1984). On associated basic sets of polynomials. Nineth Inter. Congress ,Statistics , Computer , pp. 59–83. Ain-Shams University.

[2] R.H. Makar and B.H. Makar (1951). On algebraic simple monic sets of polynomials. Proceeding of the American Mathematical Society , 2 (4), 526–537.

[3] W.F. Nassif and K.A.M. Sayyed (1979). Effectiveness of composite sets at the origin. The 14th Annual Conference on Statistics, Computer Science Operations Research and Mathematics , pp. 147–154.Cairo University, Egypt.


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[4] W.F. Newens (1953). On the representation of analytic functions by infinite series. Philosophical Transactions of the Royal Society of London , Ser. A. , 24 – 51 , 429–468.

[5] K.A.M. Sayyed and M.M. Makky (1994). Effectiveness of similar sets of polynomials. Bull. Fac. Sci. Qena(Egypt ), 2 (3), 141–156.

[6] K.A.S. Sayyed and S.M. Mena (1988). Similar sets of polynomials. Bull. Fac. Sci ., 17 (I-C), 29–38.[7] K.A.S. Sayyed and S.M. Mena (1988). Effectiveness of similar sets of polynomials at the origin. Bull. Fac.

Sci. , 17 (I-C), 39–48.

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