10_782_kumar.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 6, 2010, 1062-1069

ISSN 1307-5543 – www.ejpam.com

SPECIAL ISSUE ON COMPLEX ANALYSIS: THEORY AND APPLICATIONS

DEDICATED TO PROFESSOR HARI M. SRIVASTAVA,

ON THE OCCASION OF HIS 70TH
BIRTHDAY

Growth and Chebyshev Approximation of Entire Function

Solutions of Helmholtz Equation in R2

Devendra Kumar

Department of Mathematics [Research and PostGraduate Studies], M.M.H.College,Model Town,Ghaziabad-

201001,U.P. India

Abstract. Some bounds on growth parameters of entire function solution of Helmholtz equation in R2

have been studied in terms of Chebyshev polynomial approximation error in sup norm. Our results

extend and improve the results studied by McCoy [9].

2000 Mathematics Subject Classifications: 30A35,41A10

Key Words and Phrases: Chebyshev approximation, growth parameters, Bergman integral operator,

Helmholtz equation, entire function

1. Introduction

For classifying the entire analytic functions by their growth in function theory ,the growth

parameters order and type may be computed from the Taylor’s coefficients or Chebyshev poly-

nomial approximations.McCoy[8,9] studied the growth of entire analytic function solutions

of Helmholtz equation in R2by using function theoretic methods (see R.P.Gilbert[2,3] and Mc-

Coy[8])and obtained some bounds on growth parameters in terms of Taylor‘s coefficients and

Chebyshev polynomials approximation errors.

The Helmholtz equation be given in the form

[∂r r +
1

r
∂r +

1

r2
∂θθ + F(r2)]φ(r,θ) = 0 (1)

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D. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 1062-1069 1063

where (r,θ) are polar coordinates in R2 and F(r2) 6= 0 is a real valued entire functions with

analytic continuation as an entire function of z ∈ C .

Each solution of (1) regular at the origin has a local representation via the Bergman

operator[1,7] of the first kind

φ(r,θ) = B( f (z)) =

∫ +1

−1

E(r2, t) f (z(1− t2/2)(1− t2)−1/2d t, (2)

where

E(r2, t) = 1+

∞
∑

n=1

t2nQ(2n)(r2)

is a real valued analytic function for t ∈ [−1,+1] that is entire for r ∈ [0,∞) and is known

as Bergman E f unction. In a neighborhood of the origin the solution of (1) has an expansion

φ(r,θ) = Σ∞n=0anφn(r,θ) (3)

where

φn(r,θ) = (
reiθ

2
)nGn(r)

and

Gn(r) =

∫ +1

−1

E(r2, t)(1− t2)(n−1/2)d t, n = 0,1,2,3, . . . .

The B associate of φ is given as

f (z) = Σ∞n=0anzn. (4)

It is known from Gilbert and Colton [4] that φ(r,θ) is an entire function if and only if, the

associate f (z) is an entire function i.e.,

lim sup
n→∞

|an|
1/n = 0. (5)

The sets of polynomial solutions of Helmholtz equation are defined as

Πn = {P : P(r,θ) = Σn
κ=0aκφκ(r,θ), aκreal}

The best Chebyshev approximation error in Bernstein’s sense be given as

En(φ) = inf‖ φ − P ‖ro
: P ∈ Πn, (6)

‖ φ − P ‖r0
= M(r0,φ − P), P ∈ Πn

where

r0 = r0(K) =min{1, sup{r : E(r2, t) > 0, t ∈ [−1,+1]}}> 0,



D. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 1062-1069 1064

and the maximum modulus

M(r0,φ − P) =max{|(φ − P)(z)| : |z| < r0}.

Mccoy[9] studied the fast growth of entire function solution φ(r,θ) in terms of order δ and

type τ using the concept of index k i.e., δ(k− 1) =∞ and δ(k) <∞. Due to lack of suitable

inverse operator he obtained bounds on the order and type.It has been noticed that his results

do not give any precise information about the growth of those functions for which

δ(k− 1) =∞ and δ(k) = 0.To overcome this problem, in this paper we pick up a concept of

(p,q)-order and (p,q)-type introduced by Juneja et al.[5,6]. Roughly speeking, this concept is

a modification of the classical definition of order and type, obtained by replacing logarithms

by iterated logarithms, where the degrees of iteration are determined by p and q, p ≥ q ≥ 0.

Our approach unifies the above approach studied by those of McCoy[9] and at the same time

it is applicable to every entire function, whether of slow or fast growth. Moreover, we make

an attempt to characterize (p,q) growth of φ(r,θ) and obtained some bounds on (p,q) order

and (p,q) type in terms of Chebyshev polynomial approximation errors defined by (6).

2. Notations

1. log[m] x = exp[−m] x = log(log[m−1] x) = exp(exp[−m−1] x), m = 0, ±1,±2, . . . provided

that 0< log[m−1] x <∞ with log[0] x = exp[0] x = x .

2.

Ω(L(p,q)) =















Ω(L(p,q)) = L(p,q), if p > 2;

1+ L(p,q), if p = q = 2;

max(1, L(p,q)), if 3≤ p = q <∞;

∞, if p = q =∞.

where 0≤ Ω(p,q)≤∞.

3. (p, q)-Growth of Solutions and Chebyshev Polynomial Approximation

In this section we shall prove our main results.

Theorem 1. Let φ(r,θ) be an entire function solution of the Helmholtz equation with expansion

φ(r,θ) =

∞
∑

n=0

anφn(r,θ).

Let φ and B associate f be entire functions of (p,q)-order δ(p,q,φ) and δ(p,q, f ) for a pair of

integers (p,q), p≥2, q ≥ 1. Then the following bounds are valid

(i) δ(p,q,φ) ≥ δ(p,q, f )

(ii) δ(p,q, E) ≤ δ(p,q, f )



D. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 1062-1069 1065

where

δ(p,q,φ) = Ω(L(p,q,φ)),δ(p,q, f ) = Ω(L(p,q, f )),δ(p,q, E) = Ω(L(p,q, E))

and

L(p,q,φ) = lim sup
n→∞

log[p−1] n

log[q][En(φ)/µn(Gn)]
−1/n

,

L(p,q, f ) = lim sup
n→∞

log[p−1] n

log[q] |an|−1/n
,

L(p,q, E) = lim sup
n→∞

log[p−1] n

log[q][En(φ)]
−1/n

,

µ(Gn) =

∫ r0

0

Gn(r
2)rn+1dr > 0, n= 0,1,2,3 . . . .

Proof.

(i) Using the orthogonality argument in equation (4). we get the identity

an(
r

2
)nGn(r

2) =
1

2π

∫ 2π

0

[φ(r,θ)− P(r,θ)]e−inφdφ

for P ∈ πn−1, n=1,2,3. . . . Integration above equation over the disk we obtain

anµn(Gn)/2
n =

1

2π

∫ r0

0

∫ 2π

0

[φ(r,θ)− P(r,θ)]e−inφ rd rdφ

or

|an|µn(Gn)/2
n ≤

1

4π
r2
0 En(φ) (7)

leads to

lim sup
n→∞

log[p−1] n

log[q] |an|−1/n
≤ lim sup

n→∞

log[p−1] n

log[q][En(φ)/µn(Gn)]
−1/n

.

Using the (p,q)-order coefficient formula for the associate [5, Thm.1,pp.61] we get

δ(p,q, f )≤ δ(p,q,φ).

(ii) Let us consider

|φ(r,θ)| ≤

∫ +1

−1

|E(r2, t)|| f (z
1− t2

2
)− p(z

1− t2

2
)|(1− t2)

−1

2 d t

≤ K(r0)‖ f − p‖2/r0



D. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 1062-1069 1066

where

K(r0) =max{K(r2) : 0≤ r ≤ r0},

K(r2) =max{E(r2, t) : t ∈ [−1,+1]},

f (z)− p(z) =

∞
∑

k=n

ak(r0/2)
kzk.

From this we get for P ∈
∏

n−1

En(φ) ≤‖ φ − P ‖r0
≤ K(r0) ‖ f − p ‖2/r0

we have

en( f ) = inf{‖ f − p ‖2/r0
: p ∈ Πn−1}

and

Πn = {p : p(z) =

n
∑

k=0

ak(r0/2)
kzk, akreal}.

By using to Reddy’s [10] extension of Bernstein theorem, for given ǫ > 0 there is an

N(ǫ)> 0 such that

en( f )≤ K(r0)[2
n|an|/(r0 + ǫ)

n]

for all n ≥ N(ǫ) or

En(φ)
−1/n ≥| an |

−1/n
(r0 + ǫ)

2
K(r0)

−1/n (8)

or

log[q] En(φ)
−1/n ≥ log[q] | an |

−1/n +0(1)

or

lim sup
n→∞

log[p−1] n

log[q] En(φ)
−1/n
≤ lim sup

n→∞

log[p−1] n

log[q] | an |−1/n
.

Again using the (p,q)order coefficient formula for the associate [5, Thm.1, pp 61] we

obtain

δ(p,q, E) ≤ δ(p,q, f ).

Hence the proof of (ii) is completed.

Theorem 2. Let φ(r,θ) be an entire function solution of the Helmholtz equation with expansion

φ(r,θ) and B- associate f(z) have the same index-pair (p,q). Then the (p,q) types satisfy

(i)

[
T (p,q,φ)

M(φ)
]

1

(δ(p,q,φ)−A) ≥ [
T (p,q, f )

M( f )
]

1

(δ(p,q, f )−A) lim inf
n→∞

(log[p−2] n)
1

δ(p,q,φ)−A
− 1

δ(p,q, f )−A .



D. Kumar / Eur. J. Pure Appl. Math, 3 (2010), 1062-1069 1067

(ii)

[
T (p,q, f )

M( f )
]

1

(δ(p,q, f )−A) ≥ [
T (p,q, E)

M(E)
]

1

(δ(p,q,E)−A) .β . lim inf
n→∞

(log[p−2] n)
1

δ(p,q, f )−A
− 1

δ(p,q.E)−A ,

where

T (p,q,φ) = M(φ)υ(p,q,φ),

T (p,q, E) = M(E)υ(p,q, E),

T (p,q, f ) = M(F)υ(p,q, f )

and

1

υ(p,q,φ)
= lim inf

n→∞

[log[q−1][En(φ)/µn(rn)]
−1/n](δ(p,q,φ)−A)

log[p−2] n
,

1

υ(p,q, E)
= lim inf

n→∞

[log[q−1] En(φ)
−1/n](δ(p,q,E)−A)

log[p−2] n
,

1

υ(p,q, f )
= lim inf

n→∞

[log[q−1] | an |
−1/n](δ(p,q, f )−A)

log[p−2] n
.

Here A= 1, if p = q and A= 0 if p > q and

M(φ) =







(δ(2,2,φ)−1)δ(2,2,φ)−1

(δ(2,2,φ))δ(2,2,φ) , if (p,q) = (2,2), ;

1

eδ(2,1,φ)
, if (p,q) = (2,1), ;

1, if 3≤ p = q <∞.

M(f) and M(E) are defined similarly.Also β = (r0/2) if (p,q) = (2,1) and β=1; otherwise.

Proof.

(i) From (7) we have

[|an|
−1/n]δ(2,1, f )

n
≥ {
{2[En(φ)/µn(Gn)]

−1/n}δ(2,1,φ)

n
}
δ(2,1, f )

δ(2,1,φ) .n
δ(2,1, f )

δ(2,1,φ)
−1

.

Proceeding to limit infimum as n→∞ and using the (2,1)-type coefficient formula for

the associate [6,Thm.1,pp.181] we get

1

eδ(2,1, f )T (2,1, f )
≥ 2δ

(2,1, f )

(
1

eδ(2,1,φ)T (2,1,φ)
)δ(2,1, f )/δ(2,1,φ) lim inf

n→∞
n
δ(2,1, f )

δ(2,1,φ)
−1

or

(δ(2,1,φ)T (2,1,φ))
1

δ(2,1,φ) ≥ 2(δ(2,1, f )T (2,1, f ))
1

δ(2,1, f ) lim inf
n→∞
{(n/e)

1

δ(2,1,φ)
− 1

δ(2,1, f ) }.

(9)



REFERENCES 1068

From (7) we can obtain that

[log |an|
−1/n]δ(2,2, f )−1

n
≥ {
[log 2[En(φ)/µn(Gn)]

−1/n]δ(2,2,φ)−1

n
}
δ(2,2, f )−1

δ(2,2,φ)−1 n
(
δ(2,2, f )−1

δ(2,2,φ)−1
)−1

.

Applying the limit infimum and taking into account the (2,2)-type coefficient formula

for associate [6,Thm.1,pp.181] we obtain

(δ(2,2, f )− 1)δ(2,2, f )−1

δ(2,2, f )δ(2,2, f )
.

1

T (2,2, f )
≥ [
(δ(2,2,φ)− 1)δ(2,2,φ)−1

δ(2,2,φ)δ(2,2,φ)
.

1

T (2,2,φ)
]
δ(2,2, f )−1

δ(2,2,φ)−1

. lim inf
n→∞

n
(
δ(2,2, f )−1

δ(2,2,φ)−1
−1)

or

(T (2,2,φ))
1

δ(2,2,φ)−1 ≥ (
δ(2,2,φ)− 1

δ(2,2, f )− 1
)
(δ(2,2, f ))δ(2,2, f )/(δ(2,2, f )−1)

(δ(2,2,φ))δ(2,2,φ)/(δ(2,2,φ)−1)
(T (2,2, f ))

1

δ(2,2, f )−1

. lim inf
n→∞

n
( 1

δ(2,2,φ)−1
− 1

δ(2,2, f )−1
)
. (10)

Hence for (p,q) = (2,2) the proof is completed. Now for 3 ≤ p = q < 0 we can easily

obtain from (7) that

[log[q−1] | an |
−1/n]δ(p,q, f )

log[p−2] n
≥ {
[log[q−1][En(φ)/µ(rn)]

−1/n

]δ(p,q,φ)

log[p−2] n
}
δ(p,q, f )

δ(p,q,φ) .(log[p−2] n)
δ(p,q, f )

δ(p,q,φ)
−1

.

Proceeding to limit infimum as n→∞ and the (p,q) type coefficient formula for asso-

ciate [6, Thm.1,pp.181] taking into account we obtain

(
1

T (p,q, f )
)

1

δ(p,q, f ) ≥ (
1

T (p,q,φ)
)

1

δ(p,q,φ) lim inf
n→∞

(log[p−2] n)
( 1

δ(p,q,φ)
− 1

δ(p,q, f )
)

or

T (p,q,φ)
1

δ(p,q,φ) ≥ T (p,q, f )
1

δ(p,q, f ) lim inf
n→∞

(log[p−2] n)
( 1

δ(p,q,φ)
− 1

δ(p,q, f )
)
. (11)

Combining (9),(10) and (11) we get the required result i.e., (i).

(ii) Following the lines of proof of (i) with equation (8) the result (ii) can be prove easily.

Hence the proof is left for the reader.

References

[1] S. Bergman,Integral Operators in the Theory of Linear Partial Differential Equations,

Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 23, Springer-Verlag, New Yok,

1969.



REFERENCES 1069

[2] R.P. Gilbert, Function Theoretic Methods in Partial Differential Equations, Math. in Science

and Engineering Vol.54, Academic Press, New York, 1969.

[3] R.P. Gilbert and D.L. Colton, Integral operator methods in biaxially symmetric potential

theory,Contrib. Differential Equations 2, 441-456. 1963

[4] R.P. Gilbert and D.L. Colton, Singularities of solutions to elliptic partial differential equa-

tions,Quarterly J. Math. 19, 391-396. 1968

[5] O.P. Juneja, G.P.Kapoor and S.K. Bajpai, On the (p,q)-order and lower (p,q)-order of an

entire function,J.Reine Angew. Math. 282, 53-67. 1976.

[6] O.P.Juneja, G.P.Kapoor and S.K.Bajpai, On the (p,q) type and lower(p,q)-type of an entire

function, J.Reine Angrew. Math. 290, 180-190. 1977.

[7] E.O.Kreyszig and M.Kracht, Methods of Complex Analysis in Partial Differential Equations

with Applications, Canadian Math. Soc. Series of Monographs Adv. Texts John Wiley and

Sons, New York, 1988.

[8] P.A. McCoy, Polynomial approximation and growth of generalized axisymmetric poten-

tials,Canadian J. Math. XXXI, 1 (1579), 49-59.

[9] P.A.McCoy, Solutions of the Helmholtz equation having rapid growth, Complex Variables

and Elliptic Equations, 18,1, 91-101. 1992.

[10] A.R.Reddy, Best apporoximation of certain entire functions, J.Approx. Theory 5, 97-112.

1972.