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Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation



Huzoor H. Khan and Rifaqat Ali
 
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ABSTRACT

The Chebyshev polynomial approximation of an entire solution of Generalized Axially Symmetric Helmholtz Equation (GASHE) in Banach spaces B(p,q,m) space, Hardy space and Bergman space) have been studied. Some bounds on generalized order of GASHE functions of slow growth have been obtained in terms of the Bessel-Gegenbauer coefficients and approximation errors using function theoretic methods.

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Huzoor H. Khan and Rifaqat Ali, 2012. Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation. Asian Journal of Mathematics & Statistics, 5: 104-120.

DOI: 10.3923/ajms.2012.104.120

URL: https://scialert.net/abstract/?doi=ajms.2012.104.120
 
Received: November 01, 2011; Accepted: March 07, 2012; Published: May 18, 2012



INTRODUCTION

The partial differential equation:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(1)

is called the Generalized Axially Symmetric Helmholtz Equation (GASHE) and the solutions of Eq. 1 are called GASHE functions. A GASHE function u, regular about the origin, has the following Bessel-Gegenbauer series expansion:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(2)

where, x = rcosθ, y = rsinθ, Jv+n are Bessel function of first kind and Cvn are Gegenbauer polynomials. A GASHE function u is said to be entire if the series (2) converges absolutely and uniformly on the compact subsets of the whole (x,y)-places it is known (Gilbert, 1969) for an entire GASHE function u that:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(3)

The growth of entire function f(z) is measured by order ρ and type T defined as under:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(4)

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(5)

where:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

be the maximum modules.

In function theory, the growth parameters may be completed from the Taylor's coefficients or Chebyshev polynomial approximations. Function theoretic methods extended these properties to harmonic functions in several variables (Gilbert and Colton, 1963; Gilbert, 1969; McCoy, 1979). (McCoy, 1992) studied the rapid growth of entire function solution of Helmholtz equation in terms of order ρ and type T using the concept of index. He obtained some bounded on the order and type of entire function solution of Helmholtz equation that reflect their antecedents in the theory of analytic functions of a single complex variable. Recently, (Kumar and Arora, 2010) studied some results generalized axisymmetric potentials. In this paper we have studied the slow growth of entire GASHE function u by using the concept of generalized order (Kapoor and Nautiyal, 1981) in Banach spaces (B(p,q,m)) spaces, Hardy space and Bergman space).

Seremeta (1970) defined the generalized order and generalized type with the help of general functions as follows.

Let Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation denote the class of functions h satisfying the following conditions:

(i) h(x) is defined on [a,α) and is positive, strictly increasing differentiable and tends to ∞ as x→∞
(ii) Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
for every function n(x) such that n(x)α as x→∞.
  Let Δ denote the class of functions h satisfying condition (i) and:
(iii) Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
for every c>0 that is, h(x) is slowly increasing.
  For entire function f(z) and functions α(x)εΔ, β(x)εL*, (Seremeta, 1970), proved that:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(6)

Further, for α(x) εL0, β-1 (x)ε L0, γ (x) εL0:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(7)

where, 0<ρ<∞ is a fixed number.

It has been noticed that above relations were obtained under certain conditions which do not hold if α = β. To define this scale, (Kapoor and Nautiyal, 1981) defined generalized order ρ(α,f) of slow growth with the help of general functions as follows.

Let Ω be the class of functions h(x) satisfying (i) and (iv) there exists a δ(x) ε Ω and x0, K1 and K2 such that:

(iv) Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
  Let Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation be the class of functions h(x) satisfying (i) and (v):
(v) Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(Kapoor and Nautiyal, 1981) showed that classes Ω and Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation are contained in Δ. Further, Ω ∩ Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation= φ and they defined the generalized order ρ(α,f) for entire functions f(z) of slow growth as:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

where, α(x) either belongs to Ω or to Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation.

Vakarchuk and Zhir (2002) considered the approximation of entire functions in Banach spaces. Thus, let f(z) be analytic in the unit disc U1 = {zεC:|z|<1} and we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Let Hq denote the Hardy space of functions f(z) satisfying the condition:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

and let Hq, denote the Bergman space of functions f(z) satisfying the condition:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

For q = ∞, let ||f||H∞, = ||f||H∞ = sup{|f(z)|, zεU1}. Then Hq and Hq, are Banach spaces for q≥1. (Vakarchuk and Zhir, 2002), we say that a function f(z) which is analytic in U1 belongs to the space B(p,q,m) if:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

0<p<q≤4, 0<m<∞ and:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

It is known Gvaradge (1994) that B(p,q,m) is a Banach space for p>0 and q,m≥1, otherwise it is a Frechet space. Further Vakarchuk (1994):

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(8)

Let X denote one of the Banach spaces defined above and let:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

where, Pn consists of algebraic polynomials of degree at most n in complex variable z.

Vakarchuk and Zhir (2002) studied the generalized order of f(z) in terms of the errors En(f,x) defined above. It has been noticed that these results do not hold good when α = β = γ, i.e., for entire functions of slow growth.

It is significant to mention here that characterization of coefficient and Chebyshev approximation error of entire function GASHE, u in certain Banach spaces by generalized order of slow growth have not been studied so far. In this paper, we have made an attempt to bridge this gap. Moreover, we have obtained some bounds on generalized order of entire function GASHE u in certain Banach spaces (B(p,q,m)) space, Hardy space and Bergman spaces) in terms of coefficients and Chebyshev approximation errors.

It is important to write here that the function α(x) = logp(x), p≥1 and α(x) = exp ((logx)δ), 0<δ<1, satisfy the condition αεΔ. For α(x) = logx, our results gives the logarithmic order in place of generalized order. So if a function f has a finite logarithmic order of finite generalized order with α(x) = logpx, p≥1, then the order ρ of f is equal to zero.

NOTATIONS

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation We shall write θ(ξ) of θ1(ξ)
F[x;c] = α-1 [cα(x)], c is a positive constant
E[f[x;c]] is an integral part of the function F

MAIN RESULTS

Now we shall prove our main results.

Theorem 1: Let α (x) ∈ Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation, then the entire GASHE function u(x,y) is of generalized order ρ(u), 1≤ρ(u)≤∞, if and only if:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(9)

where:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation and r*>1.

Proof: Suppose α (x) ∈ Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation and ρ(u)<∞. Then for every ε>0, there exists r(ε) such that:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

or:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(10)

Now using the orthogonality property of Gegenbauer polynomials (Gilbert, 1969) and the uniform convergence of series (2), we have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(11)

Further, from the well known series expansion of Jv+n(kr), we have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

and so for n≥[(kr)2], where [x] denotes the integral part of x, we have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(12)

From Eq. 11 and 12 and the using the relation:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

and:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(13)

for n≥[(kr)2], we now get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(14)

where:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Since:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

as n→∞. We can choose constants K*<∞ and r*>1 such that:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Thus, for n≥[(kr)2], Eq. 14 yields:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(15)

Using Eq. 10, we obtain:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

The larger factor is minimized at:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

This leads to:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Since α (x) ∈ Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation as n→∞, we have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(16)

Conversely, let:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(17)

Suppose L(u)<∞. Then for given ε>0, there exists n0,≥n0(ε) such that:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

where, Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

The inequality:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(18)

is satisfied with some n = n(r). Then:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

From Eq. 18, we have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

We can take Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation. Let us consider the function:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

We have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(19)

As x→∞, in view of the assumption of theorem, for finite:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

is bounded. So there is an A>0 such that for x≥x1, we have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(20)

We can take A<log2. It may be seen that inequalities Eq. 18 and 19 hold for Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation Let n0, = max (n0(ε), E[x1]+1). For r>r1 (n0), we have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

From Eq. 19 and 20 it gives that:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

This leads to the fact that if for r>r1(n0), we let x*(r) designate the point where:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

then Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation where:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

We have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

For r>r1(n0),

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Since, α (x) ∈ Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation ⊆ Δ now proceeding to limits we obtain:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(21)

Combining Eq. 16 and 21 the proof is immediate.

Theorem 2: Let α (x) ∈ Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation then the entire GASHE function u(x,y) is of generalized order ρ(u) if ρ(u)≤θ(L(f*)).

where:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Proof: Using Eq. 2 with 13, we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(22)

where, K is a constant and:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

It follows from Eq. 3 that f*(z) is an entire function. Since:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

it follows from Eq. 22 that:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Now applying (Kapoor and Nautiyal, 1981) to the function f*(z) we get the required results.

Theorem 3: Let α (x) ∈ Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation and u(x,y) be a GASHE function in the disc |z|≤r0. Then the generalized order of u(x,y) satisfy:

(i) Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(ii) Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

where:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

and:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Proof: Let GASHE u(x,y) be analytic in the disc U = {zεC: |z|≤r0} and set:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

(Vakarchuk and Zhir, 2002), we say that a function which is analytic in U belongs to the space B(p,q,m) if:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

0<p<q≤∞, 0<m<∞ and:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

We have for PεPn, that:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

In view of Eq. 22 we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

or:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(23)

where:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

From Theorem 1, for any given ε>0 and all n>n0 = n0 (ε), we have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(24)

We shall prove the result in two steps. First we consider the space B(p,q,m), q = 2, 0<p<2 and m≥1. Let:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

be the nth partial sum of the Taylor series of the function f*(z). (Vakarchuk and Zhir, 2002) and using (Reddy, 1972) extension of Bernstein theorem for given ε>0 there is an n0(ε)>0 such that:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Using Eq. 23 we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

for all n≥n0(ε), where B(a,b)(a,b>0) denotes the beta function. By using Eq. 24, we obtain:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(25)

where:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Set:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(26)

Since, φ(α)<1, by virtue of Eq. 25 and 26 we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(27)

For n>n0, Eq. 27 gives:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

But:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Hence:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

It gives:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(28)

Applying the limits, we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

or:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(29)

Using the orthogonality property of Gegenbauer polynomials (Gilbert, 1969) and the uniform convergence of the series (2), for any gεπn-1, we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(30)

From Eq. 12-15 with (Vakarchuk and Zhir, 2002) in above we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Then for sufficiently large n, we have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Applying limits and using (Kapoor and Nautiyal, 1981) for, we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

or:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(31)

Now we consider the spaces B(p,q,m) for 0<p<q, q ≠ 2 and q, m≥1. (Gvaradge, 1994) showed that, for p≥p1, q≤q1 and m≤m1 if at least one of the inequalities is strict, then the strict inclusion B(p,q,m)⊂B(p1,q1,m1) holds the following relation is true:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

for any u∈B(p,q,m), the last relation gives:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(32)

Let u∈B(p,q,m) be an entire transcends function solution of Helmholtz Eq. 1 having finite generalized order ρ(u). Consider the function:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Now:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Using Eq. 23 we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(33)

For n>n0, from Eq. 33 we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Since φ(α)<1 and α∈Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation, applying the limits and using Eq. 28, we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(34)

For (ii) inequality, let 0<p<q<2 and m,q≥1. By Eq. 32, where p = p1, q = 2 and m1 = m2 and the condition Eq. 24 is already proved for the space B(p,2,m), we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(35)

Now let 0<p≤2<q. Since, we have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Therefore,

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(36)

Then for sufficiently large n, we have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

By proceeding to limits and from Kapoor and Nautiyal (1981), we obtain:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(37)

Now we assume that 2≤p<q. Set q1 = q, m1 = m and 0<p1<2 in the inequality Eq. 36, where p1 is an arbitrary fixed number, Substituting p1 for p in Eq. 36, we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(38)

Using Eq. 38 and following the same analogy as in the previous case 0<p≤2<q, for sufficiently large n, we have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Proceeding to limits and using (Kapoor and Nautiyal, 1981), we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(39)

Combining Eq. 31, 35, 37 and 39, we obtain the result (ii). This completes the proof of Theorem 3.

Theorem 4: Let u(x,y)∈Hq be a GASHE function on the disc Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation and α(x)∈ Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation. Then the generalized order of u(x,y) satisfy:

(i) Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(ii) Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

where:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

and:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Proof: We can obtain the relation:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

In view of Eq. 26, we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Thus gives:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

Applying the limits, we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

This completes the proof of (i). (ii) In view of Eq. 8, we see that:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(40)

where, ζq is a constant independent of n and u.

Using Eq. 36 with 40 we get:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

or:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation

For the Hardy space H∞, we have:

Image for - Slow Growth and Approximation of Entire Solution of Generalized Axially Symmetric Helmholtz Equation
(41)

Using Eq. 41, the inequality (40) is true for q = ∞.

Hence the proof is completed.

REFERENCES

1:  Gilbert, R.P. and D.L. Colton, 1963. Integral operator methods in biaxially symmetric potential theory. Contrib. Differ. Equ., 2: 441-456.

2:  Gilbert, R.P., 1969. Function Theoretic Methods in Partial Differential Equations. Academic Press, New York, USA., ISBN-13: 9780122830501, Pages: 311

3:  Gvaradge, M.I., 1994. On the class of spaces of analytic functions. Mat. Zanietki, 21: 141-150.

4:  Kapoor, G.P. and A. Nautiyal, 1981. Polynomial approximation of an entirefunction of slow growth. J. Approx. Theory, 32: 64-75.

5:  Kumar, D. and K.N. Arora, 2010. Growth and approximation properties generalized axisymmetric potentials. Demons. Math., 1: 107-116.

6:  McCoy, P.A., 1979. Polynomial approximation and growth of generalized axisymmetric potentials. Canad. J. Math., 31: 49-59.
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7:  McCoy, P.A., 1992. Solutions of the helmholtz equation having rapid growth. Complex Variables Theory Appl.: Int. J., 18: 91-101.
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8:  Reddy, A.R., 1972. Best approximation of certain entire functions. J. Approx. Theory, 5: 97-112.

9:  Seremeta, M.N., 1970. On the connection between the growth of the maximum modulus of an entire function and the moduli of the coefficients of its power series expansion. Am. Math. Soc. Transl., 88: 291-301.

10:  Vakarchuk, S.B., 1994. On the best polynomial approximation in some Banach spaces for functions analytical in the unit disk. Math. Notes, 55: 338-343.
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11:  Vakarchuk, S.B. and S.I. Zhir, 2002. On some problems of polynomial approximation of entire transcendental functions. Ukrainian Math. J., 54: 1393-1401.
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