On the generalized growth and approximation of entire solutions of certain elliptic partial differential equation

Devendra Kumar; Azza M. Alghamdi

doi:10.1515/dema-2022-0030

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On the generalized growth and approximation of entire solutions of certain elliptic partial differential equation

Devendra Kumar and Azza M. Alghamdi

Published/Copyright: August 22, 2022

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From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

For an entire function solution of generalized bi-axisymmetric potential equation, we obtain a relationship between the generalized growth characteristics and polynomial approximation errors in sup norm by using the general functions introduced by Seremeta [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, Amer. Math. Soc. Transl. 88 (1970), no. 2, 291–301].

Keywords: entire GBSPs functions; GBSPs polynomial approximation error; generalized order and type; Jacobi polynomials and sup norm

MSC 2010: 41A15; 30B10

1 Introduction

The partial differential equation

(1.1) ∂ 2 H ∂ x 2 + 2 μ y ∂ H ∂ y + ∂ 2 H ∂ y 2 + 2 ν x ∂ H ∂ x = 0 , μ , ν > 0

is called a generalized bi-axially symmetric potential equation (GBSPE), and the solutions of (1.1) are called generalized bi-axially symmetric potentials (GBSPs), which are even in x and y [1]. A polynomial of degree n which is even in x and y is said to be GBSP polynomial of degree n , if it satisfies (1.1). A GBSP H , regular about the origin, can be expanded as follows:

(1.2) H ≡ H ( r , θ ) = ∑ n = 0 ∞ a n P n μ − 1 2 , ν − 1 2 ( cos 2 θ ) ,

where x = r cos θ , y = r sin θ and P μ − 1 2 , ν − 1 2 ( cos 2 θ ) are Jacobi polynomials.

For the purpose of motivation, it is significant to mention that the Euler-Poisson Darboux equation, arising in gas dynamics, is viewed in terms of equation (1.1) after a transformation and has a variety of physical interpretations. The solution of equation (1.1), which satisfies a suitable radiation condition, corresponding to scattered waves, and their singularities are related to the quantum states of the scattered particles. The GBSPs play an important role in many aspects of mathematical physics, in particular, in an understanding of compressible flow in the transonic region (see [2]).

The limit μ ↓ ν produces the generalized axisymmetric potential equation. Reduction of the GBSP equation to the harmonic function follows from the limit μ ↓ 0 that also reduces the zonal harmonics to the circular harmonics. These functions form complete sets for even harmonic, respectively, analytic functions, regular at the origin. The GBSPs functions, then, are natural extensions of harmonic or analytic functions.

Let K R = { ( x , y ) : x 2 + y 2 < R 2 } , 0 < R ≤ ∞ , be the disc of radius R centered at the origin of coordinates and K R ¯ be the closure of K R . A G B S P function H is said to be regular in K R if the series (1.2) converges uniformly on compact subsets of K R . A class of GBSPs H regular in K R will be denoted by H R . The functions in the class H ∞ are called entire GBSPs.

Let Π n be a set of GBSP polynomials of degree no higher than n . Approximation of GBSP function H ∈ H R by GBSP polynomials g ∈ Π n be determined as follows:

(1.3) E n ( H , R ) = inf g ∈ Π n { max ( x , y ) ∈ K R ¯ ∣ H ( x , y ) − g ( x , y ) ∣ } .

The growth of an entire function f ( z ) can be measured in terms of the order ρ defined by

ρ = lim sup r → ∞ log [ 2 ] M ( r , f ) log r ,

where M ( r , f ) ≔ sup ∣ z ∣ ≤ r ∣ f ( z ) ∣ . If the order is a positive real number, the type T of the function is defined by

T = lim sup r → ∞ log M ( r , f ) r ρ .

For GBSPs functions, there is a large literature concerning the growth and approximation of this topic. McCoy [3] investigated classical order and type of entire function solutions of GBSP equation in terms of approximation errors in sup norm. In a subsequent paper, McCoy [4] considered the approximation of pseudo analytic functions, constructed as complex combination of real-valued analytic functions to the Stokes-Beltrami system on the disc. He obtained some coefficients and Bernstein type growth theorems on the disc in sup norm. In [5], McCoy studied growth and optimal approximation. Kasana and Kumar [6] studied the growth and approximation of solutions (not necessarily entire) of certain elliptic partial differential equations. They obtained the characterization of q -type and lower q -type ( q ≥ 2 ) of a GBSP having fast rates of growth in terms of ratio of approximation errors in L p -norm. In [7], Kasana and Kumar investigated the growth of entire function GBSP in terms of approximation error in L p -norm on Carathéodory domain. In [8], Kumar obtained some results for G B S P and the polynomial approximation of pseudo analytic functions, while in [9], Kumar considered entire function solutions of Helmholtz equation in R 2 and obtained some lower bounds on classical order and type. Also, Kumar [10] obtained the characterization of growth parameters in terms of axially symmetric harmonic polynomial and Lagrange polynomials approximation errors in n -dimensions. Srivastava [11] studied the order and the type of an entire function solution of certain elliptic partial differential equations in terms of series expansion coefficients and approximation errors. In the present article, using a different technique, we derive formulae for the generalized growth characteristics of entire GBSPs functions in terms of GBSPs polynomials approximation errors defined by (1.3) in sup norm.

The concept of order and type was generalized in the literature (see, e.g., Seremeta [12]). Here, one replaces the log function in the aforementioned formulae by more general functions α , β defined on an interval ( r 0 , ∞ ) , which are assumed to be positive, strictly increasing and tending to infinity as r → ∞ and satisfying properties of class L 0 and Λ defined below.

Let a function h ( ξ ) be defined on [ a , ∞ ) for some a ≥ 0 and which is strongly monotonically increasing and tends to ∞ as ξ → ∞ . According to Seremeta [12], this function belongs to the class L 0 if, for any real function ϕ such that ϕ ( ξ ) → ∞ as ξ → ∞ , the equality

lim ξ → ∞ h 1 + 1 ϕ ( ξ ) ξ h ( ξ ) = 1 .

It belongs to the class Λ if, for all c , 0 < c < ∞ ,

lim ξ → ∞ h ( c ξ ) h ( ξ ) = 1 .

By using functions α and β from the classes L 0 and Λ , by analogy with [12], define the generalized order ρ α , β ( H ) and generalized lower order λ α , β ( H ) of an entire GBSP function H by the formulas:

ρ α , β ( H ) = lim sup r → ∞ α ( log M ( r , H ) ) β ( r ) , λ α , β ( H ) = lim inf r → ∞ α ( log M ( r , H ) ) β ( r ) ,

where M ( r , H ) = max 0 ≤ θ ≤ 2 π ∣ H ( r , θ ) ∣ .

2 Auxiliary results

To prove our main results, the following lemmas are required.

Lemma 2.1

Let H ∈ H R , then for all n ∈ N , the following inequality holds:

∣ a n ∣ R 2 n ≤ 2 ( ( 2 n + μ + ν ) C ( μ , ν ) C ( n , μ , ν ) ) 1 2 E n − 1 ( H , R ) ,

where

C ( μ , ν ) = Γ ( μ + 1 2 ) Γ ( ν + 1 2 ) Γ ( μ + ν + 1 ) , C ( n , μ , ν ) = Γ ( n + 1 ) Γ ( n + μ + ν ) Γ n + μ + 1 2 Γ n + ν + 1 2

and E n − 1 ( H , R ) is determined by (1.3).

Proof

In view of the orthogonality property of Jacobi polynomials ([13], p. 68) with the uniform convergence of the series (1.2) on K R ¯ , we have

(2.1) a n R 2 n = 2 ( 2 n + μ + ν ) C ( n , μ , ν ) ∫ 0 π 2 H ( r , θ ) P n μ − 1 2 , ν − 1 2 ( cos 2 θ ) sin 2 μ θ cos 2 ν θ d θ .

On the basis of the addition theorem of Jacobi polynomials P n μ − 1 2 , ν − 1 2 , we obtain

(2.2) ∫ 0 π 2 g ( τ , θ ) P n μ − 1 2 , ν − 1 2 ( cos 2 θ ) sin 2 μ θ cos 2 ν θ d θ = 0 ,

where g ∈ Π n − 1 , 0 < τ < R . Considering (2.2), we can rewrite (2.1) as follows:

(2.3) a n τ 2 n = 2 ( 2 n + μ + ν ) C ( n , μ , ν ) ∫ 0 π 2 ( H ( r , θ ) − g ( τ , θ ) ) P n μ − 1 2 , ν − 1 2 ( cos 2 θ ) × sin 2 μ θ cos 2 ν θ d θ .

By using the Schwartz inequality and orthogonality of Jacobi polynomials in (2.3), we obtain

(2.4) a n τ 2 n ≤ max τ ∈ K R ¯ ∣ H ( τ , θ ) − g ( τ , θ ) ∣ ( ( 2 n + μ + ν ) C ( n , μ , ν ) C ( μ , ν ) ) 1 2 .

Next, it follows from the definition of E n ( H , R ) that there exists a GBSP polynomial g ˜ ∈ Π n − 1 , for which

(2.5) max τ ∈ K R ¯ ∣ H ( τ , θ ) − g ˜ ( τ , θ ) ∣ ≤ 2 E n − 1 ( H , R ) .

Putting g = g ˜ in (2.4) and taking into account inequality (2.5) as well as arbitrariness of τ , we obtain the lemma from (2.4).□

Lemma 2.2

For an entire GBSP function H, the following inequality holds

E n ( H , R ) ≤ K M ( r , H ) ( n + 1 ) q + 1 2 R r 2 n

for all r > e R and all sufficiently large values of n. Here, K is a constant independent of n and r and q = max ( μ , ν ) .

Proof

Let us consider the GBSP polynomial:

g n = ∑ k = 0 ∞ a k P k μ − 1 2 , ν − 1 2 ( cos 2 θ ) ,

where r > 0 and g n ∈ Π n . Considering the definition of approximation error E n ( H , R ) for all r , 0 < r < R , we obtain

(2.6) E n ( H , R ) ≤ max τ ∈ K R ¯ ∣ H ( τ , θ ) − g n ( τ , θ ) ∣ ≤ ∑ j = n + 1 ∞ ∣ a j ∣ R 2 j ∣ P j μ − 1 2 , ν − 1 2 ( cos 2 θ ) ∣ ≤ 1 Γ ( q + 1 ) ∑ j = n + 1 ∞ ∣ a j ∣ R 2 j Γ ( j + q + 1 ) Γ ( j + 1 ) .

From [14] for H ∈ H R , we have

(2.7) ∣ a k ∣ ≤ M ( r , H ) r 2 k [ ( 2 k + μ + ν ) C ( k , μ , ν ) C ( μ , ν ) ] 1 2

for every r < R .

Now combining (2.6) and (2.7), we obtain

(2.8) E n ( H , R ) ≤ M ( r , H ) Γ ( q + 1 ) ( C ( μ , ν ) ) 1 2 ∑ j = n + 1 ∞ Γ ( j + q + 1 ) Γ ( j + 1 ) [ ( 2 j + μ + ν ) C ( j , μ , ν ) ] 1 2 R r 2 j .

Since Γ ( x + a ) Γ ( x ) ∼ x a as x → ∞ , we have

Γ ( j + q + 1 ) Γ ( j + 1 ) [ ( 2 j + μ + ν ) C ( j , μ , ν ) ] 1 2 ∼ 2 j q + 1 2 as j → ∞ ,

and so,

Γ ( j + q + 1 ) Γ ( j + 1 ) [ ( 2 j + μ + ν ) C ( j , μ , ν ) ] 1 2 ∼ 2 2 j q + 1 2 for all j > j 0 .

From (2.8), we obtain

E n ( H , R ) ≤ M ( r , H ) Γ ( q + 1 ) 2 ( 2 C ( μ , ν ) ) 1 2 ∑ j = n + 1 ∞ j q + 1 2 R r 2 j = M ( r , H ) Γ ( q + 1 ) 2 ( 2 C ( μ , ν ) ) 1 2 R r 2 n ∑ j = n + 1 ∞ j q + 1 2 R r 2 ( j − n ) .

For r > e R , we have

∑ j = n + 1 ∞ j q + 1 2 R r 2 ( j − n ) ≤ e 2 n ∑ j = n + 1 ∞ j q + 1 2 ( e ) − 2 j ≤ e 2 n ( n + 1 ) q + 1 2 e − 2 ( n + 1 ) ∑ j = 0 ∞ 1 + j j 0 + 1 q + 1 2 ( e ) − 2 j , n > j 0 ≤ M ( r , H ) Γ ( q + 1 ) 2 ( 2 C ( μ , ν ) ) 1 2 e 2 R r 2 n ( n + 1 ) q + 1 2 ∑ j = 0 ∞ 1 + j j 0 + 1 q + 1 2 ( e ) − 2 j .

Hence, the proof is completed.□

3 Main results

In this section, we will prove our main results.

Theorem 3.1

The GBSP function H ∈ H R continues to an entire GBSP function H if and only if the following equality holds

(3.1) lim n → ∞ [ E n ( H , R ) ] 1 2 n = 0 .

Proof

Let H ∈ H R continues to an entire GBSP function, which is also denoted by H . Then equality (3.1) follows from Lemma 2.2. To prove the only if part, from Lemma 2.1, and the estimate ([13], p. 168) max − 1 ≤ t ≤ 1 ∣ P k μ − 1 2 , ν − 1 2 ( t ) ∣ = Γ ( n + q + 1 ) Γ ( n + 1 ) Γ ( q + 1 ) , we obtain

(3.2) ∑ n = 0 ∞ a n P n μ − 1 2 , ν − 1 2 ( cos 2 θ ) ≤ ∣ a 0 ∣ + ∑ n = 1 ∞ ∣ a n ∣ r 2 n Γ ( n + q + 1 ) Γ ( n + 1 ) Γ ( q + 1 ) ≤ ∣ a 0 ∣ + 2 ( C ( μ , ν ) ) 1 2 ∑ n = 1 ∞ ( ( 2 n + μ + ν ) C ( n , μ , ν ) ) 1 2 Γ ( n + q + 1 ) Γ ( n + 1 ) Γ ( q + 1 ) E n − 1 ( H , R ) r R 2 n ,

and hence, by (3.1), a uniform convergence of the series in the right side of equality (1.2) on compact subsets of the complex plane follows. Therefore, setting the function H ∈ H R by a series (1.2), we shall continue it over the whole complex plane C .□

Theorem 3.2

Let H ( r , θ ) be an entire GBSP function. If for all c , 0 < c < ∞ , one of the following conditions is satisfied

( i ) α , β ∈ Λ , d log F ( x , c ) d log x = O ( 1 ) , x → ∞ , ( i i ) α , β ∈ L 0 , lim x → ∞ d log F ( x , c ) d log x = p ,

where 0 < p < ∞ and the function F ( x , c ) = β − 1 ( c α ( x ) ) , then the generalized order ρ α , β ( H ) of the entire GBSP function H is given by

ρ α , β ( H ) = lim sup n → ∞ α ( 2 p n ) β ( e p R [ E n ( H , R ) ] − 1 2 n ) .

Proof

To prove Theorem 3.2, first, we consider the entire functions of complex variable z :

f 1 ( z ) = ∑ n = 0 ∞ 1 K ( n + 1 ) q + 1 2 E n ( H , R ) z R 2 n , f 2 ( z ) = ∑ n = 1 ∞ 2 ( ( 2 n + μ + ν ) C ( n , μ , ν ) C ( μ , ν ) ) 1 2 Γ ( n + q + 1 ) Γ ( n + 1 ) Γ ( q + 1 ) E n − 1 ( H , R ) z R 2 n .

For r > e R , by Lemma 2.2 and inequality (3.2), we obtain

(3.3) m ( r , f 1 ) ≤ M ( r , H ) ≤ ∣ a 0 ∣ + M ( r , f 2 ) ,

where m ( r , f 1 ) is the maximum term of power series of entire function f 1 ( z ) on circle { z : ∣ z ∣ = r } , and M ( r , f 2 ) = max ∣ z ∣ = r ∣ f 2 ( z ) ∣ . Hence, by using (3.3), we obtain

(3.4) log m ( r , f 1 ) ≤ log M ( r , H ) ≤ log M ( r , f 2 ) .

Since α , β ∈ Λ or L 0 are monotonically increasing functions, therefore from (3.4), we obtain

α ( log m ( r , f 1 ) ) β ( r ) ≤ α ( log M ( r , H ) ) β ( r ) ≤ α ( log M ( r , f 2 ) ) β ( r ) .

By using a result of Valiron [15] on the maximum term m ( r , f 1 ) , we obtain

log M ( r , f 1 ) ≃ log m ( r , f 1 ) as r → ∞ .

Hence,

(3.5) ρ α , β ( f 1 ) ≤ ρ α , β ( H ) ≤ ρ α , β ( f 2 ) .

Now taking into account the formula for the generalized order of an entire function of one complex variable in terms of its power series expansion coefficients [12] and using the fact that α , β ∈ Λ or L 0 , we obtain the following equality

(3.6) ρ α , β ( f 1 ) = ρ α , β ( f 2 ) = lim sup n → ∞ α ( 2 p n ) β ( e p R [ E n ( H , R ) ] − 1 2 n ) .

On combining (3.5) and (3.6), we complete the required proof.□

Remark 3.1

For α ( x ) = β ( x ) = log x , Theorem 3.2 gives the formula for the classical order ρ ( H ) as follows:

ρ ( H ) = lim sup n → ∞ 2 n log n log [ E n ( H , R ) ] − 1 .

Remark 3.2

For α ( x ) = x , β ( x ) = x ρ , p = 1 ρ , where ρ is the order of GBSP function H , the formula for the classical type T ( H ) is obtained from Theorem 3.2 as follows:

R T ( H ) ρ e 2 1 ρ = lim sup n → ∞ n 1 ρ [ E n ( H , R ) ] 1 2 n .

Remark 3.3

For α ( x ) = x , β ( x ) = x ρ ( x ) , where ρ ( x ) is the proximate order of GBSP function H , the formula for the generalized type T ∗ ( H ) (same orders but infinite type) with respect to proximate order ρ ( x ) :

R T ∗ ( H ) ρ e 2 1 ρ = lim sup n → ∞ ϕ ( n ) [ E n ( H , R ) ] 1 2 n ,

where t = ϕ ( τ ) is the function, inverse to τ = t ρ ( t ) .

Shah [16] characterized the generalized lower order of an entire function f ( z ) in terms of Taylor coefficients. Following the technique of Shah ([16], Thm. 2, pp. 316–317), we obtain the following theorem:

Theorem A

Let f ( z ) = ∑ n = 0 ∞ a n ( f ) z n be an entire function of one complex variable with generalized lower order of growth λ α , β ( f ) , where α , β ∈ Δ or L 0 , for c = 1 , a function F ( x , 1 ) = F ( x ) = β − 1 ( α ( x ) ) , where β − 1 is a function inverse to β , satisfies the condition:

For some function φ ( x ) → ∞ (howsoever slowly) as x → ∞ ,
β ( x φ ( x ) ) β ( e x ) → 0 as x → ∞ ,
d ( log F ( x ) ) d ( log x ) = O ( 1 ) as x → ∞ ,
a n ( f ) a n + 1 ( f ) is ultimately a nondecreasing function of n. Then
λ α , β ( f ) = liminf n → ∞ α ( ∣ n ∣ ) β ( ∣ a n ∣ − 1 n ) .

Theorem 3.3

Let H be a GBSP function. If condition (3.1) is satisfied, then GBSP function H can be continued to an entire GBSP function, for which

(3.7) λ α , β ( H ) ≥ liminf n → ∞ α ( 2 p n ) β ( e p R [ E n ( H , R ) ] − 1 2 n ) .

If, in addition, the ratio E n ( H , R ) E n + 1 ( H , R ) forms a nondecreasing function of n and the condition of Theorem 3.2 with condition (a) of Theorem A are satisfied, then the inequality (3.7) becomes an equality.

Proof

The proof follows along the lines of proof of Theorem 3.2 with Theorem A.□

Corollary 3.1

Let H ∈ H R be an entire GBSP function. Then the classical lower order of H ( r , θ ) is given by

(3.8) λ ( H ) ≥ lim inf n → ∞ log n log ( e 1 p R [ E n ( H , R ) ] − 1 2 n ) .

The inequality becomes equality when the ratio E n ( H , R ) E n + 1 ( H , R ) forms a nondecreasing function of n.

amhalghamdi@hotmail.com

Acknowledgements

The authors are thankful to learned referees for giving fruitful comments to improve the paper.

Funding information: The authors state no external funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Ethical approval: The conducted research is not related to either human or animals use.
Data availability statement: Data shearing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2021-09-18

Revised: 2022-06-03

Accepted: 2022-06-15

Published Online: 2022-08-22

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Keywords for this article

entire GBSPs functions; GBSPs polynomial approximation error; generalized order and type; Jacobi polynomials and sup norm

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