(p, q)-Growth of Entire Harmonic Functions in R^n in Terms of Approximation Errors

Vol. 12, No. 2, 2019, 486-498

ISSN 1307-5543 – www.ejpam.com Published by New York Business Global

(

p, q

)

-Growth of Entire Harmonic Functions in

_R

in

Terms of Approximation Errors

Devendra Kumar1,2_{, Rifaqat Ali}3,∗

1_{Department of Mathematics, Faculty of Sciences Al-Baha University, P.O.Box-1988, Alaqiq,}

Al-Baha-65431, Saudi Arabia, K.S.A.

2 _{Research and Post Graduate Studies, Department of Mathematics, M. M. H. College,}

Model Town, Ghaziabad-201001, U.P., India

3 _{Department of Mathematics, College of Science, King Khalid University, P.O.Box:9004,}

Postal Code:61413. Abha, Saudi Arabia, K.S.A.

Abstract. The relationship between the generalized growth parameters of an entire harmonic function in space Rn, n≥ 3, with the rate of its best harmonic polynomial approximation error

and ratios of these errors of functions harmonic in the ball of radiusRhas been studied.

2010 Mathematics Subject Classifications: 30E10, 41A15

Key Words and Phrases: Entire harmonic function, approximation errors, n-dimensional spaces, (p, q)-order and (p, q)-type, harmonic polynomials and spherical harmonics

1. Introduction

The approximation of entire functions on compact sets was studied by Srivastava and Kumar [14,15] and obtained generalized growth parameters in terms of approximation and interpolation error. Similar studies have been done for harmonic functions. The harmonic functions play an important role not only in theoretical mathematics but also in Physics and mechanics to describe different stationary processes. Therefore, it is significant to mention here that the study of generalized growth parameters of a harmonic function in ann-dimensional spaces has relevance. Harmonic functions can be expanded into series in spherical harmonics in spaceRn, n≥3 and in the adjoined Legendre polynomials in space R3. The growth characteristics of harmonic functions in terms of the coefficients of their

expansion into series as well as not related the expansion coefficients, in particular, in terms of the norm of their gradient at the origin were obtained. Also, the growth of harmonic function in terms of approximation errors by harmonic polynomials in Rn, n ≥ 3 was

considered by various authors (see,[3,6-13]). The aim of the present work is to investigate ∗

Corresponding author.

DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3373

Email addresses: d [email protected](D.Kumar), ([email protected] ; [email protected](R. Ali))

conditions under which a harmonic function in the ball ofn-dimensional space continues to the entire harmonic function, and to derive formulae for the generalized growth parameters (p, q)-order, lower(p, q)-order, (p, q)-type and lower(p, q)-type of harmonic function in space in terms of harmonic polynomial approximation errors. Here p and q are integers such thatp≥q≥1.

Let u be an entire harmonic function in Rn and has a Fourier-Laplace series expansion

[16]

u(rx) =

∞

X

k=0

Y(k)(x;u)rk, (1.1)

wherex∈Sn={x∈_Rn_{:|x|= 1}a unit sphere in}

Rncentered at the origin

Y(k)(x;u) =a(k)₁ Y₁(k)(x) +a(k)₂ Y₂(k)(x) +· · ·+a(k)_γ kY

(k) γk (x), a(k)_j = (u, Y_j(k)) = Γ(n/2)

2(π)n2

Z

Sn

u(x)Y_j(k)(x)dS, j= 1, γk, γk=

(2k+n−2)(k+n−3)! k!(n−2)! .

HeredSis the element of the surface area on the sphereSn,(u, Y_j(k)) is the scalor product inL2(Sn) andY(k)is a spherical harmonic of degreek, k∈Z+={0,1,2, . . . ,}on the unit sphereSn(n≥2) [16].

Let B_Rn ={y ∈_Rn _{:|y| ≤R} be the ball of radius R in space}

Rn, n ≥3 centered at

the origin, and Bn

R be the closure ofBRn. We denote HR, the class of harmonic functions inBn_Rand continuous on B_Rn,0< R <∞.

Let πk be the set of harmonic polynomials of degree≤ k. The approximation error of functionu∈HR by harmonic polynomialsP ∈πk be defined as

E_R(k)(u) = inf P∈πk

{max y∈Bn R

|u(y)−P(y)|}. (1.2)

Foru∈HR continue to the entire harmonic function ofn-dimensional spaceRn, n≥3, it

is known [17 ,p.45] that

lim k→∞(E

(k) R (u))

1

k = 0. (1.3)

The concept of orderρ(F) and lower orderλ(F) of an entire function F(z) =P∞

n=0anzn was introduced by R.P. Boas [1] as

ρ(F) = lim r→∞sup

log logM(r, F)

logr , λ(F) = limr→∞inf

log logM(r, F)

logr .

The concept of type T(F) and lower type t(F) has been introduced when the entire functions have same nonzero finite order. An entire function of orderρ,0< ρ <∞, is said to be of type T(F) and lower typet(F) if

T(F) = lim r→∞sup

logM(r, F)

rρ(F) , t(F) = lim_r→∞inf

where 0< ρ(F)<∞, M(r, F) = max0≤θ≤2π|F(r, θ)|.

For the class of order ρ(F) = 0 and ρ(F) = ∞, the type can not be defined. To refine the above concept of order and type, Juneja et. al., [4,5] introduced the concept of (p, q)-orders and (p, q)-types.

Therefore, we define the (p, q)-order and lower(p, q)-order as ρ(p, q, F) = lim sup

r→∞

log[p]M(r, F)

log[q]r , λ(p, q, F) = lim infr→∞

log[p]M(r, F)

log[q]r , (1.4) b≤ρ(p, q, F)≤ ∞, b= 0 ifp > q and b= 1 if p=q. The (p, q)-type and lower(p, q)-type are defined as

T(p, q, F) = lim sup r→∞

log[p−1]M(r, F)

log[q−1]rρ(p,q,F), t(F) = lim infr→∞

log[p−1]M(r, F)

log[q−1]rρ(p,q,F), (1.5) where log[0](x) =xand log[m](x) = log[m−1]log(x) for m≥1.

Notations:

P(α) =P(α, p, q) ={α if p > q,

{1 +α if p=q= 2,

{max(1, α) if 3≤p=q <∞,

{∞ if p=q=∞,}

and

M(α) =M(α, p, q) ={ 1

eα if (p, q) = (2,1),

{(α−1)

(α−1)

αα if (p, q) = (2,2),

{1 if p≥3,}

From [4] we define the relations between(p, q)-order, lower(p, q)-order, the coefficients of F(z) and ratios of these successive coefficients as following:

Theorem A. LetF(z) =P∞

n=0anzn be an entire function of (p, q)-orderρ(p, q, F), then ρ(p, q, F) =P(L(p, q, F))

where

L(p, q, F) = lim sup n→∞

log[p−1]n log[q]|an|−n1

.

Theorem B.LetF(z) =P∞

where

L∗(p, q, F) = lim sup n→∞

log[p−1]n log[q]| an

an+1|

.

Theorem C. Let F(z) = P∞

n=0anzn be an entire function of (p, q)-order ρ(p, q, F) and (| an

an+1|) a nondecreasing function of nforn > n0 then

λ(p, q, F) =P(l(p, q, F)) where

l(p, q, F) = lim inf n→∞

log[p−1]n log[q]|an|−

1 n .

Theorem D. Let F(z) =P∞

n=0anzn be an entire function of (p, q)-order ρ(p, q, F) and (| an

an+1|) a nondecreasing function of nforn > n0 then

λ(p, q, F) =P(l∗(p, q, F)) where

l∗(p, q, F) = lim inf n→∞

log[p−1]n log[q]| an

an+1|

.

From [5] we define the relation between (p, q)-type, lower(p, q)-type and the coefficients of F(z) as:

Theorem E. Let F(z) = P∞

n=0anzn be an entire function of (p, q)-order ρ(p, q, F) and (p, q)-typeT(p, q, F) if and only if T =M V, where

V(p, q, F) = lim sup n→∞

log[p−2]n (log[q−1]|an|−n1)ρ−A

.

withA= 1 if (p, q) = (2,2) and A= 0 if (p, q)6= (2,2). Theorem F. Let F(z) = P∞

n=0anzn be an entire function of (p, q)-order ρ(p, q, F), lower(p, q)-type t(p, q, F) and (| an

an+1|) a nondecreasing function of n for n > n0 then

t=M v, where

v(p, q, F) = lim inf n→∞

log[p−2]n (log[q−1]|an|−n1)ρ−A

.

2. Auxiliary Results

In this section we will prove some auxiliary results which will be used in the sequel. Consider the two functions f and g of complex variablez:

f(z) =

∞

X

k=0

p

(2ν)!

√

2(2ν+ 1)!(k+ 2ν)2νE (k) R (u)(

z R)

and

g(z) =

∞

X

k=1 4

(2ν)!(k+ 2ν) 2ν_E(k)

R (u)( z R)

k_{. (2.2)}

In view of [17, pp. 47] we see that if u is an entire function thenf and g are also entire functions of the complex variable z. Using Lemma 3 with inequality (8) of [17], we get

m(r, f)≤M(r, u)≤ |Y(0)(ξ, u)|+M(r, g) (2.3) wherem(r, f) is the maximum term of power series of functionf(z) on the circle{z:|z|= r}, and M(r, g) = max|z|=r|g(z)|.

Lemma 2.1. Let f and g be defined by (2.1) and (2.2). Then the (p, q)-orders and (p, q)-types off and grespectively are equal.

Proof. First we consider the case (p, q) = (2,1),

1

ρ(2,1, f) = lim infk→∞ −1

klog(

√

(2ν)!

√

2(2ν+1)!(k+2ν)2νR

−k_E(k) R (u))

logk ,

= lim inf k→∞

logRk(E_R(k)(u))−1+ log√2(2ν+ 1)!(k+ 2ν)2ν −logp(2ν)! klogk , = lim inf

k→∞

logRk(E_R(k)(u))−1 klogk + 1,

then _ρ(2,1,f)1 ≥ 1 and ρ(2,1, f) ≤ 1. This implies necessarily we have ρ(2,1, f) = 0 to defineρ(p, q, f).

Now

ρ(2,1, f) = 0⇒lim inf k→∞

1

ρ(2,1, f) = +∞

⇒lim inf k→∞

logRk(E_R(k)(u))−1

klogk = +∞

⇒lim inf k→∞

klogk logRk_(E(k)

R (u))−1 = 0

⇒lim inf k→∞

logk! logRk_(E(k)

R (u))−1 = 0

Forp≥q >1, we have

1

L(p, q, f) = lim infk→∞

log[q−1](−1 klog(

√

(2ν)!

√

2(2ν+1)!(k+2ν)2νR

−k_E(k) R (u))) log[p−1]k

= lim inf k→∞

log[q−1](log[R −k_(E(k)

R (u))] −1

k +

log√2(2ν+1)!(k+2ν)2ν

k )−

log√(2ν)! k log[p−1]k

= lim inf k→∞

1

log[p−1]klog

[q−1]₍logR−k(E (k) R (u))

−1

k )× (1 + log

√

2(2ν+ 1)!(k+ 2ν)2ν log[R−k_(E(k)

R (u))]−1

− log

p

(2ν)! log[R−k_(E(k)

R (u))]−1 )

= lim inf k→∞

log[q−1](−1 klog(R

−k_E(k) R (u))) log[p−1]k +

log(1 +o(1)) log[p−1]k = lim inf

k→∞

log[q−1](−1_klog(R−kE_R(k)(u))) log[p−1]k , and

1

L(p, q, g) = lim infk→∞

log[q−1](−1_klog(_(2ν)!4 (k+ 2ν)2ν_E(k−1) R (u)R

−k₎₎ log[p−1]k

= lim inf k→∞

log[q−1](log[R −k_(E(k)

R (u))] −1 k + log(2ν)! k − 1

klog 4(k+ 2ν) 2ν₎ log[p−1]k

= lim inf k→∞

log[q−1](−1_klog(R−kE_R(k)(u))) log[p−1]k +

log(1 +o(1)) log[p−1]k = lim inf

k→∞

log[q−1](−1 klog(R

−k_E(k) R (u))) log[p−1]k ,

Using Theorem A, we see that the functionf and ghave same (p, q)-order, it leads to the fact that

ρ(p, q, f) =ρ(p, q, g) =ρ.

1

v(p, q, f) = lim infk→∞

(−1 klog(

√

(2ν)!

√

2(2ν+1)!(k+2ν)2νR

−k_E(k) R (u)))ρ

−1

log[p−2]k = lim inf

k→∞

(log[R−k(E

(k) R (u))]

−1

k +

log√2(2ν+1)!(k+2ν)2ν

k −

log√(2ν)! k )ρ

−1

log[p−2]k = lim inf

k→∞

1 log[p−2]k(

logR−k(E_R(k)(u))−1 k )

ρ−1_×

(1 +log

√

2(2ν+ 1)!(k+ 2ν)2ν log[R−k_(E(k)

R (u))]−1

− log

p

(2ν)! log[R−k_(E(k)

R (u))]−1 )ρ−1

= lim inf k→∞

(−1_klog(R−kE_R(k)(u)))ρ−1 log[p−2]k +

log(1 +o(1))ρ−1 log[p−2]k = lim inf

k→∞

(−1 klog(R

−k_E(k)

R (u)))ρ−1 log[p−2]k . Similarly for gwe have

1

v(p,2, g) = lim infk→∞

(−1 klog(R

−k_E(k) R (u)))ρ log[p−2]k . Now for the caseq ≥3, we have

1

v(p, q, f) = lim infk→∞

log[q−2](−1 klog(

√

(2ν)!

√

2(2ν+1)!(k+2ν)2νR

−k_E(k) R (u)))ρ log[p−2]k

= lim inf k→∞

log[q−2](log[R −k_(E(k)

R (u))] −1

k +

log√2(2ν+1)!(k+2ν)2ν

k −

log√(2ν)! k )ρ log[p−2]k

= lim inf k→∞

1

log[p−2]klog

[q−2]₍logR−k(E (k) R (u))−1 k )

ρ_×

(1 + log

√

2(2ν+ 1)!(k+ 2ν)2ν log[R−k_(E(k)

R (u))]−1

− log

p

(2ν)! log[R−k_(E(k)

R (u))]−1 )ρ−1

= lim inf k→∞

log[q−2](−1_klog(R−kE_R(k)(u)))ρ log[p−2]k +

log(1 +o(1))ρ log[p−2]k = lim inf

k→∞

(−1 klog(R

−k_E(k) R (u)))ρ log[p−2]k . In the same manner for the functiong, we obtain

1

v(p, q, g) = lim infk→∞

(−1 klog(R

Lemma 2.2. Letube an entire harmonic function of an n-dimensional spacen≥3 with (p, q)-orderρ(p, q, u), lower (p, q)-orderλ(p, q, u), (p, q)-typeT(p, q, u) and lower (p, q)-type t(p, q, u). If f and g are entire functions defined as in (2.1) and (2.2), then

ρ(p, q, f) =ρ(p, q, u) =ρ(p, q, g), (2.4) λ(p, q, f)≤λ(p, q, u)≤λ(p, q, g), (2.5) T(p, q, f) =T(p, q, u) =T(p, q, g), (2.6) t(p, q, f)≤t(p, q, u)≤t(p, q, g). (2.7) Proof. From (2.3) with (1.4) and (1.5) we have

ρ(p, q, f)≤ρ(p, q, u)≤ρ(p, q, g). (2.8) Since ρ(p, q, f) =ρ(p, q, g), now (2.4) easily obtain by using (2.8). From (2.3) we can get (2.5) immediately. We denote the common value of (p, q)-order of f, g and u and using (2.3) we get

log[p−1]m(r, f) (log[q−1]r)ρ ≤

log[p−1]m(r, u) (log[q−1]r)ρ ≤

log[p−1]m(r, g) (log[q−1]r)ρ It proves (2.6) and (2.7).

Now let us define αk = maxx∈Sn|Y(k)(x;u)|, β_k =

√

(2ν)!

√

2(2ν+1)!(k+2ν)2νE (k) R R

−k _{and γ} k = 4

(2ν)!(k+ 2ν)2νE (k−1) R R

−k_.

3. Main Results

Theorem 3.1. Let u be an entire harmonic function in Rn, n ≥ 3 with (p, q)-order

ρ(p, q, u) and (p, q)-typeT(p, q, u). If ( E

(k) R (u)

E_R(k−1)(u)) is a nondecreasing function ofkfork > k0, then

ρ(p, q, u) =P(L(p, q, u)) where

L(p, q, u) = lim sup k→∞

log[p−1]k log[q](E_R(k)(u)R−k₎−1

k

(3.1) and

T(p, q, u) =M v∗(p, q, u) where

v∗(p, q, u) = lim sup k→∞

log[p−2]k (log[q−1](E_R(k)(u)R−k₎−1

k)ρ−A

Proof. Corresponding to an entire harmonic function u(rx) = P∞

k=0Y(k)(x;u)rk we define the entire function u(ζx) = P∞

k=0max|Y(k)(x;u)|ζk,|x| = 1 [2], now applying Theorem A, we have

ρ(p, q, u) =P(L(p, q, u)) where

L(p, q, u) = lim sup k→∞

log[p−1]k log[q](αk)−k1

.

We know that ( αk

αk+1) is a nondecreasing function ofk, k > k0if (

E(_Rk)(u)

E(_Rk−1)(u)) is a nondecreas-ing function of k fork > k0. This implies that (_ββ_kk₊₁) and (_γγ_kk₊₁) are also nondecreasing function of k, k > k0.

Using [17, Lemma 1] we obtain αk αk+1

≤ (k+ 2ν)

2ν_E(k−1) R (u)R (k+ 1 + 2ν)2ν_E(k)

R (u) .

Letp(x) = (_x+2ν+1x+2ν )2ν,

logp(x) = 2νlog(x+ 2ν)−2νlog(x+ 1 + 2ν), p0(x)

p(x) = 2ν x+ 2ν −

2ν x+ 2ν+ 1,

taking w(x) = _x+2ν2ν , w(x)−w(x+ 1) > 0 for any x > 0. Hence w(x) is a decreasing function and subsequentlyp0(x)>0 forx >0. Hence ( αk

αk+1) is nondecreasing if (

E_R(k)(u) E_R(k−1)(u)) is nondecreasing function ofk fork > k0. The result (3.1) is obtain by using the relation (2.4) and the result (3.2) is found from (2.6).

Theorem 3.2. Let u be an entire harmonic function in Rn, n ≥ 3 with (p, q)-order

ρ(p, q, u) and ( E

(k) R (u)

E_R(k−1)(u)) is a nondecreasing function ofk fork > k0, then

ρ(p, q, u) =P(L(p, q, u)) (3.3) where

L(p, q, u) = lim sup k→∞

log[p−1]k log[q](E

(k−1) R (u)R

E(_Rk)(u) ) Proof. For an entire functionu(zx) =P∞

where

L(p, q, u) = lim sup k→∞

log[p−1]k log[q]( αk

αk+1)

if ( αk

αk+1) is a nondecreasing function ofkfork > k0. Applying above relation for the entire

functionf, we obtain

ρ(p, q, f) =P(L(p, q, f)), (3.4)

L(p, q, f) = lim sup k→∞

log[p−1]k log[q]( βk

βk+1)

= lim sup k→∞

log[p−1]k log[q](E

(k−1) R (u)R

E_R(k)(u) ( k+1+2ν

k+2ν )2ν) = lim sup

k→∞

log[p−1]k(log[q−1](log(E (k) R (u)R E_R(k+1)(u)

) + 2νlog(k+ 1 + 2ν k+ 2ν )

−1₎₎

= lim sup k→∞

log[p−1]k log[q](E

(k) R (u)R E(_Rk+1)(u))

.

Similarly for the entire function g, we have L(p, q, g) = lim sup

k→∞

log[p−1]k log[q](E

(k) R (u)R E_R(k+!)(u))

.

ρ(p, q, g) =P(L(p, q, g)) Now (3.3)follows from (2.4).

Theorem 3.3. Let u be an entire harmonic function in Rn, n ≥ 3 with (p, q)-order

ρ(p, q, u), lower (p, q)-order λ(p, q, u), lower (p, q)-type t(p, q, u) and let ( E

(k) R (u) E_R(k+1)(u)) is a nondecreasing function ofk fork > k0, then

λ(p, q, u) =P(l∗(p, q, u)) (3.5) where

l∗(p, q, u) = lim inf k→∞

log[p−1]k log[q](R−k_E(k)

R (u))

−1 k

and

t(p, q, u) =M v(p, q, u) (3.6) where

v(p, q, u) = lim inf k→∞

log[p−2]k (log[q−1](R−k_E(k)

R (u))

Proof. Applying Theorem D to the functionf and gwe can easily obtain λ(p, q, f) =P(l∗(p, q, f))

and

λ(p, q, g) =P(l∗(p, q, g)) where

l∗(p, q, f) = lim inf k→∞

log[p−1]k log[q](R−k_E(k)

R (u))

−_k1,

l∗(p, q, g) = lim inf k→∞

log[p−1]k log[q](R−k_E(k)

R (u))

−1 k

Now the result (3.6) follows from (2.5). Similarly applying Theorem F to the entire functionf and g, we get

t(p, q, f) =M v(p, q, f), t(p, q, g) =M v(p, q, g)

where

v(p, q, f) = lim inf k→∞

log[p−2]k (log[q−1](R−k_E(k)

R (u))

−1 k)ρ−A

v(p, q, g) = lim inf k→∞

log[p−2]k (log[q−1](R−k_E(k)

R (u))

−1 k)ρ−A The result (3.7) follows from (2.7) and above relations.

Acknowledgements

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