# On the Approximation of Entire Harmonic Functions inℝn Having Slow Growth

Authors: Devendra Kumar; A. Ghareeb
Journal: Journal of Mathematics
 (2022)

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## Abstract

The generalized growth of entire transcendental functions in terms of polynomial approximation errors in some Banach spaces has been studied by various authors. The main purpose of this paper is to study the harmonic polynomial approximation of entire harmonic functions in spaceℝn,n≥3, in certain Banach spaces. Moreover, the generalized type of harmonic functions of slow growth has been characterized in terms of best harmonic polynomial approximation errors. Our results add new aspects for the case of order zero.


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## Body

## 1. Introduction

For entire harmonic functions of one variable (identifyingℝ2 with ℂ), there is a large literature concerning the growth of this topic. For entire harmonic functions (i.e., harmonic functions defined on ℝn), elementary characterizations of the growth order and type in terms of Taylor coefficients can be found in the work of Temliakow [1], Fryant [2], Fugard [3], Kapoor and Nautiyal [4], Fryant and Shankar [5], and Veselovska [6].Several papers about generalized order and type for entire harmonic functions have been published (see, e.g., [3, 7–12]). The historical background of our work is an old result of Bernstein characterization when a continuous real-valued function f on the interval −1,1 can be extended to an entire (holomorphic) function in terms of the best approximation by polynomials of degree ≤k. In 1968, Varga [13] obtained results giving the order and type of the entire extension.Analogues of these results for harmonic functions were proved by Kapoor and Nautiyal [4] for n=3. These results were generalized by Veselovska [6] for general dimension. Recently, Veselovska [14] obtained the generalized order and lower order in terms of best approximation errors in the sup norm. The main objective of the present paper is to find an analogue of results of Vakarchuk and Zhir [15] for harmonic functions. We use a new class of functions introduced by Kapoor and Nautiyal [16] to study the slow growth of entire harmonic functions. Our results add new aspects in the results of Veselovska [14], namely, in the case ρ=0.The harmonic functions have a significant role to study different stationary processes in theoretical mathematics, physics, and mechanics. Therefore, the study of generalized growth parameters of a harmonic function inn-dimensional space ℝn, n≥3, has significance.LetBRn=y∈ℝn:y≤R be the ball of radius R and BRn¯ be its closure. We designate by HR, 0<R<∞, the class of functions harmonic in BRn and continuous on BRn¯.Letu∈HR∀r, 0<r<R; then, the Fourier–Laplace series expansion [17] is given as(1)urx=∑k=0∞Ykx;urk,where Yk are the spherical harmonics in the unit sphere Sn=y∈ℝn:y=1 in the space ℝn,n≥3 ([2], pp.157–174, [15]):(2)Ykx;u=a1kY1kx+a2kY2kx+…+aγkkYγkkx,ajk=u,Yjk,j=1,γk¯,x∈Sn,and u,Yk is the scalar product in L2Sn. Here, γk=2k+n−2k+n−3!/k!n−2!, and the scalar product in L2Sn is defined as(3)f,g=1ωn∫Snfxgxds,where ds is an element of area of the sphere Sn and ωn=2πn/2/Γn/2 is the area of the surface.


## 2. Preliminaries and Notions

The growth of a continuous functionu:ℝn⟶ℂ can be measured in terms of the order ρ defined by(4)ρ=limsupr⟶∞loglogMr,ulogr,where Mr,u:=supx≤rux. If the order is a positive real number, the type T of the function is defined by(5)T=limsupr⟶∞logMr,urρ.A function has slow growth if the order is equal to 0.The concept of order and type was generalized in the literature (see, e.g., Seremeta [18]), in particular, to discuss subclasses of the entire function of order ρ=0 (functions of slow growth). Here, one replaces the log function in the above formulae by more general functions α,β defined on an interval r0,∞, which are assumed to be positive, strictly increasing and tending to infinity as r⟶∞, and satisfying properties of classes L0 and Λ defined below.Lethξ be a monotonically increasing function defined on a,∞ for some a≥0 and hξ⟶∞ as ξ⟶∞. From Seremeta [18], hξ∈L0 if, ϕξ⟶∞ as ξ⟶∞, the equality(6)limξ⟶∞h(1+(1/ϕξ))ξhξ=1.The functionhξ∈Λ if, for all c, 0<c<∞,(7)limξ⟶∞hcξhξ=1.A functionu is said to be an entire harmonic function if it is harmonic in the entire space ℝn. We denote Mr,u=maxx∈Snurx.Considering the functionsα and β from the classes L0 and Λ, following [18], Veselovska [14] defined the generalized order ρα,βu and lower order λα,βu of an entire harmonic function u∈ℝn by the following formulae:(8)ρα,βu=limsupr⟶∞αlogMr,uβr,λα,βu=liminfr⟶∞αlogMr,uβr.Kapoor and Nautiyal [16] defined generalized order ρα,u of slow growth as(9)ρα,u=limsupr⟶∞αlogMr,uαlogr,where αξ either belongs to Ω or Ω¯. Here, Ω is the class of functions hξ defined on a,∞, strictly increasing, and differentiable, and hξ⟶∞ as ξ⟶∞ with the condition(10)0<K1≤dhξdδlogξ≤K2<∞,for all ξ>ξ0 and δξ∈Ω, and Ω¯ is the class of functions hξ which satisfies(11)limξ⟶∞dhξdlogξ=K,0<K<∞.Kapoor and Nautiyal [16] showed that classes Ω and Ω¯ are contained in Λ and Ω∩Ω¯=ϕ.From Vakarchuk and Zhir [15] for u∈HR, we set(12)Mq(r,u)=1ωn∫Snu(reiθ)qdθ1/q,q∈[1,∞),0<r<R,0<R<∞.We assume the Hardy space of functionsurx denoted by Hq∗ which satisfy the condition(13)uHq∗=limr⟶R−0Mqr,u<∞,and the Bergman space of functions urx denoted by Hq∗∗ satisfy the following relation:(14)uHq∗∗=k+ννωn∬Snuyqdsy1/q<∞,ν=n−22.Forq=∞, let uHq∗∗=uHq∗=supuy,y∈Sn. Then, Hq∗ and Hq∗∗ are Banach spaces for q≥1. The spaces Bp,q,λ, 0<p<q≤∞, λ>0, of functions u∈HR with the norm(15)up,q,λ=∫SnR−rRλ1/p−1/q−1Mqλr,udr1/λ,λ<∞,up,q,∞=sup0<r<RR−rR(1/p−1/q)Mqr,udr,λ=∞,for minq,λ≥1, Bp,q,λ are Banach spaces. It is known from [19] that Bp,q,λ is a Banach space for p>0 and q,λ≥1; otherwise, it is a Frechet space. From [20], we have(16)Hq∗⊆Hq∗∗=Bq2,q,q,1≤q<∞.LetX denote one of the Banach spaces defined above, and the approximation error of functions u∈HR by harmonic polynomials P∈πk is defined as(17)ERku,X=infuy−PyX,y∈B¯Rn,where πk denotes a set of harmonic polynomials of maximum degree k.Veselovska [14] studied generalized α,β-orders and generalized α,β-types of u∈HR in terms of the approximation errors ERku,X and also the α,β-lower order in terms of ratios of approximation errors. The generalized type Tα,u of function u∈HR with 0<ρα,u<∞ is defined as(18)Tα,u=limsupr⟶∞αlogMr,uαlogrρ,where αξ either belongs to Ω or Ω¯.This paper is organized as follows. First, the generalized type has been characterized in terms of coefficients occurring in the Fourier–Laplace series expansion ofurx. Finally, the necessary and sufficient conditions of Tα,u for the function of order zero in certain Banach spaces have been investigated. Our results are different from those of Veselovska [14].Notations are as follows:(1)
Fξ;T,ρ=α−1T¯αξ1/ρ where ρ is nonzero finite and T¯=T+ε(2)
EFξ;T,ρ denotes an integer part of F


## 3. Main Results

Theorem 1.
Letαξ∈Ω¯; then, the entire harmonic function u∈ℝn, n≥3, with nonzero finite ρ, is of generalized type T if and only if(19)T=limsupr⟶∞αlogMr,uαlogrρ=limsupk⟶∞αk/ραρ/ρ−1logmaxx∈SnYkx;u−1/kρ−1,provided dFξ;T,ρ/dlogξ=O1 as ξ⟶∞∀ nonzero finite T.Proof.
Let(20)limsupr⟶∞αlogMr,uαlogrρ=T.
ConsiderT<∞. Then, for every ε>0, there exists rε such that(21)αlogMr,uαlogrρ≤T+ε=T¯∀r≥rεorlogMr,u≤α−1T¯αlogrρ.
Letr=rk satisfy the equation(22)k=ρlogrFlogr;T¯,1ρ.
Then,(23)logr=α−11T¯αkρ1/ρ−1=Fkρ;1T¯,ρ−1.
Using Lemma 2 of [14],(24)maxx∈SnYkx;u≤2k+2ν2ν2ν!Mr,ur−k,r>0,k∈ℤ≤exp−klogr+α−1T¯α(logr)ρ+2k+2ν2ν(2ν)!.
Taking equations (22) and (23) into account, we get(25)maxx∈SnYkx;u≤exp−kF+kρF+2k+2ν2ν2ν!orρρ−1logmaxx∈SnYkx;u−1/k≥α−11T¯αkρ1/ρ−1+O1,orT¯≥limsupk⟶∞αk/ραρ/ρ−1logmaxx∈SnYkx;u−1/kρ−1.
Proceeding to limits, we get(26)T≥limsupk⟶∞αk/ραρ/ρ−1logmaxx∈SnYkx;u−1/kρ−1.
Inequality (26) obviously holds when T=∞.
In order to prove the reverse inequality, let(27)limsupk⟶∞αk/ραρ/ρ−1logmaxx∈SnYkx;u−1/kρ−1=η.
Supposeη<∞. Then, for every ε>0 and ∀k≥Nε, we have(28)αk/ραρ/ρ−1logmaxx∈SnYkx;u−1/kρ−1=η+ε=η¯.maxx∈SnYkx;u≤1expρ−1k/ρFk/ρ;1/η¯,ρ−1.
The inequality(29)maxx∈SnYkx;u−1/k≤re−ρ−1/ρFk/ρ;1/η¯,ρ−1≤12is fulfilled beginning with some k=kr. Then,(30)∑k=kr+1∞maxx∈SnYkx;urk≤∑k=kr+1∞12k≤1.
From (29), we have(31)2r≤eρ−1/ρk/ρFk/ρ;1/η¯,ρ−1,and choose kr=Eρα−1T¯αlogr+log2ρ−1. Consider the function ϕξ=rξe−ρ−1/ρFξ/ρ;1/η¯,ρ−1. We have(32)ϕ′ξϕξ=logr−ρ−1ρFξρ;1η¯,ρ−1−dFξ/ρ;1/η¯,ρ−1dlogξ=0.
Asξ⟶∞, in view of the theorem, for η0<η<∞, dFξ/ρ;1/η¯,ρ−1/dlogξ is bounded. So, there is b>0 such that, for ξ≥ξ1, we have(33)dFξ/ρ;1/η¯,ρ−1dlogξ≤b.
Chooseb>log2; then, inequalities (29) and (30) hold for k>k1r=Eρα−1η¯αlogr+Aρ−1+1. Let k0=maxNε,Eξ1+1. For r>r1k0, we have ϕ′k0/ϕk0>0. From (32) and (33), it follows that (ϕ′(k1r)/ϕk1r)<0. Finally, for r>r1k0, we assume that ξ∗r denotes the point such that ϕξ∗r=maxk0≤ξ≤k1rϕξ; then, k0<ξ∗r<k1r and ξ∗r=ρα−1η¯αlogr−arρ−1, and it gives(34)−b<ar=dFξ/ρ;1/η¯,ρ−1dlogξξ=ξ∗r≤b.
Furthermore,(35)maxk0<k<k1rmaxx∈SnYkx;urk≤maxk0<ξ<k1rϕξ=rρα−1η¯αlogr−arρ−1eρα−1η¯αlogr−arρ−1logr−ar=exparρα−1η¯αlogr−arρ−1≤expAρα−1η¯αlogr+Aρ−1.
Forr>r1k0,(36)Mr,u≤∑k=0∞maxx∈SnYkx;urk=∑k=0k0maxx∈SnYkx;urk+∑k=k0+1k1rmaxx∈SnYkx;urk+∑k=k1r+1∞maxx∈SnYkx;urk≤Ork0+k1rmaxk0<k<k1rmaxx∈SnYkx;urk+1orMr,u1+o1≤exp(bρ+o1)α−1η¯αlogr+bρ−1,αlogMr,u≤η¯αlogr+bρ−1≤η¯αlogr+bρ.
Now, we have(37)α(bρ+o1)−1logMr,uαlogr+bρ≤η¯=η+ε.
Sinceαξ∈Ω¯⊂Λ, proceeding to limits, we get(38)αlogMr,uαlogrρ≤η.
Combining inequalities (26) and (38), we get the required result.Theorem 2.
Supposeαξ∈Ω¯; then, a necessary and sufficient condition for u∈Bp,q,λ to be of generalized type T with nonzero finite order ρ is(39)T=limsupk⟶∞αk/ραρ/ρ−1logRERkBp,q,λ;u−1/kρ−1.Proof.
In order to prove our theorem, first, we take the spaceBp,q,λ, q=2, 0<p<2, and λ≥1. Assume that u∈Bp,q,λ having T with order ρ. Using Theorem 1, it gives(40)limsupk⟶∞αk/ραρ/ρ−1logmaxx∈SnYkx;u−1/kρ−1=T.
For givenε>0 and all k>k0=k0ε, we have(41)maxx∈SnYkx;u≤1expρ−1k/ρFk/ρ;1/T¯,ρ−1.
Let us consider the functionQkrζ=∑j=0kYjζ;urj, r>0 and ζ∈Sn. It is kth-partial sum of the Fourier–Laplace series of the harmonic function in ℝn, n≥3. Following pp.1396 of [21], we have(42)ERkBp,q,λ;u≤B1/λk+1λ+1;λ1p−12∑j=k+1∞maxζ∈SnYjζ;uRj21/2.
Here,Ba,ba,b>0 has been used as the beta function. Using (41), we have(43)ERkBp,2,λ;u≤B1/λk+1λ+1;λ(1/p−1/2)expρ−1k/ρFk/ρ;1/T¯,ρ−1∑j=k+1∞R2jψj2α1/2,where(44)ψjα≃expk+1/ρρ−1α−1αk+1/ρ/T+ε1/ρ−1expj/ρρ−1α−1αj/ρ/T+ε1/ρ−1Rk+1−j.
Set(45)ψα≃exp−1ρρ−1α−1α1/ρT+ε1/ρ−1.
Takingj≥k+1 and since αξ is an increasing function, it gives(46)ψjα≤expk+1−j)ρρ−1α−1αk+1/ρT+ε1/ρ−1≤ψj−k+1α.
Sinceψα<1, we get the following from equations (43) and (46):(47)R−kERk(B(p,2,λ);u)≤B1/λk+1)λ+1;λ(1/p)−(1/2(1−ψ2α))1/2exp(k+1/ρ)(ρ−1)α−1α(k+1/ρ)/T+ε1/ρ−1.
Fork>k0, equation (47) gives(48)T+ε≥αk+1/ραρ/1+1/nlogRERk−1/k+logB1/λk+1λ+1;λ1/p−1/2/1−ψ2α1/21/kρ−1.
We have(49)Bk+1λ+1;λ1p−12=Γk+1λ+1Γλ(1/p−1/2)Γk+1/2+1/pλ+1.
Hence,(50)Bk+1λ+1;λ1p−12≃e−k+1λ+1k=1λ+1k+1λ+3/2Γ(1/p−1/2)ek+1/2+1/pλ+1k+1/2+1/pλ+1k+1/2+1/p+3/2.
Thus,(51)Bk+1λ+1;λ1p−121/k+1≃1.
In view of equation (51), we have(52)T≥limsupk⟶∞αk/ραρ/ρ−1logRERkBp,q,λ;u−1/kρ−1.
To obtain the reverse inequality in (52), from equation (42), we have(53)ERkBp,2,λ;u≥maxζ∈SnYk+1ζ;uRk+1B1/λk+1λ+1;λ1p−12.
Then, for sufficiently largek, we obtain(54)αk/ραρ/ρ−1logRERkBp,q,λ;u−1/kρ−1≥αk/ραρ/ρ−1logmaxx∈SnYk+1ζ;u−1/k+S∗ρ−1≥αk/ραρ/ρ−1logmaxx∈SnYkζ;u−1/k+S∗ρ−1,where S∗=log(B−ρ/kλ[k+1λ+1;λ(1/p−1/2)]). Now, taking equation (40) into account, we get(55)limsupk⟶∞αk/ραρ/ρ−1logRERkBp,q,λ;u−1/kρ−1≥T.
Combining equations (52) and (55), we obtain the result(56)T=limsupk⟶∞αk/ραρ/ρ−1logRERkBp,q,λ;u−1/kρ−1.
Now, we discuss the spacesBp,q,λ for 0<p<q, q≠2, and q, λ≥1. For p≥p1, q≤q1, and λ≤λ1, Gvaradze [19] observed that if at least one of the inequalities is strict, then Bp,q,λ⊂Bp1,q,λ1 holds, and it gives(57)up1,q1,λ1≤21/q−1/q1λ1p−1q1/λ−1/λ1up,q,λ.
For any harmonic functionurx∈Bp,q,λ, the last relation yields(58)ERkBp1,q1,λ1;u≤21/2−1/q1λ1p−1q1/λ−1/λ1ERkBp,q,λ;u.
We will prove the necessity of condition (39) for Bp,q,λ, q≠2.
Leturx∈Bp,q,λ be an entire harmonic function with finite ρα;u whose type T is given by equation (40). In view of relation (41), for k>k0, we get(59)ERkBp,q,λ;u=uτζ−Qτζp,q,λ≤∫SnR−rRλ1/p−1/q−1Mqλr,udr1/λ.
Now,(60)urxq=∑k=0∞Ykx;urkq≤rk+1∑j=k+1∞Yjx;uq.
Hence,(61)ERkBp,q,λ;u≤B1/λk+1λ+1;λ1p−1q∑j=k+1∞Yjx;u≤B1/λk+1λ+1;λ(1/p−1/q)(1−ψα)expk+1/ρρ−1α−1αk+1/ρ/T+ε1/ρ−1.
Fork>k0, from (61), we have(62)T+ε≥αk+1/ραρ/1+1/nlogRERk−1/k+logB1/λk+1λ+1;λ1/p−1/2/1−ψα1/kρ−1.
Asψα<1 and α∈Ω¯, in view of equation (51), we obtain(63)T≥limsupk⟶∞αk/ραρ/ρ−1logERkBp,q,λ;u−1/kρ−1.
In order to prove the reverse inequality, let0<p<q<2 and λ,q≥1. Using equation (58), where p1=p, q1=2, and λ1=λ, and condition (51) which is already proved for the space Bp,2,λ, we obtain(64)limsupk⟶∞αk/ραρ/ρ−1logERkBp,q,λ;u−1/kρ−1≥limsupk⟶∞αk/ραρ/ρ−1logERkBp,2,λ;u−1/kρ−1=T.
Now, consider0<p≤2<q, and we know that(65)M2r,u≤Mqr,u,0<r<R,and it gives(66)ERkBp,q,λ;u≥maxζ∈SnYk+1ζ;uRk+1B1/λk+1λ+1;λ1p−1q.
For a sufficiently large value ofk, we have(67)αk/ραρ/ρ−1logRERkX;u−1/kρ−1≥αk/ραρ/ρ−1logmaxx∈SnYk+1x;uRk+1−1/k+S∗∗ρ−1≥αk/ραρ/ρ−1logmaxx∈SnYkx;u−1/k+S∗∗ρ−1,where S∗∗=log(B−ρ/kλ[k+1λ+1;λ(1/p−1/q)]).
Now, proceeding to limits and using equation (40), we have(68)limsupk⟶∞αk/ραρ/ρ−1logRERkX;u−1/kρ−1≥limsupk⟶∞αk/ραρ/ρ−1logYkx;u−1/kρ−1=T.
Now, assume that2≤p<q. Set q1=q,λ1=λ, and 0<p1<2 in equation (58), and putting p1 for p in (66), we obtain(69)ERkBp,q,λ;u≥Yk+1x;uRk+1B1/λk+1λ+1;λ1p1−1q.
Using equation (69) and following on the lines of the previous case 0<p≤2<q, for sufficiently large k, we obtain(70)limsupk⟶∞αk/ραρ/ρ−1logRERkX;u−1/kρ−1≥T.
Combining inequalities (52) and (55) with (70), we get the required result.Theorem 3.
Letαξ∈Ω¯. A necessary and sufficient condition for urx∈Hq∗ to be an entire harmonic function in space ℝn, n≥3, of generalized type Wα>0 with 0<ρ<∞ is(71)limsupk⟶∞αk/ραρ/ρ−1logRERkHq∗;u−1/kρ−1=Wα.Proof.
Leturx=∑k=0∞Ykx;urk be an entire harmonic function with nonzero finite order ρ and generalized type T. Since(72)limk⟶∞Ykx;u1/k=0,urx∈Bp,q,λ, with 0<p<q≤∞ and q,λ≥1. In view of [1], it gives(73)ERkBq2,q,q;u≤ηq∗ERkHq∗;u,1≤q<∞,where ηq∗ is a constant independent of k and u.
ForH∞∗, we have(74)ERkBp,∞,∞;u≤ERkH∞∗;u,1<p<∞.
Set(75)Wα,u=limsupk⟶∞αk/ραρ/ρ−1logRERkHq∗;u−1/kρ−1≥limsupk⟶∞αk/ραρ/ρ−1log(RERk(Bq/2,q,q;u−1/k)ρ−1≥T,1≤q<∞.
In view of equation (74), we can easily prove inequality (75) for q=∞. To prove the reverse inequality(76)Wα;u≤T,we use relation (41) as follows:(77)ERkHq∗;u≤uy−QkyHq∗≤∑j=k+1∞maxζ∈SnYjζ;uRj≤1expk+1/ρρ−1α−1αk+1/ρ/T+ε1/ρ−1∑j=k+1∞Rjψjα.
In view of equation (46), we have(78)ERkHq∗;u≤uy−QkyHq∗≤1(1−ψα)expk+1/ρρ−1α−1αk+1/ρ/T+ε1/ρ−1∑j=k+1∞Rjψjα,or1ERkHq∗;u≥1−ψαexpk+1/ρρ−1α−1αk+1/ρ/T+ε1/ρ−1.
This gives(79)T+ε≥αk+1/ρ[αρ/ρ−1logRERkHq∗;u−1/k+1+log1−ψα−1/k+1]ρ−1.
Proceeding to limits ask⟶∞ in (79), we obtain equation (76). Hence, we have(80)Wα,u=T.
This completes the proof of Theorem3.The case of Bergman spaces for1≤q<∞ follows from Theorem 3 and for 1≤q<∞ from Theorem 2.


## 4. Conclusion

Kapoor and Nautiyal [4, 16] studied the growth of functions harmonic in ℝ3. These results were generalized by Veselovska [6, 14] for general dimension. In this paper, we have investigated the slow growth of entire harmonic functions in general dimension and added new aspects in the results of Veselovska.



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