Approximation of entire transcendental functions of several complex variables in some Banach spaces for slow growth

1 Introduction

By ℂ we denote a space of complex numbers z=x+i⁢y, and

is the m-dimensional complex space. Let f be an entire function of m complex variables 𝐳=(z1,…,zm)∈ℂm and let {DR}∈ℂm, R>0, be a family of complete m-circular domains depending on the parameter R and such that 𝐳∈DR if and only if

Suppose that Mf,D⁢(R)=max⁡{|f⁢(𝐳)|:𝐳∈DR}. Gol’dberg [4] introduced the order and type of growth of f⁢(𝐳) as (see also [2])

and, for 0<ρD<∞,

The quantities ρD and TD are called the D-order and D-type of the entire function f, respectively.

We now write the expansion of the entire function f in the Taylor series

where 𝐤=(k1⁢…,km)∈ℤ+m, |𝐤|=k1+…+km, 𝐳𝐤=z1k1⁢…⁢zmkm and c𝐤⁢(f)=ck1,…,km⁢(f) are the Taylor coefficients of f. Gol’dberg [4] established the following relationship between ρ and T with the moduli of the coefficients |c𝐤⁢(f)|:

and

Sheremeta [17, 18] generalized Gol’dberg’s results with the help of general functions as follows.

Let L denote the class of functions h satisfying the following conditions:

1. h⁢(x) is defined on [a,∞) and is positive, strictly increasing and differentiable, and tends to ∞ as x→∞.

2. For any function φ⁢(x) such that φ⁢(x)→∞ as x→∞, we have

   limx→∞⁡h⁢{(1+1φ⁢(x))⁢x}h⁢(x)=1.

Let Δ denote the class of functions h satisfying condition (i) and

that is, h⁢(x) is slowly increasing.

Sheremeta [17] introduced the following notions of generalized order and generalized type of growth for entire functions of several complex variables:

for α⁢(x)∈Δ and β⁢(x)∈L. Further, for α⁢(x)∈L, β-1⁢(x)∈L and γ⁢(x)∈L,

Setting α⁢(x)=log⁡x and β⁢(x)=x in (1.5) and α⁢(x)=β⁢(x)=γ⁢(x)=x in (1.6), we obtain the Gol’dberg definitions of order and type of entire functions (1.1) and (1.2), respectively.

Sheremeta also established the relationship between the generalized order of growth (1.5) and (1.6) of an entire function f and its Taylor coefficients. We now summarize these results in the form of the following theorems.

It has been noticed that the above characterizations of generalized order and generalized type of f⁢(𝐳) in terms of Taylor coefficients do not hold when α=β=γ, i.e., entire functions of slow growth. To overcome this difficulty, Kapoor and Nautiyal [9] introduced a new class of functions Ω for m=1, defined as follows. A function ϕ⁢(x) belongs to Ω if ϕ⁢(x) satisfies (i) and the following condition:

1. There exists a function δ⁢(x)∈Δ and xo,K1 and K2 such that for all x>xo ,

   0<K1≤d⁢(ϕ⁢(x))d⁢(δ⁢(log⁡x))≤K2<∞.

Further, a function ϕ⁢(x) belongs to Ω¯ if ϕ⁢(x) satisfies (i) and

Kapoor and Nautiyal [9, p. 66] showed that Ω,Ω¯⊆Δ and Ω∩Ω¯=ϕ. Let α⁢(x)∈Ω or Ω¯. Then, following [9, p. 66], for an entire function f⁢(𝐳), we define the generalized order and generalized type as

and

respectively, where α⁢(x) either belongs to Ω or Ω¯.

Recently Vakarchuk and Zhir [24] studied the best polynomial approximations of entire transcendental functions of several complex variables in Banach spaces. Let Um={𝐳∈ℂm:|zj|<1,j=1,…,m} be a unit polydisk in ℂm and let Γm={𝐳∈ℂm:|zj|=1,j=1,…,m} be its skeleton. Also, let ℝm be an m-dimensional real space. Further, by

and

we denote m-dimensional cubes in the space ℝm. Let A⁢(Um) be the set of all analytic functions in the set Um. For any function f∈A⁢(Um), we get

where f⁢(𝐫⁢ei⁢𝐭)=f⁢(r1⁢ei⁢t1,…,rm⁢ei⁢tm), d⁢𝐭=d⁢t1⁢⋯⁢d⁢tm and

Let Hq⁢(Um), 0<q≤∞, denote the Hardy space of functions f⁢(𝐳)∈A⁢(Um) satisfying the condition

and let Hq′⁢(Um) denote the Bergman space of functions f⁢(𝐳)∈A⁢(Um) satisfying the condition

For q=∞, let ∥f∥H∞′=∥f∥H∞=sup⁡{|f⁢(𝐳)|:𝐳∈Um}. Then Hq and Hq′ are Banach spaces for q≥1. Following [24, p. 1794], we say that a function f∈A⁢(Um) belongs to the space Bm⁢(p,q,λ) if for 0<p<q≤∞,

and

It is known (see [6, 7]) that Bm⁢(p,q,λ) is a Banach space for p>0 and q,λ≥1, otherwise it is a Frechet space.

Let Pn denote a subspace of algebraic polynomials of m complex variables of the form

where n∈ℤ+. Let X be one of the Banach spaces of analytic functions of m complex variables listed earlier. Let En⁢(f,X) denote the value of the best polynomial approximation of the function f∈X by elements of the subspace Pn, i.e.,

Several authors established the relationship between the order and type of an entire function and the rate of its best polynomial approximation in different domains, see [1, 15, 3, 8, 16]. Kumar [10, 11] investigated the growth and approximation of entire function solutions of the Helmholtz equation. Vakarchuk and Zhir [22, 20, 21, 23] studied some problems of approximation of entire transcendental functions in some Banach spaces. It has been noticed that the study of growth of entire transcendental functions in terms of En⁢(f,X) in several complex variables has not been done extensively (see [12, 13, 14, 19, 24, 25]) as in a single complex variable. Vakarchuk and Zhir [24] obtained the necessary and sufficient condition for f∈X to be an entire transcendental function of the generalized order of growth ρm⁢(f;α,β) in terms of En⁢(f,X), where X is one of the Banach spaces of functions analytic in Um. It is significant to mention here that these results do not hold for α=β. So, in the present paper we have tried to refine this scale. To the best of our knowledge, the generalized type Tm⁢(f;α,α) has not been characterized in terms of En⁢(f,X) in m-complex variables so far. The results of Vakarchuk and Zhir [24] can be improved by our results.

2 Main results