Approximation of Entire Functions of Slow Growth

Abstract

In this paper, first of all, we defined the generalized order and generalized type of Taylor entire function; Secondly, we show some interesting relationship on the maximum modulus, the maximum term and the coefficients of Taylor entire function; Finally, we study the polynomial approximation of entire functions in Banach spaces ((B(p, q, k); f ), Hardy spaces, Bergman spaces), the coefficient characterization of generalized type of Taylor entire function of slow growth has been obtained in terms of the approximation errors.

Coefficients Characterization of Entire Harmonic Functions in Terms of Norm of Gradients at Origin in R^n, n≥ 3

“…Ning Juhong and Chen Qing [9] improved above results by introducing a new class Ω * (the extension of Ω).…”

“…Ning Juhong and Chen Qing [9] obtained the following coefficient characterization: Let α(x) ∈ Ω * , then some necessary and sufficient conditions of the entire function f (z) having generalized order ρ is…”

Generalized growth and approximation errors of entire harmonic functions in \(R^n\), \(n \geq 3\)

“…where α(x) ∈ Ω. N. Juhong and C. Qing [4] extended the range of α(x) by defining a new class Ω * as the extension of Ω and obtained some results concerning above generalized growth parameters of entire function f (z).…”

“…Theorem D. [4] Let α(x) ∈ Ω * , then some necessary and sufficient conditions of the entire function f (z) with generalized order ρ is…”

“…Similar characterizations had been investigated for entire harmonic functions in R n , n ≥ 3 in terms of harmonic polynomial approximation errors. When we discuss time dependent problems in R 3 it leads to study the entire harmonic functions in R 4 . Therefore, it is significant to mention here that the harmonic functions play an important role in theoretical mathematical research, physics and mechanics to describe different stationary processes.…”