Approximation via Power Series Method in Two-Dimensional Weighted Spaces
- PDF / 602,096 Bytes
- 13 Pages / 439.37 x 666.142 pts Page_size
- 90 Downloads / 208 Views
Approximation via Power Series Method in Two-Dimensional Weighted Spaces Kamil Demirci1 · Sevda Yıldız1
· Fadime Dirik1
Received: 22 July 2019 / Revised: 12 December 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this work, we obtain a Korovkin-type approximation theorem for double sequences of real-valued functions by using the power series method in two-dimensional weighted spaces. We also study the rate of convergence by using the weighted modulus of continuity, and in the last section, we present an application that satisfies our new Korovkin-type approximation theorem but does not satisfy classical one. Keywords Korovkin theorem · Weighted space · Double sequence · Power series method Mathematics Subject Classification 40B05 · 40C15 · 41A36
1 Introduction The Korovkin theory has a big impact on the development of approximation theory [10]. The first study that considers summability theory in Korovkin-type approximation theory is Gadjiev and Orhan’s paper [9]. Within this scope, many mathematicians investigate and improve this theory via different spaces and convergence methods [1,4,7,18]. The concept of power series method is more effective than ordinary convergence, which includes Abel and Borel methods which are many well-known summability methods. Both methods’ definitions are based on power series, and they are not matrix methods. Power series methods are considered in the Korovkintype approximation theory first time with the Abel summability method in [15],
Communicated by Rosihan M. Ali.
B
Sevda Yıldız [email protected] Kamil Demirci [email protected]
1
Department of Mathematics, Sinop University, Sinop, Turkey
123
K. Demirci et al.
and many authors show how this concept can be applied to approximation theory [11,12,14,16,17]. The subject of this paper is to give Korovkin theorem for double sequences of real-valued functions by using the power series method in twodimensional weighted spaces. We study the rate of convergence by using the weighted modulus of continuity and present an application that satisfies our new Korovkintype approximation theorem but does not satisfy classical one. Let ( pmn ) be a double sequence of nonnegative numbers with p00 > 0 and such that the following power series ∞ p (t, s) := pmn t m s n m,n=0
has radius of convergence R with R ∈ (0, ∞] and t, s ∈ (0, R) . If, for all t, s ∈ (0, R) , ∞ 1 pmn t m s n xmn = A, lim t,s→R − p (t, s) m,n=0
then we say that the double sequence x = (xmn ) is convergent to A in the sense of power series method [5]. The power series method for double sequences is regular if and only if ∞ lim
t,s→R −
m=0 pmν t
p (t, s)
m
= 0 and lim
∞
t,s→R −
n=0 pμn s
p (t, s)
n
= 0, for any μ, υ,
(1)
hold [5]. Throughout the paper, we suppose that power series method is regular. 1 Remark 1 Note that in the case of R = 1, if pmn = 1 and pmn = (m+1)(n+1) , the power series methods coincide with Abel summability method and logarithmic summability 1 , the power series met
Data Loading...
