Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 850760, 8 pages
http://dx.doi.org/10.1155/2013/850760
Research Article
Bivariate Positive Operators in Polynomial Weighted Spaces
Octavian Agratini
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Street Kogălniceanu 1, 400084 Cluj-Napoca, Romania
Correspondence should be addressed to Octavian Agratini; agratini@math.ubbcluj.ro
Received 25 February 2013; Accepted 26 March 2013
Academic Editor: Sung Guen Kim
Copyright © 2013 Octavian Agratini. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper aims to two-dimensional extension of some univariate positive approximation processes expressed by series. To be
easier to use, we also modify this extension into finite sums. With respect to these two new classes designed, we investigate
their approximation properties in polynomial weighted spaces. The rate of convergence is established, and special cases of our
construction are highlighted.
1. Introduction
The approximation of functions by using linear positive operators is currently under research. Usually, two types of positive approximation processes are used: the discrete, respectively, continuous form. In the first case, they often are
designed through a series. Since the construction of such
operators requires an estimation of infinite sums, this restricts
the operator usefulness from the computational point of view.
In this respect, in order to approximate a function, it is interesting to consider partial sums, the number of terms considered in sum depending on the function argument. Roughly
speaking, these discrete operators are truncated fading away
their “tails.” Thus, they become usable for generating software approximation of functions. Among the pioneers who
approached this direction we mention Gróf [1] and Lehnhoff
[2]. In the same direction a class of univariate linear positive
operators is investigated in [3].
This work focuses on a general bivariate class of discrete
positive linear operators expressed by infinite sums. This class
acts in polynomial weighted spaces of continuous functions
of two variables defined by R+ × R+ , where R+ = [0, ∞). By
using a certain modulus of smoothness we give theorems on
the degree of approximation. Further, we replace the infinite
sum by a truncated one, and we study the approximation
properties of the new defined family of operators. Compared
to what has been done so far, the strengths of this paper
consist in using general classes of two-dimensional discrete
operators, implying an arbitrary network of nodes. Finally
we present some particular classes of operators that can be
obtained from our family.
2. The Operators
Our linear and positive operators have the role to approximate functions defined on R+ × R+ . Therefore, on this domain
we define for every (𝑚, 𝑛) ∈ N × N a net of form Δ 𝑚,𝑛 =
Δ 1,𝑚 × Δ 2,𝑛 , where Δ 1,𝑚 (0 = 𝑥𝑚,0 < 𝑥𝑚,1 < ⋅ ⋅ ⋅ ) and
Δ 2,𝑛 (0 = 𝑦𝑛,0 < 𝑦𝑛,1 < ⋅ ⋅ ⋅ ). Set N0 = {0} ∪ N, and 𝐶(R+ )
stands for the space of all real-valued continuous functions
on R+ .
Products of parametric extensions of two univariate
operators are appropriate tools to approximate functions of
two variables. For this reason, the starting point is given by
the following one-dimensional operators:
∞
(𝐴 𝑚 𝑓) (𝑥) = ∑ 𝑎𝑚,𝑖 (𝑥) 𝑓 (𝑥𝑚,𝑖 ) ,
𝑖=0
(1)
∞
(𝐵𝑛 𝑓) (𝑦) = ∑ 𝑏𝑛,𝑗 (𝑦) 𝑓 (𝑦𝑛,𝑗 ) ,
𝑗=0
where 𝑎𝑚,𝑖 , 𝑏𝑛,𝑗 are nonnegative functions belonging to 𝐶(R+ ),
(𝑖, 𝑗) ∈ N0 × N0 , such that the following identities
∞
∞
𝑖=0
𝑗=0
∑ 𝑎𝑚,𝑖 (𝑡) = ∑ 𝑏𝑛,𝑗 (𝑡) = 1,
𝑡 ≥ 0,
(2)
2
Abstract and Applied Analysis
take place. These conditions mean that the operators 𝐴 𝑚 and
𝐵𝑛 preserve the monomial 𝑒0 , 𝑒0 (𝑡) = 1, a property often seen
at classical linear positive operators.
For each 𝑧 ∈ R+ , define the function 𝜑𝑧 by 𝜑𝑧 (𝑡) = 𝑡 − 𝑧,
𝑡 ∈ R+ . Also, for each 𝑟 ∈ N, set
M𝑟 (𝐴 𝑚 ; 𝑥) = (𝐴 𝑚 𝜑𝑥𝑟 ) (𝑥) ,
𝑡 ≥ 0, deg (Γ𝑟 ) ≤ 𝑟.
−1
𝑤𝑝 (𝑡) = (1 + 𝑡𝑝 )
for 𝑝 ∈ N.
(5)
The space is endowed with the norm ‖ ⋅ ‖𝑝 , ‖𝑓‖𝑝 =
sup𝑥≥0 𝑤𝑝 (𝑥)|𝑓(𝑥)|.
For bivariate operators we consider the space
𝐶𝑝,𝑞 (R2+ )
= {𝑓 ∈
𝐶 (R2+ )
: 𝑤𝑝,𝑞 𝑓 is uniformly continuous
and bounded on R2+ }
(6)
associated to the weighted function 𝑤𝑝,𝑞 (𝑥, 𝑦) = 𝑤𝑝 (𝑥)𝑤𝑞 (𝑦),
(𝑝, 𝑞) ∈ N0 × N0 . The norm of this space is denoted by ‖ ⋅ ‖𝑝,𝑞
and is defined by
󵄩󵄩󵄩𝑓󵄩󵄩󵄩 = sup 𝑤𝑝,𝑞 (𝑥, 𝑦) 󵄨󵄨󵄨𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 .
󵄨
󵄩 󵄩𝑝,𝑞
󵄨
(7)
(𝑥,𝑦)∈R2+
These spaces are ordered with respect to inclusion as follows: if (𝑝, 𝑞) ≤ (𝑝󸀠 , 𝑞󸀠 ), then 𝐶𝑝,𝑞 (R2+ ) ⊂ 𝐶𝑝󸀠 ,𝑞󸀠 (R2+ ). Moreover, ‖𝑓‖𝑝󸀠 ,𝑞󸀠 ≤ ‖𝑓‖𝑝,𝑞 for any 𝑓 ∈ 𝐶𝑝,𝑞 (R2+ ). Examining the
weight 𝑤𝑝,𝑞 , it is easy to verify the inequality
󵄨󵄨
󵄨󵄨 𝑡
1
𝑑𝑢
1
󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨∫
󵄨󵄨 𝑥 𝑤𝑝,𝑞 (𝑢, 𝑧) 󵄨󵄨󵄨 ≤ |𝑡 − 𝑥| ( 𝑤𝑝,𝑞 (𝑥, 𝑧) + 𝑤𝑝,𝑞 (𝑡, 𝑧) ) ,
󵄨
󵄨
for any 𝑥 ≥ 0 and 𝑡 ≥ 0.
≤ |𝑡 − 𝑥| (1 + 𝑥𝑝 + 1 + 𝑡𝑝 ) (1 + 𝑧𝑞 )
= |𝑡 − 𝑥| (
(8)
(9)
1
1
+
).
𝑤𝑝,𝑞 (𝑥, 𝑧) 𝑤𝑝,𝑞 (𝑡, 𝑧)
Starting from (1), for each (𝑚, 𝑛) ∈ N × N we introduce
a linear positive operator in polynomial weighted space
𝐶𝑝,𝑞 (R2+ ) as follows:
∞ ∞
(𝐿 𝑚,𝑛 𝑓) (𝑥, 𝑦) = ∑ ∑ 𝑎𝑚,𝑖 (𝑥) 𝑏𝑛,𝑗 (𝑦)𝑓 (𝑥𝑚,𝑖 , 𝑦𝑛,𝑗 ) ,
𝑖=0 𝑗=0
(𝑥, 𝑦) ∈ R2+ .
(4)
Apparently is a tough condition, but the examples that we
give in the last section show that it is carried out by different
classes of operators.
In what follows we specify the function spaces in which
the operators act.
For univariate operators 𝐴 𝑠 , 𝐵𝑠 we consider the space
𝐶𝑝 (R+ ), 𝑝 ∈ N0 fixed, consisting of all real-valued functions
𝑓 continuous on R+ such that 𝑤𝑝 𝑓 is uniformly continuous
and bounded on R+ , where the weight 𝑤𝑝 is defined as
follows:
𝑤0 (𝑡) = 1,
󵄨󵄨
󵄨󵄨 𝑡
𝑑𝑢
󵄨󵄨
󵄨󵄨
𝑝
𝑞
󵄨
󵄨󵄨∫
󵄨󵄨 𝑥 𝑤𝑝,𝑞 (𝑢, 𝑧) 󵄨󵄨󵄨 = |𝑡 − 𝑥| (1 + 𝜉𝑥,𝑡 ) (1 + 𝑧 )
󵄨
󵄨
M𝑟 (𝐵𝑛 ; 𝑦) = (𝐵𝑛 𝜑𝑦𝑟 ) (𝑦) ,
(3)
representing the 𝑟th central moment of the specified operators.
For a simplified writing, we will use the common notation
𝐿 𝑠 , where 𝐿 𝑠 = 𝐴 𝑠 for all 𝑠 ∈ N or 𝐿 𝑠 = 𝐵𝑠 for all 𝑠 ∈ N.
For our purposes, relative to the central moments, we
require additional conditions. For each 𝑟 ∈ N, M𝑟 (𝐿 𝑠 ; 𝑡) as
a function of 𝑡 is bounded by a polynomial of degree at most
𝑟. Moreover, 𝑠𝑟/2 M𝑟 (𝐿 𝑠 ; ⋅) is bounded with respect to 𝑠. These
requirements can be brought together and redrafted in the
following way: for each 𝑟 ∈ N, a polynomial Γ𝑟 exists such
that
𝑠𝑟/2 M𝑟 (𝐿 𝑠 ; 𝑡) ≤ Γ𝑟 (𝑡) ,
Indeed, based on the first mean value theorem for integration, between 𝑥 and 𝑡, a point 𝜉𝑥,𝑡 exists such that
(10)
If the function 𝑓 ∈ 𝐶𝑝,𝑞 (R2+ ) can be decomposed in the
following manner 𝑓(𝑥, 𝑦) = 𝑓1 (𝑥)𝑓2 (𝑦), (𝑥, 𝑦) ∈ R2+ , then
one has
(𝐿 𝑚,𝑛 𝑓) (𝑥, 𝑦) = (𝐴 𝑚 𝑓1 ) (𝑥) (𝐵𝑛 𝑓2 ) (𝑦) .
(11)
For example, we get 𝐿 𝑚,𝑛 𝑤𝑝,𝑞 = (𝐴 𝑚 𝑤𝑝 )(𝐵𝑛 𝑤𝑞 ).
In order to present the rate of convergence for our bivariate operators, we use a modulus of smoothness associated to
any function 𝑓 belonging to 𝐶𝑝,𝑞 (R2+ ). It is given by the formula
󵄩
󵄩
𝜔𝑓 (ℎ, 𝛿) = sup 󵄩󵄩󵄩Δ 𝑢,V 𝑓󵄩󵄩󵄩𝑝,𝑞 ,
(ℎ, 𝛿) ∈ R2+ ,
(12)
Δ 𝑢,V 𝑓 (𝑥, 𝑦) = 𝑓 (𝑥 + 𝑢, 𝑦 + V) − 𝑓 (𝑥, 𝑦)
(13)
0≤𝑢≤ℎ
0≤V≤𝛿
where
for (𝑥, 𝑦) and (𝑢, V) belonging to 𝐶𝑝,𝑞 (R2+ ). Alternative notation is 𝜔(𝑓; 𝑥, 𝑦). More information about moduli of smoothness can be found in the monograph [4].
Further, we indicate a truncated variant of operators
defined at (10). Let (𝑢𝑠 )𝑠≥1 , (V𝑠 )𝑠≥1 be strictly increasing
sequences of positive numbers such that
lim √𝑠𝑢𝑠 = lim √𝑠V𝑠 = ∞.
𝑠→∞
𝑠→∞
(14)
Taking in view the net Δ 1,𝑚 , we partitioned the set N0 into
two parts
𝐼 (𝑥, 𝑢𝑚 ) = {𝑖 ∈ N0 : 𝑥𝑚,𝑖 ≤ 𝑥 + 𝑢𝑚 } ,
𝐼 (𝑥, 𝑢𝑚 ) = N0 \ 𝐼 (𝑥, 𝑢𝑚 ) .
(15)
Similarly, via the network Δ 2,𝑛 , we introduce 𝐽(𝑦, V𝑛 ) and
𝐽(𝑦, V𝑛 ).
Abstract and Applied Analysis
3
For each (𝑚, 𝑛) ∈ N × N and any 𝑓 ∈ 𝐶𝑝,𝑞 (R2+ ) we define
the linear positive operators
𝑚 ,V𝑛 ⟩
𝑓) (𝑥, 𝑦)
(𝐿⟨𝑢
𝑚,𝑛
=
∑
∑ 𝑎𝑚,𝑖 (𝑥) 𝑏𝑛,𝑗 (𝑦) 𝑓 (𝑥𝑚,𝑖 , 𝑦𝑛,𝑗 ) ,
𝑖∈𝐼(𝑥,𝑢𝑚 ) 𝑗∈𝐽(𝑦,V𝑛 )
(𝑥, 𝑦) ∈
(16)
Throughout the paper, by 𝑐(⋅) we denote different real constants, in the brackets specifying the parameter(s) that the
indicated constant depends.
At first we collect some useful results relative to the onedimensional operators 𝐿 𝑠 where 𝐿 𝑠 = 𝐴 𝑠 (𝑠 ∈ N) or 𝐿 𝑠 =
𝐵𝑠 (𝑠 ∈ N).
Lemma 1. Let 𝑝 ∈ N0 , and let the weight 𝑤𝑝 be given by (5).
The operator 𝐿 𝑠 satisfies
1
; 𝑡) ≤ 𝑐 (𝑝) ,
𝑤𝑝
󵄩 󵄩
󵄨
󵄨
(ii) 𝑤𝑝 (𝑡) 󵄨󵄨󵄨(𝐿 𝑠 𝑓) (𝑡)󵄨󵄨󵄨 ≤ 𝑐 (𝑝) 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑝 ,
󵄩
󵄩 󵄩
󵄩󵄩
󵄩󵄩𝐿 𝑠 𝑓󵄩󵄩󵄩𝑝 ≤ 𝑐 (𝑝) 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑝 ,
Lemma 2. Let 𝑝 ∈ N0 , and let the weight 𝑤𝑝 be given by (5).
For any 𝑠 ∈ N the operator 𝐿 𝑠 satisfies
(i) 𝑤𝑝 (𝑡) 𝐿 𝑠 (
R2+ .
3. Auxiliary Results
(i) 𝑤𝑝 (𝑡) 𝐿 𝑠 (
By using (17) we obtain the first inequality of the relation
(18). Further, applying sup𝑡≥0 , the second inequality is proved.
󵄨󵄨 󵄨󵄨
√Γ̃ (𝑡)
󵄨𝜑 󵄨
(ii) 𝑤𝑝 (𝑡) 𝐿 𝑠 ( 󵄨 𝑡 󵄨 ; 𝑡) ≤ 𝑐 (𝑝)
,
√𝑠
𝑤𝑝
𝑡 ≥ 0,
(21)
(22)
Proof. Let 𝑝 ∈ N0 and 𝑠 ∈ N be fixed.
(i) Using the identity
𝑝
𝑝
1 + 𝑡𝑝 = 1 + ∑ ( ) (𝑥 − 𝑡)𝑘 𝑡𝑝−𝑘 ,
𝑘
(23)
𝑘=0
we can write the following:
(18)
𝑤𝑝 (𝑡) 𝐿 𝑠 (
𝜑𝑡2
1
; 𝑡) =
𝐿 (𝜑2 ; 𝑡)
𝑤𝑝
1 + 𝑡𝑝 𝑠 𝑡
𝑝
𝑡𝑝−𝑘 𝑝
( ) 𝐿 𝑠 (𝜑𝑡𝑘+2 ; 𝑡)
𝑝 𝑘
1
+
𝑡
𝑘=0
+∑
Proof. Let 𝑝 ∈ N0 and 𝑠 ∈ N be fixed.
(i) By using (5), (2), (3), and (4) we can write
𝑝
𝑝
≤ M2 (𝐿 𝑠 ; 𝑡) + ∑ ( ) M𝑘+2 (𝐿 𝑠 ; 𝑡)
𝑘
1
𝑤𝑝 (𝑡) 𝐿 𝑠 ( ; 𝑡)
𝑤𝑝
𝑘=0
𝑝
= 𝑤𝑝 (𝑡) 𝐿 𝑠 (1 + ((𝑥 − 𝑡) + 𝑡)𝑝 ; 𝑡)
≤
𝑝
𝑝
= 𝑤𝑝 (𝑡) (1 + 𝑡𝑝 + ∑ ( ) 𝐿 𝑠 ((𝑥 − 𝑡)𝑘 𝑡𝑝−𝑘 ; 𝑡))
𝑘
(19)
𝑘=1
𝑝−𝑘
𝑝 𝑡
=1+ ∑ ( )
𝑘
𝑘=1
𝑡 ≥ 0,
𝑝
where 𝜑𝑡 (𝑥) = 𝑥 − 𝑡 and Γ̃ = Γ2 + ∑𝑘=0 ( 𝑝𝑘 ) Γ𝑘+2 . The polynomials Γ] , ] ≥ 2, are introduced by (4).
(17)
where 𝑡 ≥ 0 and 𝑠 ∈ N.
𝑝
𝜑𝑡2
Γ̃ (𝑡)
; 𝑡) ≤
,
𝑤𝑝
𝑠
Γ2 (𝑡)
𝑝 Γ (𝑡)
.
+ ∑ ( ) 𝑘+2
𝑘 𝑠1+𝑘/2
𝑠
𝑘=0
(24)
During the previous relations we used notation (3) and
̃ relation (21)
hypothesis (4). Considering the significance of Γ,
follows.
M𝑘 (𝐿 𝑠 ; 𝑡)
1 + 𝑡𝑝
𝑝
𝑝−𝑘
𝑝 𝑡 Γ𝑘 (𝑡)
≤1+ ∑ ( )
.
𝑘
1 + 𝑡𝑝
𝑘=1
(ii) Taking into consideration relation (1), we apply the
Cauchy-Schwarz inequality, and this allows us to
write
Since deg(𝑡𝑝−𝑘 Γ𝑘 (𝑡)) ≤ 𝑝, the previous expression is
bounded with respect to 𝑡 ∈ R+ , and inequality (17) follows.
(ii) To be more explicit we consider 𝐿 𝑠 = 𝐴 𝑠
∞
󵄨
󵄨 󵄨
󵄨
𝑤𝑝 (𝑡) 󵄨󵄨󵄨(𝐴 𝑠 𝑓) (𝑡)󵄨󵄨󵄨 ≤ 𝑤𝑝 (𝑡) ∑ 𝑎𝑠,𝑘 (𝑡) (𝑤𝑝 󵄨󵄨󵄨𝑓󵄨󵄨󵄨) (𝑥𝑠,𝑘 )
𝑘=0
1
𝑤𝑝 (𝑥𝑠,𝑘 )
󵄨󵄨 󵄨󵄨
󵄨𝜑 󵄨
𝑤𝑝 (𝑡) 𝐿 𝑠 ( 󵄨 𝑡 󵄨 ; 𝑡)
𝑤𝑝
1/2
≤ (𝑤𝑝 (𝑡) 𝐿 𝑠 (
1
; 𝑡))
𝑤𝑝
(𝑤𝑝 (𝑡) 𝐿 𝑠 (
𝜑𝑡2
; 𝑡))
𝑤𝑝
1/2
.
(25)
1
󵄩 󵄩
≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑝 𝑤𝑝 (𝑡) 𝐴 𝑠 ( ; 𝑡) .
𝑤
Relations (17) and (21) imply (22).
𝑝
(20)
4
Abstract and Applied Analysis
We mention that with the help of the same CauchySchwarz inequality, relations (2) and (4) lead us to the following inequality:
Γ (𝑡)
󵄨 󵄨𝑟
𝐿 𝑠 (󵄨󵄨󵄨𝜑𝑡 󵄨󵄨󵄨 ; 𝑡) ≤ √ 2𝑟 𝑟 ,
𝑠
1
𝑡 ≥ 0.
0≤𝜏2 ≤𝛿
(26)
Lemma 3. Let (𝑝, 𝑞) ∈ N0 × N0 . For any (𝑚, 𝑛) ∈ N × N, the
operator 𝐿 𝑚,𝑛 given by (10) verifies
󵄩󵄩
󵄩󵄩
1
󵄩
󵄩
(i) 󵄩󵄩󵄩󵄩𝐿 𝑚,𝑛 (
; ⋅)󵄩󵄩󵄩󵄩 ≤ 𝑐 (𝑝, 𝑞) ,
𝑤𝑝,𝑞 󵄩󵄩𝑝,𝑞
󵄩󵄩
(27)
󵄨
󵄨 󵄩
󵄩
sup 𝑤𝑝,𝑞 (𝑥, 𝑦) 󵄨󵄨󵄨󵄨Δ 𝜏1 ,𝜏2 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨󵄨 = 󵄩󵄩󵄩󵄩Δ 𝜏1 ,𝜏2 𝑓󵄩󵄩󵄩󵄩𝑝,𝑞 .
(𝑥,𝑦)∈R2+
𝑓 ∈ 𝐶𝑝,𝑞 (R2+ ) .
Keeping in mind (31), we consequently obtain
ℎ
𝛿
1
󵄩
󵄩
󵄩󵄩
󵄩
sup 󵄩󵄩󵄩󵄩Δ 𝜏1 ,𝜏2 𝑓󵄩󵄩󵄩󵄩𝑝,𝑞 ∫ 𝑑𝑢 ∫ 𝑑V = 𝜔𝑓 (ℎ, 𝛿)
󵄩󵄩𝑓ℎ,𝛿 − 𝑓󵄩󵄩󵄩𝑝,𝑞 ≤
ℎ𝛿 0≤𝜏1 ≤ℎ
0
0
1
󵄩 󵄩
≤ 𝑤𝑝,𝑞 (𝑥, 𝑦) 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑝,𝑞 𝐿 𝑚,𝑛 (
; 𝑥, 𝑦) .
𝑤𝑝,𝑞
(36)
(28)
󵄨
󵄨
𝑤𝑝,𝑞 (𝑥, 𝑦) 󵄨󵄨󵄨(𝐿 𝑚,𝑛 𝑓) (𝑥, 𝑦)󵄨󵄨󵄨
(29)
and (32) is completed.
(ii) We justify only the first inequality, and the second
inequality can be proven in the same manner.
Occurs
ℎ
𝜕
(∫ 𝑓 (𝑥 + 𝑢, 𝑦 + V) 𝑑𝑢) = 𝑓 (𝑥 + ℎ, 𝑦 + V)
𝜕𝑥 0
− 𝑓 (𝑥, 𝑦 + V)
= Δ ℎ,V 𝑓 (𝑥, 𝑦) − Δ 0,V 𝑓 (𝑥, 𝑦)
(37)
Applying sup(𝑥,𝑦)∈R2+ and taking into account (27), one
obtains (28).
In our investigation we appeal to the Steklov function.
This can be used to approximate continuous functions by
smoother functions. The Steklov function associated with 𝑓 ∈
𝐶(R2+ ) is given as follows:
𝛿
1 ℎ
∫ 𝑑𝑢 ∫ 𝑓 (𝑥 + 𝑢, 𝑦 + V) 𝑑V,
ℎ𝛿 0
0
(𝑥, 𝑦) ∈ R2+ ,
𝛿
1 ℎ
𝑓ℎ,𝛿 (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦) =
∫ 𝑑𝑢 ∫ Δ 𝑢,V 𝑓 (𝑥, 𝑦) 𝑑V.
ℎ𝛿 0
0
(see (13)). Further, we can write
󵄩󵄩 𝜕
󵄩󵄩
󵄩󵄩
󵄩
󵄩󵄩 𝑓ℎ,𝛿 󵄩󵄩󵄩
󵄩󵄩 𝜕𝑥
󵄩󵄩𝑝,𝑞
󵄨󵄨 𝛿
󵄨󵄨
1
󵄨
󵄨
=
sup 𝑤𝑝,𝑞 (𝑥, 𝑦) 󵄨󵄨󵄨󵄨∫ (Δ ℎ,V 𝑓 (𝑥, 𝑦) − Δ 0,V 𝑓 (𝑥, 𝑦)) 𝑑V󵄨󵄨󵄨󵄨
ℎ𝛿 𝑥 ≥ 0
󵄨󵄨 0
󵄨󵄨
𝑦≥0
(30)
≤
where ℎ > 0 and 𝛿 > 0. By using (13) we deduce
𝛿
1
󵄨
󵄨
sup 𝑤𝑝,𝑞 (𝑥, 𝑦) ∫ 󵄨󵄨󵄨Δ ℎ,V 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 𝑑V
ℎ𝛿 𝑥 ≥ 0
0
𝑦≥0
(31)
In the next lemma we have gathered some known properties of Steklov function 𝑓ℎ,𝛿 , where 𝑓 ∈ 𝐶𝑝,𝑞 (R2+ ). These properties establish connections between 𝑓ℎ,𝛿 and the modulus 𝜔𝑓
indicated at (12). For the sake of completeness we present the
proofs of these inequalities.
Lemma 4. Let 𝑓 belong to 𝐶𝑝,𝑞 (R2+ ), and let 𝑓ℎ,𝛿 be defined by
(30). The following relations take place:
󵄩
󵄩
(i) 󵄩󵄩󵄩𝑓ℎ,𝛿 − 𝑓󵄩󵄩󵄩𝑝,𝑞 ≤ 𝜔𝑓 (ℎ, 𝛿) ,
(32)
󵄩󵄩 𝜕
󵄩󵄩 𝜕
󵄩󵄩
󵄩󵄩 2
2
󵄩󵄩
󵄩
󵄩
󵄩
(ii) 󵄩󵄩󵄩 𝑓ℎ,𝛿 󵄩󵄩󵄩 ≤ 𝜔𝑓 (ℎ, 𝛿) ,
󵄩󵄩 𝑓ℎ,𝛿 󵄩󵄩󵄩 ≤ 𝜔𝑓 (ℎ, 𝛿) ,
󵄩󵄩 𝜕𝑦
󵄩󵄩 𝜕𝑥
󵄩󵄩𝑝,𝑞 ℎ
󵄩󵄩 𝛿
(33)
where ℎ > 0, 𝛿 > 0, and 𝜔𝑓 is defined by (12).
(35)
0≤𝜏2 ≤𝛿
Proof. The first statement follows immediately from the definition of the weight 𝑤𝑝,𝑞 and relations (11), (17).
Regarding the second statement, based on (10), we get
𝑓ℎ,𝛿 (𝑥, 𝑦) =
(34)
On the other hand,
Lemma 1 leads to the following result.
󵄩
󵄩
󵄩 󵄩
(ii) 󵄩󵄩󵄩𝐿 𝑚,𝑛 𝑓󵄩󵄩󵄩𝑝,𝑞 ≤ 𝑐 (𝑝, 𝑞) 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑝,𝑞 ,
Proof. Let ℎ > 0 and 𝛿 > 0 be arbitrarily fixed.
(i) For 𝑢 ∈ [0, ℎ] and V ∈ [0, 𝛿] we deduce
󵄨
󵄨
󵄨
󵄨󵄨
󵄨󵄨Δ 𝑢,V 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 ≤ sup 󵄨󵄨󵄨󵄨Δ 𝜏1 ,𝜏2 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨󵄨 .
0≤𝜏 ≤ℎ
+
𝛿
1
󵄨
󵄨
sup 𝑤𝑝,𝑞 (𝑥, 𝑦) ∫ 󵄨󵄨󵄨Δ 0,V 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 𝑑V
ℎ𝛿 𝑥 ≥ 0
0
𝑦≥0
:= 𝐼1 + 𝐼2 .
(38)
Since 0 ≤ V ≤ 𝛿, it is clear that |Δ ℎ,V 𝑓(𝑥, 𝑦)| ≤
sup 0≤𝜏1 ≤ℎ |Δ 𝜏1 ,𝜏2 𝑓(𝑥, 𝑦)|, and we obtain
0≤𝜏2 ≤𝛿
𝐼1 ≤
𝛿
1
󵄨
󵄨
sup ∫ sup 𝑤𝑝,𝑞 (𝑥, 𝑦) 󵄨󵄨󵄨󵄨Δ 𝜏1 ,𝜏2 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨󵄨 𝑑V
ℎ𝛿 0≤ 𝜏1 ≤ ℎ 0 𝑥 ≥ 0
0≤𝜏2 ≤𝛿
𝑦≥0
1
1
󵄩
󵄩󵄩
󵄩󵄩 = 𝜔 (ℎ, 𝛿) .
≤
sup 󵄩󵄩󵄩Δ
ℎ 0≤ 𝜏1 ≤ ℎ󵄩 𝜏1 ,𝜏2 󵄩𝑝,𝑞 ℎ 𝑓
(39)
0≤ 𝜏2 ≤ 𝛿
In the same manner we show 𝐼2 ≤ (1/ℎ)𝜔𝑓 (ℎ, 𝛿). Returning at (38), the proof is ended.
Abstract and Applied Analysis
5
1
In the following we denote by 𝐶𝑝,𝑞
(R2+ ) the space of all
2
functions 𝑔 : R+ → R having the first order partial
derivatives such that the functions 𝜕𝑔/𝜕𝑥, 𝜕𝑔/𝜕𝑦, and 𝑔
belong to 𝐶𝑝,𝑞 (R2+ ).
1
(R2+ ), then for any
Lemma 5. Let (𝑝, 𝑞) ∈ N0 × N0 . If 𝑔 ∈ 𝐶𝑝,𝑞
(𝑚, 𝑛) ∈ N × N the operator 𝐿 𝑚,𝑛 given by (10) verifies
󵄨
󵄨
𝑤𝑝,𝑞 (𝑥, 𝑦) 󵄨󵄨󵄨(𝐿 𝑚,𝑛 𝑔) (𝑥, 𝑦) − 𝑔 (𝑥, 𝑦)󵄨󵄨󵄨
󵄩󵄩󵄩 𝜕𝑔 󵄩󵄩󵄩 Φ (𝑥) 󵄩󵄩󵄩 𝜕𝑔 󵄩󵄩󵄩 Φ (𝑦)
≤ 𝑐 (𝑝, 𝑞) (󵄩󵄩󵄩 󵄩󵄩󵄩
+ 󵄩󵄩󵄩 󵄩󵄩󵄩
),
(40)
󵄩󵄩 𝜕𝑥 󵄩󵄩𝑝,𝑞 √𝑚 󵄩󵄩 𝜕𝑦 󵄩󵄩𝑝,𝑞 √𝑛
(𝑥, 𝑦) ∈ R2+ ,
where
𝑝
𝑝
Φ (𝑡) = √Γ2 (𝑡) + √ Γ2 (𝑡) + ∑ ( ) Γ𝑘+2 (𝑡),
𝑘
(41)
see (8). Applying 𝐿 𝑚,𝑛 and by using successively (11), (26),
(17), and (22) we have
󵄩󵄩 𝜕𝑔 󵄩󵄩
|𝑡 − 𝑥|
󵄩 󵄩
𝐽1 ≤ 󵄩󵄩󵄩 󵄩󵄩󵄩 (𝐿 𝑚,𝑛 (
; 𝑥, 𝑦)
󵄩󵄩 𝜕𝑥 󵄩󵄩𝑝,𝑞
𝑤𝑝,𝑞 (𝑥, 𝑧)
+ 𝐿 𝑚,𝑛 (
󵄩󵄩 𝜕𝑔 󵄩󵄩
1
1
󵄩 󵄩
󵄨 󵄨
𝐴 (󵄨󵄨𝜑 󵄨󵄨 ; 𝑥) 𝐵𝑛 ( ; 𝑦)
= 󵄩󵄩󵄩 󵄩󵄩󵄩 (
󵄩󵄩 𝜕𝑥 󵄩󵄩𝑝,𝑞 𝑤𝑝 (𝑥) 𝑚 󵄨 𝑥 󵄨
𝑤𝑞
󵄨󵄨 󵄨󵄨
󵄨𝜑 󵄨
1
+𝐴 𝑚 ( 󵄨 𝑥 󵄨 ; 𝑥) 𝐵𝑛 ( ; 𝑦))
𝑤𝑝
𝑤𝑞
󵄩󵄩 𝜕𝑔 󵄩󵄩
𝑐 (𝑝) √ Γ̃ (𝑥)
𝑐 (𝑞)
1 √ Γ2 (𝑥)
󵄩 󵄩
≤ 󵄩󵄩󵄩 󵄩󵄩󵄩 (
+
)
󵄩󵄩 𝜕𝑥 󵄩󵄩𝑝,𝑞 𝑤𝑝 (𝑥)
𝑚
𝑤𝑝 (𝑥)
𝑚
𝑤𝑞 (𝑦)
𝑘=0
≤
the polynomials Γ] , ] = 2, 𝑝 + 2, being indicated at (4).
Proof. Let (𝑥, 𝑦) ∈ R2+ and (𝑚, 𝑛) ∈ N2 be arbitrarily fixed.
1
Since 𝑔 ∈ 𝐶𝑝,𝑞
(R2+ ), for any (𝑡, 𝑧) ∈ R2+ we can write
𝑔 (𝑡, 𝑧) − 𝑔 (𝑥, 𝑦) = ∫
𝑡
𝑥
𝑧
𝜕
𝜕
𝑔 (𝑢, 𝑧) 𝑑𝑢 + ∫
𝑔 (𝑥, V) 𝑑V.
𝜕𝑢
𝑦 𝜕V
(42)
Since 𝐿 𝑚,𝑛 is linear monotone and reproduces the constants, from the previous identity we obtain
󵄨󵄨󵄨(𝐿 𝑚,𝑛 𝑔) (𝑥, 𝑦) − 𝑔 (𝑥, 𝑦)󵄨󵄨󵄨
󵄨
󵄨
󵄨󵄨
𝑡
𝜕
󵄨
= 󵄨󵄨󵄨󵄨𝐿 𝑚,𝑛 (∫
𝑔 (𝑢, 𝑧) 𝑑𝑢; 𝑥, 𝑦)
󵄨󵄨
𝑥 𝜕𝑢
󵄨󵄨
𝑧
𝜕
󵄨
+𝐿 𝑚,𝑛 (∫
𝑔 (𝑥, V) 𝑑V; 𝑥, 𝑦)󵄨󵄨󵄨󵄨
󵄨󵄨
𝑦 𝜕V
(43)
󵄨󵄨 𝑡 𝜕
󵄨󵄨
󵄨
󵄨
󵄨
󵄨
≤ 𝐿 𝑚,𝑛 (󵄨󵄨∫
𝑔 (𝑢, 𝑧) 𝑑𝑢󵄨󵄨 ; 𝑥, 𝑦)
󵄨󵄨 𝑥 𝜕𝑢
󵄨󵄨
󵄨󵄨 𝑧
󵄨󵄨
𝜕
󵄨
󵄨
+ 𝐿 𝑚,𝑛 (󵄨󵄨󵄨󵄨∫
𝑔 (𝑥, V) 𝑑V󵄨󵄨󵄨󵄨 ; 𝑥, 𝑦)
󵄨󵄨 𝑦 𝜕V
󵄨󵄨
:= 𝐽1 + 𝐽2 .
Further, one has
󵄨󵄨
󵄨󵄨 𝑡 𝜕
󵄨
󵄨󵄨
𝑔 (𝑢, 𝑧) 𝑑𝑢󵄨󵄨󵄨
󵄨󵄨∫
󵄨󵄨
󵄨󵄨 𝑥 𝜕𝑢
󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑡
󵄨󵄨 𝜕
𝑑𝑢
󵄨󵄨
󵄨
󵄨
󵄨
󵄨󵄨
≤ 󵄨󵄨󵄨∫ 𝑤𝑝,𝑞 (𝑢, 𝑧)󵄨󵄨󵄨 𝑔 (𝑢, 𝑧) 󵄨󵄨󵄨󵄨
󵄨󵄨 𝑤𝑝,𝑞 (𝑢, 𝑧) 󵄨󵄨󵄨
󵄨󵄨 𝑥
󵄨󵄨 𝜕𝑢
󵄨󵄨
󵄩󵄩 𝜕𝑔 󵄩󵄩 󵄨󵄨󵄨 𝑡
𝑑𝑢
󵄨󵄨
󵄩 󵄩
󵄨
≤ 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄨󵄨󵄨󵄨∫
󵄩󵄩 𝜕𝑥 󵄩󵄩𝑝,𝑞 󵄨󵄨 𝑥 𝑤𝑝,𝑞 (𝑢, 𝑧) 󵄨󵄨󵄨󵄨
󵄩󵄩 𝜕𝑔 󵄩󵄩
1
1
󵄩 󵄩
≤ 󵄩󵄩󵄩 󵄩󵄩󵄩 (
+
) |𝑡 − 𝑥| ,
󵄩󵄩 𝜕𝑥 󵄩󵄩𝑝,𝑞 𝑤𝑝,𝑞 (𝑥, 𝑧) 𝑤𝑝,𝑞 (𝑡, 𝑧)
|𝑡 − 𝑥|
; 𝑥, 𝑦))
𝑤𝑝,𝑞 (𝑡, 𝑧)
𝑐 (𝑝, 𝑞) 󵄩󵄩󵄩󵄩 𝜕𝑔 󵄩󵄩󵄩󵄩 √Γ2 (𝑥) + √Γ̃ (𝑥)
,
󵄩 󵄩
√𝑚
𝑤𝑝,𝑞 (𝑥, 𝑦) 󵄩󵄩󵄩 𝜕𝑥 󵄩󵄩󵄩𝑝,𝑞
(45)
where 𝑐(𝑝, 𝑞) is a suitable constant. Following the same pathway, we find
𝐽1 ≤
𝑐 (𝑝, 𝑞) 󵄩󵄩󵄩󵄩 𝜕𝑔 󵄩󵄩󵄩󵄩 √Γ2 (𝑦) + √Γ̃ (𝑦)
.
󵄩 󵄩
√𝑛
𝑤𝑝,𝑞 (𝑥, 𝑦) 󵄩󵄩󵄩 𝜕𝑦 󵄩󵄩󵄩𝑝,𝑞
(46)
Considering the increases established for 𝐽1 , 𝐽2 and
returning to the relation (43), the inequality (40) is completely proven.
4. Main Results
The rate of convergence for 𝐿 𝑚,𝑛 operator will be read as
follows.
Theorem 6. Let (𝑝, 𝑞) ∈ N0 × N0 . For any (𝑚, 𝑛) ∈ N × N, the
operator 𝐿 𝑚,𝑛 given by (10) satisfies
󵄨
󵄨
𝑤𝑝,𝑞 (𝑥, 𝑦) 󵄨󵄨󵄨(𝐿 𝑚,𝑛 𝑓) (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨
(47)
Φ (𝑥) Φ (𝑦)
≤ 𝑐 (𝑝, 𝑞) 𝜔𝑓 (
,
),
√𝑚 √𝑛
(𝑥, 𝑦) ∈ R2+ , where Φ is given by (41) and 𝑐(𝑝, 𝑞) is a suitable
constant.
Proof. Setting
󵄨
󵄨
𝑇1 := 𝑤𝑝,𝑞 (𝑥, 𝑦) 󵄨󵄨󵄨𝐿 𝑚,𝑛 (𝑓 − 𝑓𝑛,𝛿 ; 𝑥, 𝑦)󵄨󵄨󵄨 ,
(44)
󵄨
󵄨
𝑇2 := 𝑤𝑝,𝑞 (𝑥, 𝑦) 󵄨󵄨󵄨(𝐿 𝑚,𝑛 𝑓ℎ,𝛿 ) (𝑥, 𝑦) − 𝑓ℎ,𝛿 (𝑥, 𝑦)󵄨󵄨󵄨 ,
(48)
󵄨
󵄨
𝑇3 := 𝑤𝑝,𝑞 (𝑥, 𝑦) 󵄨󵄨󵄨𝑓ℎ,𝛿 (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 ,
we can write
󵄨
󵄨
𝑤𝑝,𝑞 (𝑥, 𝑦) 󵄨󵄨󵄨(𝐿 𝑚,𝑛 𝑓) (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 ≤ 𝑇1 + 𝑇2 + 𝑇3 . (49)
6
Abstract and Applied Analysis
We establish upper bounds for these three quantities.
Relations (28) and (32) imply that
󵄩
󵄩
󵄩
󵄩
𝑇1 ≤ 󵄩󵄩󵄩𝐿 𝑚,𝑛 (𝑓 − 𝑓ℎ,𝛿 ; 𝑥, 𝑦)󵄩󵄩󵄩𝑝,𝑞 ≤ 𝑐 (𝑝, 𝑞) 󵄩󵄩󵄩𝑓 − 𝑓ℎ,𝛿 󵄩󵄩󵄩𝑝,𝑞
≤ 𝑐 (𝑝, 𝑞) 𝜔𝑓 (ℎ, 𝛿) .
Setting
⟨𝑢𝑚 ,V𝑛 ⟩
𝑓) (𝑥, 𝑦)
(1 𝑅𝑚,𝑛
(50)
=
∑
∑ 𝑎𝑚,𝑖 (𝑥) 𝑏𝑛,𝑗 (𝑦) 𝑓 (𝑥𝑚,𝑖 , 𝑦𝑛,𝑗 ) ,
𝑖∈𝐼(𝑥,𝑢𝑚 ) 𝑗∈𝐽(𝑦,V𝑛 )
1
(R2+ );
For 𝑇2 we use Lemma 5 choosing 𝑔 = 𝑓ℎ,𝛿 ∈ 𝐶𝑝,𝑞
see definition (31).
One has
󵄩󵄩 𝜕𝑓 󵄩󵄩 Φ (𝑥) 󵄩󵄩 𝜕𝑓 󵄩󵄩 Φ (𝑦)
󵄩 ℎ,𝛿 󵄩󵄩
󵄩 ℎ,𝛿 󵄩󵄩
+ 󵄩󵄩
)
𝑇2 ≤ 𝑐 (𝑝, 𝑞) (󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩 𝜕𝑥 󵄩󵄩󵄩𝑝,𝑞 √𝑚 󵄩󵄩󵄩 𝜕𝑦 󵄩󵄩󵄩𝑝,𝑞 √𝑛
2Φ (𝑥) 2Φ (𝑦)
≤ 𝑐 (𝑝, 𝑞) (
+
) 𝜔𝑓 (ℎ, 𝛿)
ℎ√𝑚
𝛿√𝑛
⟨𝑢𝑚 ,V𝑛 ⟩
𝑓) (𝑥, 𝑦)
(2 𝑅𝑚,𝑛
=
∑
∑ 𝑎𝑚,𝑖 (𝑥) 𝑏𝑛,𝑗 (𝑦) 𝑓 (𝑥𝑚,𝑖 , 𝑦𝑛,𝑗 ) ,
𝑖∈𝐼(𝑥,𝑢𝑚 ) 𝑗∈𝐽(𝑦,V𝑛 )
⟨𝑢𝑚 ,V𝑛 ⟩
𝑓) (𝑥, 𝑦)
(3 𝑅𝑚,𝑛
(51)
=
∑
∑ 𝑎𝑚,𝑖 (𝑥) 𝑏𝑛,𝑗 (𝑦) 𝑓 (𝑥𝑚,𝑖 , 𝑦𝑛,𝑗 ) ,
𝑖∈𝐼(𝑥,𝑢𝑚 ) 𝑗∈𝐽(𝑦,V𝑛 )
(56)
(see also (33)). Finally, inequality (32) implies
󵄩
󵄩
𝑇3 ≤ 󵄩󵄩󵄩𝑓ℎ,𝛿 − 𝑓󵄩󵄩󵄩𝑝,𝑞 ≤ 𝜔𝑓 (ℎ, 𝛿) .
we can write
(52)
Setting ℎ = Φ(𝑥)/√𝑚, 𝛿 = Φ(𝑦)/√𝑛 and coming back to
(49), we can affirm that a certain constant 𝑐(𝑝, 𝑞) exists such
that (47) holds.
󵄨󵄨 ⟨𝑢𝑚 ,V𝑛 ⟩
󵄨
󵄨󵄨(𝐿 𝑚,𝑛 𝑓) (𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨󵄨(𝐿 𝑚,𝑛 𝑓) (𝑥, 𝑦) − 𝑓 (𝑥)󵄨󵄨󵄨󵄨
󵄨
󵄨
3
󵄨 ⟨𝑢𝑚 ,V𝑛 ⟩
󵄨
+ ∑ 󵄨󵄨󵄨󵄨(𝑘 𝑅𝑚,𝑛
𝑓) (𝑥, 𝑦)󵄨󵄨󵄨󵄨 .
𝑘=1
(57)
Knowing that the modulus 𝜔𝑓 enjoys the property
lim(ℎ,𝛿) → (0+ ,0+ ) 𝜔𝑓 (ℎ, 𝛿) = 0, from Theorem 6 we deduce the
following result.
Theorem 7. Let (𝑝, 𝑞) ∈ N0 × N0 , and let the operators 𝐿 𝑚,𝑛 ,
(𝑚, 𝑛) ∈ N × N, be defined by (10).
For any (𝑥, 𝑦) ∈ R2+ the pointwise convergence takes place
lim (𝐿 𝑚,𝑛 𝑓) (𝑥, 𝑦) = 𝑓 (𝑥, 𝑦) ,
𝑚,𝑛 → ∞
𝑓 ∈ 𝐶𝑝,𝑞 (R2+ ) . (53)
If 𝐾1 , 𝐾2 are compact intervals included in R+ , then (53)
holds uniformly on the domain 𝐾1 × 𝐾2 .
𝑚 ,V𝑛 ⟩
,
Theorem 8. Let (𝑝, 𝑞) ∈ N0 × N0 , and let the operators 𝐿⟨𝑢
𝑚,𝑛
2
(𝑚, 𝑛) ∈ N × N, be defined by (16). For any (𝑥, 𝑦) ∈ R+ the
pointwise convergence takes place
𝑚 ,V𝑛 ⟩
𝑓) (𝑥, 𝑦) = 𝑓 (𝑥, 𝑦) ,
lim (𝐿⟨𝑢
𝑚,𝑛
𝑚,𝑛 → ∞
𝑓 ∈ 𝐶𝑝,𝑞 (R2+ ) .
(54)
If 𝐾1 , 𝐾2 are compact intervals included in R+ , then (54)
holds uniformly on the domain 𝐾1 × 𝐾2 .
Proof. Let (𝑥, 𝑦) ∈ R2+ be arbitrarily fixed. Taking in view the
partitions of N0 (see (15)), we use the following decomposition:
N0 × N0 = (𝐼 × 𝐽) ∪ (𝐼 × 𝐽) ∪ (𝐼 × 𝐽) ∪ (𝐼 × 𝐽) .
(55)
⟨𝑢𝑚 ,V𝑛 ⟩
If we show lim𝑚,𝑛 → ∞ (𝑘 𝑅𝑚,𝑛
𝑓)(𝑥, 𝑦) = 0, 𝑘 ∈ {1, 2, 3},
then, based on (53), our statement (54) follows, and the proof
is ended.
Further, we prove the previous limit only for 𝑘 = 1, other
two following similar routes.
Since 𝑓 ∈ 𝐶𝑝,𝑞 (R2+ ), a constant 𝑀𝑓 exists such that
󵄨
󵄨󵄨
𝑝
𝑞
󵄨󵄨𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 ≤ 𝑀𝑓 (1 + 𝑥 ) (1 + 𝑦 ) ,
(𝑥, 𝑦) ∈ R2+ .
(58)
Based on the classical inequality (𝑎 + 𝑏)𝑠 ≤ 2𝑠−1 (𝑎𝑠 + 𝑏𝑠 ),
𝑎 ≥ 0, 𝑏 ≥ 0, 𝑠 ∈ N0 , we deduce
𝑡𝑠 ≤ (|𝑡 − 𝑧| + 𝑧)𝑠 ≤ 2𝑠−1 (|𝑡 − 𝑧|𝑠 + 𝑧𝑠 ) ,
𝑡 ≥ 0,
𝑧 ≥ 0,
𝑠 ∈ N0 .
(59)
Consequently, (58) implies that
󵄨󵄨
󵄨
󵄨󵄨𝑓 (𝑥𝑚,𝑖 , 𝑦𝑛,𝑗 )󵄨󵄨󵄨
󵄨
󵄨
󵄨
󵄨𝑝
≤ 𝑀𝑓 (1 + 2𝑝−1 (󵄨󵄨󵄨𝑥𝑚,𝑖 − 𝑥󵄨󵄨󵄨 + 𝑥𝑝 ))
󵄨
󵄨𝑞
× (1 + 2𝑞−1 (󵄨󵄨󵄨󵄨𝑦𝑛,𝑗 − 𝑦󵄨󵄨󵄨󵄨 + 𝑦𝑞 )) ,
where (𝑖, 𝑗) ∈ 𝐼(𝑥, 𝑢𝑚 ) × 𝐽(𝑦, V𝑛 ).
(60)
Abstract and Applied Analysis
7
Using this relation we have
󵄨󵄨 ⟨𝑢𝑚 ,V𝑛 ⟩
󵄨
󵄨󵄨(1 𝑅𝑚,𝑛 𝑓) (𝑥, 𝑦)󵄨󵄨󵄨
󵄨
󵄨
where 𝛿𝑟 = 1 if 𝑟 is odd, 𝛿𝑟 = 0 if 𝑟 is even, and 𝑏𝑛,𝑟,𝑗 are
positive coefficients bounded with respect to 𝑛. In particular,
M𝑟 (𝑉𝑛 ; 𝑥) is a polynomial of degree 𝑟 without a constant
term. These properties ensure that condition (4) is achieved.
The study of these operators in polynomial weighted
spaces was carried out in [6]. Choosing in (10) 𝐴 𝑚 = 𝑉𝑚
and 𝐵𝑛 = 𝑉𝑛 we obtain the Baskakov operator for functions
of two variables. The net is Δ 𝑚,𝑛 = (𝑖/𝑚, 𝑗/𝑛)𝑖,𝑗≥0 . Our
results indicated at (47) and (53) are identified with the results
established by Gurdek et al. [7, Equations (22), (28)].
The univariate truncated operators has been approached
in [8]. The truncated version specified in (16) coincides with
the operators studied by Walczak [9, Equation (17)]. In this
case 𝐼(𝑥, 𝑢𝑚 ) from (15) becomes {𝑖 ∈ N0 : 𝑖 ≤ [𝑚(𝑥 + 𝑢𝑚 )]}.
Here [𝜆] indicates the largest integer not exceeding 𝜆.
(2) Szász operators [10] are of the form:
≤ 𝑀𝑓
×
󵄨
󵄨𝑝
∑ 𝑎𝑚,𝑖 (𝑥) (1 + 2𝑝−1 𝑥𝑝 + 2𝑝−1 󵄨󵄨󵄨𝑥𝑚,𝑖 − 𝑥󵄨󵄨󵄨 )
𝑖∈𝐼(𝑥,𝑢𝑚 )
󵄨
󵄨𝑞
∑ 𝑏𝑛,𝑗 (𝑦) (1 + 2𝑞−1 𝑦𝑞 + 2𝑞−1 󵄨󵄨󵄨󵄨𝑦𝑛,𝑗 − 𝑦󵄨󵄨󵄨󵄨 )
𝑗∈𝐽(𝑦,V𝑛 )
:= 𝑀𝑓 𝑆1 𝑆2 .
(61)
We establish an upper bound for 𝑆1 . Since 𝑖 ∈ 𝐼(𝑥, 𝑢𝑚 ),
clearly
󵄨𝑝
−𝑝 󵄨󵄨
(62)
1 < 𝑢𝑚
󵄨󵄨𝑥𝑚,𝑖 − 𝑥󵄨󵄨󵄨 , 𝑝 ≥ 1,
and we can write
𝑆1 ≤ (1 + 2𝑝−1 𝑥𝑝 )
+ 2𝑝−1
∞
𝑘
(𝑆𝑛 𝑓) (𝑥) = ∑ 𝑝𝑛,𝑘 (𝑥) 𝑓 ( ) ,
𝑛
𝑘=0
1
󵄨
󵄨𝑝
∑ 𝑎𝑚,𝑖 (𝑥) 󵄨󵄨󵄨𝑥𝑚,𝑖 − 𝑥󵄨󵄨󵄨
𝑝
𝑢𝑚 𝑖∈𝐼 𝑥,𝑢
( 𝑚)
󵄨
󵄨𝑝
∑ 𝑎𝑚,𝑖 (𝑥) 󵄨󵄨󵄨𝑥𝑚,𝑖 − 𝑥󵄨󵄨󵄨
𝑖∈𝐼(𝑥,𝑢𝑚 )
1 + 2𝑝−1 𝑥𝑝
≤ (
+ 2𝑝−1 ) M1/2
𝑝
2𝑝 (𝐴 𝑚 ; 𝑥)
𝑢𝑚
𝑝−1 𝑝
𝑒−𝑛𝑥 (𝑛𝑥)𝑘
𝑝𝑛,𝑘 (𝑥) =
,
𝑘!
(63)
𝑟−1
1+2 𝑥
≤ (
+ 2𝑝−1 ) √
𝑝
𝑚𝑝
𝑢𝑚
𝑟−1
M2𝑟+1 (𝑆𝑛 ; 𝑥) = ∑
Γ2𝑞 (𝑦)
1 + 2𝑞−1 𝑦𝑞
+ 2𝑞−1 ) √
.
𝑞
𝑛𝑞
V𝑛
𝑗=0𝑛
(64)
Considering (14), relation (61) leads to the claimed result.
5. Particular Cases
In presenting these cases, we are looking for one-dimensional
linear operators that verify conditions (2) and (4).
(1) Baskakov operators [5] are given by the formula
∞
𝑘
(𝑉𝑛 𝑓) (𝑥) = ∑ 𝑏𝑛,𝑘 (𝑥) 𝑓 ( ) ,
𝑛
𝑘=0
𝑛+𝑘−1 𝑘
𝑏𝑛,𝑘 (𝑥) = (
) 𝑥 (1 + 𝑥)−𝑛−𝑘 ,
𝑘
𝑞𝑗
𝑟+𝑗
𝑗=0𝑛
(see (26) and (4)). Arguing similarly we get
𝑆2 ≤ (
𝑥 ≥ 0.
One has 𝑆𝑛 𝑒0 = 𝑒0 and M1 (𝑆𝑛 ; 𝑥) = 0. The central moments
have the following form:
M2𝑟 (𝑆𝑛 ; 𝑥) = ∑
Γ2𝑝 (𝑥)
(65)
𝑥 ≥ 0,
where 𝑉𝑛 𝑒0 = 𝑒0 , hence (2) fulfilled. Since 𝑉𝑛 reproduces the
monomial 𝑒1 , 𝑒1 (𝑡) = 𝑡, its first central moment M1 (𝑉𝑛 ; ⋅) is
null.
For 𝑟 ≥ 2, the 𝑟th central moment is given as follows [6,
Lemma 4]:
[𝑟/2]
𝑥 (1 + 𝑥) 𝑗 1 + 2𝑥 𝛿𝑟
)(
) ,
M𝑟 (𝑉𝑛 ; 𝑥) = ∑ 𝑏𝑛,𝑟,𝑗 (
𝑛
𝑛
𝑗=1
(66)
(67)
𝑥𝑟−𝑗 ,
𝑝𝑗
𝑟+𝑗+1
(68)
𝑥
𝑟−𝑗
,
where 𝑞𝑗 , 𝑝𝑗 are constants; see, for example, [11, Equations
(9.5.10)-(9.5.11)]. For this class, conditions (2) and (4) are
achieved.
The research of 𝑆𝑛 , 𝑛 ∈ N, operators in polynomial
weighted spaces has appeared in [6]. The truncated univariate
Szász operators and another extension to functions of two
variables in weighted spaces have been considered in [2] and
[12], respectively. In the latter paper instead 𝑤𝑝,𝑞 was used the
weight 𝜌, 𝜌(𝑥, 𝑦) = 1 + 𝑥2 + 𝑦2 .
Our theorems of the previous section lead us to twodimensional versions of genuine Szász operators and of their
truncated form. In this case the net is Δ 𝑚,𝑛 = (𝑖/𝑚, 𝑗/𝑛)𝑖,𝑗≥0 .
The next example comes from the world of Quantum
Calculus which, in the past two decades, has gained popularity in the construction of linear approximation processes.
We choose a 𝑞-analogue of Szász-Mirakjan operators recently
introduced and studied by Mahmudov [13].
(3) 𝑞-Szász-Mirakjan-Mahmudov operators. Let 𝑞 > 1
and 𝑛 ∈ N. For 𝑓 : R+ → R one defines the operator
∞
(𝑀𝑛,𝑞 𝑓) (𝑥) = ∑ 𝑓 (
𝑘=0
[𝑘]𝑞
[𝑛]𝑞
)
[𝑛]𝑘𝑞 𝑥𝑘
𝑞𝑘(𝑘−1)/2 [𝑘]𝑞 !
𝑒𝑞 (−[𝑛]𝑞 𝑞−𝑘 𝑥) ,
(69)
8
Abstract and Applied Analysis
𝑗+1
where 𝑒𝑞 (𝑧) = ∏∞
)). We recall the
𝑗 = 0 (1 + (𝑞 − 1)(𝑧/𝑞
standard notations in 𝑞-calculus. For 𝑛 ∈ N consider
1 − 𝑞𝑛
{
, if 𝑞 ∈ (0, 1) ∪ (1, ∞) ,
[𝑛]𝑞 = { 1 − 𝑞
if 𝑞 = 1;
{𝑛,
(70)
[𝑛]𝑞 ! = [1]𝑞 [2]𝑞 ⋅ ⋅ ⋅ [𝑛]𝑞 .
Also, [0] = 0 and [0]! = 1. In [13] it was proved that
𝑀𝑛,𝑞 𝑒0 = 𝑒0 and 𝑀𝑛,𝑞 is a linear positive operator from
𝐶𝑝 (R+ ) to 𝐶𝑝 (R+ ) for any 𝑝 ∈ N0 . Withal, for all moments
𝑀𝑛,𝑞 𝑒𝑟 , 𝑒𝑟 (𝑡) = 𝑡𝑟 , explicit formulas were given as follows:
𝑟
(𝑀𝑛,𝑞 𝑒𝑟 ; 𝑥) = ∑ 𝑠𝑞 (𝑟, 𝑘)
𝑘=1
𝑥𝑘
,
[𝑛]𝑟−𝑗
𝑞
(71)
where
𝑠𝑞 (0, 0) = 1,
𝑠𝑞 (𝑟, 0) = 0 for 𝑟 > 0,
𝑠𝑞 (𝑟, 𝑘) = 0 for 𝑟 < 𝑘,
𝑠𝑞 (𝑟 + 1, 𝑘) = [𝑘]𝑞 𝑠𝑞 (𝑟, 𝑘) + 𝑠𝑞 (𝑟, 𝑘 − 1)
for 𝑟 ≥ 0,
(72)
𝑘 ≥ 1,
represent 𝑞-analogue of Stirling numbers; see [13, Lemma
2.6]. One can see that (𝑀𝑛,𝑞 𝑒𝑟 )(𝑥) is a polynomial in 𝑥 of
degree 𝑟 without a constant term, and it is bounded with
respect to [𝑛]𝑞 . These properties are transferred to the central
moments 𝑀𝑛,𝑞 (𝜑𝑥𝑟 ; 𝑥), and consequently (4) takes place, provided to replace 𝑛 with [𝑛]𝑞 . To construct two-dimensional
operators of the form (10) and (16) we choose 𝐴 𝑚 = 𝑀𝑚,𝑞 ,
𝐵𝑛 = 𝑀𝑛,𝑞 , and the network will be
Δ 𝑚,𝑛 = (
[𝑖]𝑞 [𝑗]𝑞
,
) .
[𝑚]𝑞 [𝑛]𝑞 𝑖,𝑗≥0
(73)
In time were carried out 𝑞-analogues of these operators
not only for 𝑞 > 1 but for the case 𝑞 ∈ (0, 1); see, for
example, [14, 15]. Extensions of these classes of operators by
our method also work there.
References
[1] J. Gróf, “Über approximation durch polynome mit belegungsfunktion,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 35, no. 1-2, pp. 109–116, 1980.
[2] H. G. Lehnhoff, “On a modified Szasz-Mirakjan-operator,” Journal of Approximation Theory, vol. 42, no. 3, pp. 278–282, 1984.
[3] O. Agratini, “On the convergence of a truncated class of operators,” Bulletin of the Institute of Mathematics. Academia Sinica,
vol. 31, no. 3, pp. 213–223, 2003.
[4] G. A. Anastassiou and S. G. Gal, Approximation Theory. Moduli
of Continuity and Global Smoothness Preservation, Birkhäuser,
Boston, Mass, USA, 2000.
[5] V. A. Baskakov, “An example of a sequence of linear positive operators in the space of continuous functions,” Doklady
Akademii Nauk SSSR, vol. 113, pp. 249–251, 1957.
[6] M. Becker, “Global approximation theorems for Szász-Mirakjan
and Baskakov operators in polynomial weight spaces,” Indiana
University Mathematics Journal, vol. 27, no. 1, pp. 127–142, 1978.
[7] M. Gurdek, L. Rempulska, and M. Skorupka, “The Baskakov
operators for functions of two variables,” Collectanea Mathematica, vol. 50, no. 3, pp. 289–302, 1999.
[8] J. Wang and S. Zhou, “On the convergence of modified Baskakov operators,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 28, no. 2, pp. 117–123, 2000.
[9] Z. Walczak, “Baskakov type operators,” The Rocky Mountain
Journal of Mathematics, vol. 39, no. 3, pp. 981–993, 2009.
[10] O. Szász, “Generalization of S. Bernstein’s polynomials to the
infinite interval,” Journal of Research of the National Bureau of
Standards, vol. 45, pp. 239–245, 1950.
[11] Z. Ditzian and V. Totik, Moduli of Smoothness, vol. 9, Springer,
New York, NY, USA, 1987.
[12] N. İspir and C. Atakut, “Approximation by modified SzaszMirakjan operators on weighted spaces,” Proceedings of the
Indian Academy of Sciences, vol. 112, no. 4, pp. 571–578, 2002.
[13] N. I. Mahmudov, “Approximation by the q-Szász-Mirakjan
operators,” Abstract and Applied Analysis, vol. 2012, Article ID
754217, 16 pages, 2012.
[14] A. Aral, “A generalization of Szász-Mirakyan operators based
on q-integers,” Mathematical and Computer Modelling, vol. 47,
no. 9-10, pp. 1052–1062, 2008.
[15] A. Aral and V. Gupta, “The q-derivative and applications to qSzász Mirakyan operators,” Calcolo, vol. 43, no. 3, pp. 151–170,
2006.
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The Scientific
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International Journal of
Differential Equations
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International Journal of
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Mathematical Physics
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Journal of
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International
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Stochastic Analysis
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International Journal of
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Discrete Dynamics in
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Optimization
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