Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 217464, 12 pages
doi:10.1155/2012/217464
Research Article
Asymptotic Formulae via a Korovkin-Type Result
Daniel Cárdenas-Morales,1 Pedro Garrancho,1 and Ioan Raşa2
1
Departamento de Matemáticas, Universidad de Jaén, Campus Las Lagunillas s/n.,
23071 Jaén, Spain
2
Department of Mathematics, Technical University, Str. C. Daicoviciu 15,
400020 Cluj-Napoca, Romania
Correspondence should be addressed to Daniel Cárdenas-Morales, cardenas@ujaen.es
Received 31 December 2011; Accepted 15 February 2012
Academic Editor: Gabriel Turinici
Copyright q 2012 Daniel Cárdenas-Morales et al. 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.
We present a sort of Korovkin-type result that provides a tool to obtain asymptotic formulae for
sequences of linear positive operators.
1. Introduction
This paper deals with the approximation of continuous functions by sequences of positive
linear operators. In this setting, on studying a sequence of operators, say Ln , one usually
intends to prove firstly that the sequence defines an approximation process; that is, for each
function f of some space, Ln f converges to f, in a certain sense as n tends to infinity;
afterwards, one searches for quantitative results that estimate the degree of convergence, and
finally one measures the goodness of the estimates, mainly through inverse and saturation
results. An outstanding tool to achieve these saturation results is given by the asymptotic
formulae that provide information about the so-called optimal degree of convergence. The
most representative expression of this type was stated for the classical Bernstein operators
by Voronovskaja in 1 and reads as follows: for f ∈ C0, 1 and x ∈ 0, 1, if f x
exists, then
x1 − x f x.
lim n Bn fx − fx 2
n→∞
This work dwells upon this type of expressions.
1.1
2
Abstract and Applied Analysis
A classical key ingredient to prove an asymptotic formula for a sequence of positive
linear operators Ln is Taylor’s theorem. From it, the formula appears after some minor work
if one is able to find easy-to-use expressions for the first moments of the operators, namely,
Ln eix x Ln e1 − xe0 i x,
i 0, 1, 2, 4,
1.2
ei and eix denote the monomials ei t ti and eix t t − xi . On the contrary, if the
calculation of the moments gets complicated, then a drawback appears.
The main purpose of this paper is to present a tool that helps to overcome these
situations. The basic ideas behind the main result Theorem 2.1 below lies in 2, Chapter
5, the novelty here being that instead of using the Taylor formula of the function f to be
approximated, we consider the Taylor formula of f ◦ ϕ−1 for a certain function ϕ.
It is the intention of the authors that the paper offers a clear and quick procedure to
obtain asymptotic expressions for a wide variety of sequences of linear positive operators.
2. The Main Result
Let ϕ be any ∞-times continuously differentiable function on 0, 1, such that ϕ0 0, ϕ1 x
the function
1, and ϕ x > 0 for x ∈ 0, 1. We denote by eϕ,i
i
x
eϕ,i
t ϕt − ϕx ,
2.1
and Di denotes the usual ith differential operator, though we keep on using the common
notation f and f for the first and second derivatives of a function f.
Although we restrict our attention to the space C0, 1 of all continuous functions
defined on 0, 1, the following main result of the paper remains valid for any compact real
interval with the obvious modifications.
Theorem 2.1. Let Ln : C0, 1 → C0, 1 be a sequence of linear positive operators, and let x ∈ 0, 1
be fixed. Let us assume that there exist a sequence of positive real numbers λn → ∞ (as n → ∞)
and two functions p, q ∈ C0, 1, p being strictly positive on 0, 1, such that for all i ∈ {0, 1, 2, 4}
x
x
x
x
lim λn Ln eϕ,i
x − eϕ,i
x pxD2 eϕ,i
x qxD1 eϕ,i
x.
n → ∞
2.2
Then for each f ∈ C0, 1, twice differentiable at the point x,
lim λn Ln fx − fx pxf x qxf x.
n → ∞
2.3
Remark 2.2. Notice that for i 0, identity 2.2 becomes limn → ∞ λn Ln e0 x − 1 0, which
is obviously fulfilled if the operators Ln preserve the constants. This property is satisfied by
most classical sequences of linear operators see, e.g., 3, among them being the ones that
we study in the present paper.
Abstract and Applied Analysis
3
Moreover, for i 1, 2, 4, identity 2.2 becomes, respectively,
x
lim λn Ln eϕ,1
x pxϕ x qxϕ x,
n → ∞
x
lim λn Ln eϕ,2
x 2pxϕ x2 ,
2.4
n → ∞
x
lim λn Ln eϕ,4
x 0.
n → ∞
These are, actually, the identities that we will explicitly use throughout the paper. However,
we have decided to write the hypotheses of the theorem as in 2.2 for the sake of brevity
and to put across that we can think of the result as if it were of Korovkin type, since we
can guarantee the convergence of λn Ln fx − fx and obtain its limit for any f ∈ C0, 1,
whenever we have it for four test functions.
Proof. The classical Taylor theorem, applied to the function f ◦ ϕ−1 , yields for t ∈ 0, 1 that
ft f ◦ ϕ−1 ϕt f ◦ ϕ−1 ϕx D1 f ◦ ϕ−1 ϕx ϕt − ϕx
D2 f ◦ ϕ−1 ϕx 2
2
ϕt − ϕx ht − x ϕt − ϕx ,
2
2.5
where h is a continuous function which vanishes at 0. Equivalently we can write for t ∈ 0, 1
D2 f ◦ ϕ−1 ϕx x
x
x
ft fx D1 f ◦ ϕ−1 ϕx eϕ,1
eϕ,2 t hx teϕ,2
t t,
2
2.6
where hx t : ht − x. Applying the operator Ln and then evaluating at the fixed point x, we
obtain the equality
Ln fx x
fxLn eϕ,0
x
D
1
−1
f ◦ϕ
ϕx
x
Ln eϕ,1
x
D2 f ◦ ϕ−1 ϕx
x
Ln eϕ,2
x
2
x
Ln hx eϕ,2
x.
2.7
Now we subtract fx from both sides and multiply by λn to get
x
x
λn Ln fx − fx − λn Ln hx eϕ,2
x − 1
x fxλn Ln eϕ,0
x
D1 f ◦ ϕ−1 ϕx λn Ln eϕ,1
x
2
−1
D f ◦ϕ
ϕx
x
λn Ln eϕ,2
x.
2
2.8
4
Abstract and Applied Analysis
Taking into account the basic identities
f x
D1 f ◦ ϕ−1 ϕx ,
ϕ x
f xϕ x − f xϕ x
1
2
−1
ϕx D f ◦ϕ
2
ϕ x
ϕ x
2.9
and using 2.4, we derive that
lim
n → ∞
x
λn Ln fx − fx − λn Ln hx eϕ,2
x pxf x qxf x.
2.10
x
x → 0 as n → ∞.
The proof will be over once we prove that λn Ln hx eϕ,2
To this purpose let > 0 and let θx be an open set containing x such that for t ∈ θx ,
x
t, we have that for all t ∈ 0, 1
|hx t| < . Then if we define wt : max{0, |hx t| − }eϕ,2
x
x
x
x
t ≤ eϕ,2
t max{0, |hx t| − }eϕ,2
t eϕ,2
t wt.
hx teϕ,2
2.11
On the other hand, w vanishes on θx , so there is a constant M such that for all t ∈ 0, 1,
x
t.
|wt| ≤ Meϕ,4
2.12
Finally, the linearity and positivity of Ln allow us to write, from 2.11 and 2.12,
x
x
x
x λn MLn eϕ,4
x,
x ≤ λn Ln eϕ,2
λn Ln hx eϕ,2
2.13
from where taking limits and using again 2.4,
x
lim sup λn Ln hx eϕ,2
x ≤ 2pxϕ x2 .
n → ∞
2.14
This ends the proof, as px and ϕ x are strictly positive and was arbitrary.
3. Applications
The first two applications correspond to sequences of operators recently introduced in 4, 5.
They represent respective modifications of the classical Bernstein operators and the wellknown modified Meyer-König and Zeller operators see 6 which, instead of preserving
the linear functions, hold fixed e0 and e2 . Thus they are inside this new line of work which
originated with the paper 7 and found further development in a long list of papers see,
e.g., 8–14.
The aforementioned preserving property usually makes it quite difficult to compute
the first moments and consequently to obtain asymptotic formulae. Here our result, applied
with λn n and ϕ e2 , enters the scene.
Abstract and Applied Analysis
5
Two further applications with different values of ϕ are presented afterwards in less
detail.
3.1. Modified Bernstein Operators Which Preserve x2
This section deals with the following sequence of operators presented in 4 we use the same
notation as a byproduct of some interesting results, defined for f ∈ C0, 1 and n > 1 as
⎞
⎛
kk
−
1
n k
⎠
⎝
x 1 − xn−k .
f
Bn,0,2 fx k
nn
−
1
k0
n
3.1
As pointed out in 13, this provides an example of a sequence of positive linear polynomial
operators that preserve e0 and e2 and represents an approximation process for functions f ∈
C0, 1.
The presence of the square root in the definition makes it difficult to obtain easy-tohandle expressions for the moments Bn,0,2 eix x, i 0, 1, 2, 4, and consequently to obtain an
asymptotic formula. We will apply our theorem to get it, though we first prove a quantitative
result missed in 4.
Proposition 3.1. Let f ∈ C0, 1, x ∈ 0, 1, and let δ > 0. Then
⎛
⎞
n−1
1
1
−
−
x
1
Bn,0,2 fx − fx ≤ ω f, δ ⎝1 ⎠.
2x1 − x
δ
n−1
3.2
Proof. From the usual quantitative estimate in terms of ωf, · stated in 15, we can write
Bn,0,2 fx − fx ≤ ω f, δ 1 1 2xx − Bn,0,2 e1 x .
δ
Now, for n > 1, k − 1/n − 1 ≤ k/n, so
Thus
3.3
kk − 1/nn − 1 ≤ k/n and Bn,0,2 e1 ≤ Bn e1 e1 .
x − Bn,0,2 e1 x ≥ 0.
On the other hand, for n > 1, 1/2k/nk−1/n−1 ≥
so
3.4
kk − 1/nn − 1 ≥ k−1/n−1,
n n
k−1 n k
1
k k−1
n k
x 1 − xn−k ≥ Bn,0,2 e1 x ≥
x 1 − xn−k ,
k
k
2 k1 n n − 1
n
−
1
k1
3.5
6
Abstract and Applied Analysis
and using the equalities
n
k n
n
k−1 n k
x 1 − xn−k
k
n
−
1
k1
xk 1 − xn−k x,
k
n
k1
n n nx − 1 1 − xn
k 1
n k
−
,
x 1 − xn−k k
n − 1 k1 n n
n−1
n−1
3.6
we get
1−x 1 − x
1 − 1 − xn−1 ≤ x − Bn,0,2 e1 x ≤
1 − 1 − xn−1 ,
2n − 1
n−1
3.7
from which the result follows.
Corollary 3.2. For all f ∈ C0, 1 and x ∈ 0, 1, whenever f x exists,
1−x 1 − xx f x f x.
lim n Bn,0,2 fx − fx −
n→∞
2
2
3.8
Proof. We will apply the theorem with ϕ e2 , λn n, qx −1 − x/2, and px 1 −
xx/2. As the operators Bn,0,2 preserve the constants, it suffices to check that 2.4 holds true.
First computations yield
Bn,0,2 e0 x 1,
Bn,0,2 e2 x x2 ,
4n − 2 3 n − 3n − 2 4
2
x2 x x ,
nn − 1
nn − 1
nn − 1
32n − 2 3 38n − 3n − 2 4
4
Bn,0,2 e6 x x2 x x
2
2
n n − 1
n2 n − 12
n2 n − 12
12n − 4n − 3n − 2 5 n − 5n − 4n − 3n − 2 6
x x ,
n2 n − 12
n2 n − 12
208n − 2 3 652n − 2n − 3 4
8
Bn,0,2 e8 x x2 x x
3
3
n n − 1
n3 n − 13
n3 n − 13
576n − 4n − 3n − 2 5 188n − 5n − 4n − 3n − 2 6
x x
n3 n − 13
n3 n − 13
24n − 6n − 5n − 4 n − 3n − 2 7
x
n3 n − 13
n − 7n − 6n − 5n − 4n − 3n − 2 8
x .
n3 n − 13
Bn,0,2 e4 x 3.9
Abstract and Applied Analysis
7
x
appear after some calculations using the following
Thus for i 1, 2, 4 the quantities Bn,0,2 eϕ,i
identities:
x
eex2 ,1 e2 − x2 e0 ,
eϕ,1
x
eϕ,2
eex2 ,2 e4 − 2x2 e2 x4 e0 ,
x
eex2 ,4 e8 − 4x2 e6 6x4 e4 − 4x6 e2 x8 e0 ,
eϕ,4
x
Bn,0,2 eϕ,1
x 0,
x
Bn,0,2 eϕ,2
x x
Bn,0,2 eϕ,4
x 21 − xx2
1 − 3x 2nx,
n − 1n
3.10
41 − xx2 n −
13 n3
n4 12 − 12xx4 n3 112 − 324x 216x2 x3
n2 159 − 1041x 1881x2 − 1011x3 x2
n 52 − 759x 2921x2 − 4089x3 1887x4 x
2 − 102x 876x2 − 2580x3 3060x4 − 1260x5 .
Finally we are in a position to apply Theorem 2.1 and then prove the corollary, since
we show below that assumptions in 2.4 are fulfilled:
x
lim nBn,0,2 eϕ,1
x 0 2xqx 2px qxϕ x pxϕ x,
n→∞
x
lim nBn,0,2 eϕ,2
x 41 − xx3 8x2 px 2pxϕ x2 ,
n→∞
3.11
x
lim nBn,0,2 eϕ,4
x 0.
n→∞
3.2. The Modified Meyer-König and Zeller Operators
For f ∈ C0, 1, x ∈ 0, 1, and n > 1, we consider the operators defined as
⎛
∞
Rn fx f⎝
k0
⎞
kk − 1
⎠ n k xk 1 − xn1 ,
k
n kn k − 1
3.12
for x < 1 and Rn f1 f1.
They were introduced in 5 as a modification of the well-known modified MeyerKönig and Zeller operators see 6.
It turns to be another example of a sequence of positive linear operators that preserve
e0 and e2 and represents an approximation process for functions f ∈ C0, 1.
8
Abstract and Applied Analysis
Here again, the presence of the square root in the definition makes it difficult to obtain
easy-to-handle expressions for Rn eix x, i 0, 1, 2, 4, and consequently to obtain by usual
means an asymptotic formula. With this aim we shall apply our theorem.
Corollary 3.3. For all f ∈ C0, 1 and x ∈ 0, 1, whenever f x exists,
x1 − x2 1 − x2 f x f x.
lim n Rn fx − fx −
n→∞
2
2
3.13
Proof. We will apply the theorem with ϕ e2 , λn n, qx −1 − x2 /2 and px x1 −
x2 /2. As the operators Rn preserve the constants, it suffices to check that 2.4 holds true.
Direct computations with the use of mathematical software Mathematica give
Rn e0 x 1,
Rn e2 x x2 ,
Rn e4 x n2 1 nx4
4nx3
2 n3 n 2 n3 n4 n
Rn e6 x 12n8 x5
1 n2 n2 3 n2 4 n2 5 n2
Rn e8 x 4n3 x5
O n−2 ,
2 n3 n4 n5 n
n10 17 nx6
1 n2 n2 3 n2 4 n2 5 n2 6 n2
12n12 x7
2
2
2
2
2
1 n2 n 3 n 4 n 5 n 6 n 7 n
2
O n−2 ,
24n19 x7
1 n2 2 n3 3 n3 4 n3 5 n3 6 n3 7 n3
n22 59 nx8
1 n2 2 n3 3 n3 4 n3 5 n3 6 n3 7 n3 8 n3
24n25 x9
1 n2 2 n3 3 n3 4 n3 5 n3 6 n3 7 n3 8 n3 9 n3
Now, using again 3.10 we can write
x
Rn eϕ,1
x 0,
x
Rn eϕ,2
x 4x3 n3 1 − x2
O n−2 ,
2 n3 n4 n5 n
O n−2 .
3.14
Abstract and Applied Analysis
x
Rn eϕ,4
x 9
n25 x7 24 59x 24x2
1 n2 2 n3 3 n3 4 n3 5 n3 6 n3 7 n3 8 n3 9 n3
−
3n22 x8
1 n 2 n 3 n 4 n3 5 n3 6 n3 7 n3 8 n3
4n12 x7 12 17x 12x2
2
3
3
1 n2 n2 3 n2 4 n2 5 n2 6 n2 7 n2
6n3 x7 4 x 4x2
O n−2
2 n3 n4 n5 n
3.15
and then the following identities which prove the corollary:
x
lim nRn eϕ,1
x 0 2xqx 2px qxϕ x pxϕ x,
n→∞
x
lim nRn eϕ,2
x 4x3 1 − x2 8x2 px 2pxϕ x2 ,
n→∞
3.16
x
lim nMn,4 eϕ,4
x 0.
n→∞
3.3. The Modified Bernstein Operators Which Preserve a General Function τ
Let τ be any function fulfilling the same properties as the general function ϕ considered in
the paper. For f ∈ C0, 1, x ∈ 0, 1, and n > 0, we consider the following operators defined
from the classical Bernstein operators Bn :
Bnτ fx n k n
.
τxk 1 − τxn−k f ◦ τ −1
k
n
k0
3.17
This sequence of linear operators was studied by the authors in 13. Here we show a nice
way to obtain its asymptotic formula.
Corollary 3.4. For all f ∈ C0, 1 and x ∈ 0, 1, whenever f x exists,
τx1 − τxτ x f x
lim n Bnτ fx − fx −
n → ∞
2τ x3
τx1 − τx f x.
2τ x2
3.18
10
Abstract and Applied Analysis
Proof. It follows the same pattern as the proof of the previous corollaries. It suffices to make
use of the following identities which one can obtain directly from the corresponding ones for
the Bernstein operators i.e., Bn eix x, i 1, 2, 4; see, e.g., 2:
x
Bnτ eτ,1
x 0,
x
Bnτ eτ,2
x x
Bnτ eτ,4
x τx1 − τx
,
n
31 − τx2 τx2
n2
τx1 − τx 1 − 6τx 6τx2
.
n3
3.19
3.4. The Modified Bernstein Operators Which Preserve xj
This section deals with the family of sequences of positive linear polynomial operators
Bn,0,j , j 1, 2, . . ., presented in 4 and defined for f ∈ C0, 1 and n ≥ j as
Bn,0,j fx n
k0
⎛
f⎝
1/j ⎞ kk − 1 · · · k − j 1
⎠ n xk 1 − xn−k .
k
nn − 1 · · · n − j 1
3.20
The first two elements Bn,0,1 and Bn,0,2 are, respectively, the Bernstein operators and those ones
studied in Section 3.1. The operator Bn,0,j holds fixed the functions e0 and ej .
The next corollary deals with Bn,0,3 and shows an application of Theorem 2.1 with ϕ e3 .
Corollary 3.5. For all f ∈ C0, 1 and x ∈ 0, 1, whenever f x exists,
x1 − x f x − 1 − xf x.
lim n Bn,0,3 fx − fx n → ∞
2
3.21
Proof. It follows the same pattern as Corollary 3.2 although some more cumbersome
calculations, which we have carried out with the use of Mathematica, are required. We detail
Abstract and Applied Analysis
11
below the identities that allow us to end the proof. We make use of the notation
n − 3i n − 3n − 4 · · · n − i − 2,
Bn,0,3 e0 x 1,
Bn,0,3 e3 x x3 ,
Bn,0,3 e6 x Bn,0,3 e9 x 9n − 32
18n − 3
6
n − 33
x6 x5 x4 x3 ,
nn − 1n − 2
nn − 1n − 2
nn − 1n − 2
nn − 1n − 2
n − 36
n2 n − 12 n − 2
Bn,0,3 e12 x 862n − 33
n2 n
2
− 1 n − 2
n2 n − 12 n − 22
n3 n − 13 n − 2
n2 n − 12 n − 2
n3 n − 13 n − 2
186876n − 33
3
n3 n − 1 n − 2
3
− 1 n − 2
n2 n
2
− 1 n − 2
n3 n − 13 n − 2
x6 3
3
1242n − 32
54n − 38
x9 3
216
x8 2
2
243n − 34
n2 n − 12 n − 22
x5 x7
540n − 3
n2 n
− 12 n − 22
x4
x3 ,
x12 3
11025n − 36
n3 n
27n − 35
x6 2
36
n − 39
x9 2
x11 3
56808n − 35
n3 n − 13 n − 2
x8 3
94284n − 32
3
n3 n − 1 n − 2
3
x5 1107n − 37
n3 n − 13 n − 23
x10
149580n − 34
n3 n − 13 n − 23
13608n − 3
n3 n − 13 n − 23
x7
x4
x3 .
3.22
x
Bn,0,3 eex3 ,i , required to apply Theorem 2.1, appear after
For i 1, 2, 4 the quantities Bn,0,3 eϕ,i
some calculations using the following identities:
x
eϕ,1
eex3 ,1 e3 − x3 e0 ,
x
eϕ,2
eex3 ,2 e6 − 2x3 e3 x6 e0 ,
x
eϕ,4
eex3 ,4
3.23
e12 − 4x e9 6x e6 − 4x e3 x e0 .
3
6
9
12
Finally, motivated by the well-known Voronovskaja formula for the classical Bernstein
operators and by the results in Corollaries 3.2 and 3.5, we close this section and the paper
stating the following conjecture.
Conjecture 3.6. For all f ∈ C0, 1, x ∈ 0, 1, and j ∈ {4, 5, . . .}, whenever f x exists,
x1 − x 1 − x lim n Bn,0,j fx − fx f x − j − 1
f x.
n → ∞
2
2
3.24
12
Abstract and Applied Analysis
Acknowledgments
The authors thank the referees for their suggestions. The first two authors are partially
supported by the Junta de Andalucı́a Research Group FQM-0178 and by Universidad de
Jaén and Caja Rural de Jaén Project UJA 2009/12/07.
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