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EAST JOURNAL ON APPROXIMATIONS
Volume 17, Number 3 (2011), 299-317
COMMUTATIVITY, DIRECT AND STRONG
CONVERSE RESULTS FOR PHILIPS OPERATORS
Margareta Heilmann and Gancho Tachev
Dedicated to the memory of Borislav Bojanov
We study the so-called Phillips operators which can be considered
as genuine Sz´asz-Mirakjan-Durrmeyer operators. As main results we
prove the commutativity of the operators as well as their commutativity
with an appropriate differential operator and establish a strong converse
inequality of type A for the approximation of real valued continuous
bounded functions 𝑓on [0,∞). Together with the corresponding direct
theorem we derive an equivalence result for the error of approximation
and an appropriate K-functional and modulus of smoothness.
Mathematics Subject Classification (2010): 41A36, 41A25, 41A27,
41A17.
Key words and phrases: Phillips operators, genuine Sz´asz-Mirakjan-
Durrmeyer operators, commutativity, commutativity with differential
operators, Bernstein type inequality, direct result, strong converse re-
sult.
1. Introduction
We consider a sequence
𝑆𝑛of positive linear operators (see [15, 13]) which are
known in the literature as Phillips operators. These operators can also be con-
sidered as genuine Sz´asz-Mirakjan-Durrmeyer operators in the same meaning
as the genuine Bernstein-Durrmeyer operators (see e. g. [14]) and the genuine
Baskakov-Durrmeyer operators (see e. g. [11]), i. e., they commute, preserve
linear functions and commute with an appropriate differential operator.
In [6, Theorem 2] Z. Finta and V. Gupta proved a strong converse inequal-
ity of type B in the terminology of K. G. Ivanov and Z. Ditzian [4] for the
Phillips operators. They use a general theorem developed by V. Totik [17,
Theorem 1] where a direct and a strong converse result of type B is proved
300 Commutativity, direct and strong converse results...
for positive linear operators satisfying certain conditions. In [7] Z. Finta also
proved a general converse result of type B under certain conditions for the
considered operators involving some general weight functions and applied his
results (among others) to the Phillips operators. In [6, 7] estimates for the
difference of Phillips operators and the classical Sz´asz-Mirakjan operators are
used extensively.
The aim of our paper is to establish a direct and a strong converse result
of type A with explicit constants for the Phillips operators. To do so, we
proceed in a similar way as H.-B. Knoop and X. L. Zhou in [12] for the
Bernstein operators. A crucial step in this method is an appropriate strong
Voronovskaja type theorem similar to [4, Lemma 8.3] and good estimates
for the norms of weighted derivatives. In the proof of the strong converse
inequality of type A we take advantage of a nice representation for the iterates
of the operators (see Theorem 3.1) which is specific for the Phillips and the
Sz´asz-Mirakjan-Durrmeyer operators.
Let us mention that we first used a different approach to establish a strong
converse inequality of type A based on the strong converse result of type B
in [6] and especially using Theorem 3.1 and (16) to estimate ∥
𝑆𝑛𝑓−𝑓∥by
𝐾∥
𝑆𝐾𝑛𝑓−𝑓∥,𝐾being a constant greater than 1. However, the constants
appearing in the result of the present paper are much better.
Let 𝑓∈𝐶[0,∞) be a real valued continuous function on [0,∞) satisfying
an exponential growth condition, i. e.,
𝑓∈𝐶𝛼[0,∞) := {𝑓∈𝐶[0,∞) : ∣𝑓(𝑡)∣ ≤ 𝑀 𝑒𝛼𝑡 , 𝑡 ∈[0,∞)}
with some constants 𝑀 > 0 and 𝛼 > 0. Then the Phillips operators
𝑆𝑛,
𝑛 > 𝛼, are defined by
(1)
𝑆𝑛(𝑓, 𝑥) = 𝑛∞
𝑘=1
𝑠𝑛,𝑘(𝑥)∞
0
𝑠𝑛,𝑘−1(𝑡)𝑓(𝑡)𝑑𝑡 +𝑒−𝑛𝑥𝑓(0),
where
𝑠𝑛,𝑘(𝑥) = (𝑛𝑥)𝑘
𝑘!𝑒−𝑛𝑥, 𝑘 ∈ℕ0, 𝑥 ∈[0,∞).
Throughout this paper 𝑓will mostly be considered as a function in 𝐶𝐵[0,∞),
the space of real valued continuous bounded functions on [0,∞) endowed with
the norm ∥𝑓∥= sup𝑥≥0∣𝑓(𝑥)∣. We also need the space
𝑊2
∞(𝜑) = {𝑔∈𝐶𝐵[0,∞) : 𝑔′∈𝐴𝐶𝑙𝑜𝑐[0,∞), 𝜑2𝑔′′ ∈𝐶𝐵[0,∞)},
where here and in what follows 𝜑(𝑥) = √𝑥,𝑥∈[0,∞).
We point out that 𝑛is considered as a natural number in [6, 7], but in our
statements here 𝑛may be considered also as an arbitrary positive number.
Margareta Heilmann and Gancho Tachev 301
The Phillips operators are closely related to the Sz´asz-Mirakjan operators [16]
defined by
(2) 𝑆𝑛(𝑓, 𝑥) = ∞
𝑘=0
𝑠𝑛,𝑘(𝑥)𝑓𝑘
𝑛,
to its Kantorovitch variants
𝑆𝑛(𝑓, 𝑥) = 𝑛∞
𝑘=0
𝑠𝑛,𝑘(𝑥)𝑘+1
𝑛
𝑘
𝑛
𝑓(𝑡)𝑑𝑡
and the Durrmeyer version
𝑆𝑛(𝑓, 𝑥) = 𝑛∞
𝑘=0
𝑠𝑛,𝑘(𝑥)∞
0
𝑠𝑛,𝑘(𝑡)𝑓(𝑡)𝑑𝑡.
All these operators preserve constants and
𝑆𝑛and 𝑆𝑛also preserve linear
functions and interpolate 𝑓at 0.
The operators
𝑆𝑛and 𝑆𝑛are connected in the same way as the operators
𝑆𝑛and
𝑆𝑛, i. e.,
(𝑆𝑛𝑓)′=
𝑆𝑛𝑓′and (
𝑆𝑛𝑓)′=𝑆𝑛𝑓′
(3)
if 𝑓∈𝐶1
𝛼[0,∞) = {𝑓∈𝐶1[0,∞) : 𝑓, 𝑓 ′∈𝐶𝛼[0,∞)}. For the proof of
(
𝑆𝑛𝑓)′=𝑆𝑛𝑓′see (19).
The paper is organized as follows. In Section 2 we give some basic and
auxiliary results such as the moments, the image of the Phillips operators
for monomials and some identities which will be used throughout the paper.
Section 3 is devoted to the proof of the commutativity results. It turns out
that the commutativity of the Phillips operators can be derived as a corollary
from a nice representation of
𝑆𝑛(
𝑆𝑚) as a Phillips operator. The strong
Voronovkaja type result is proved in Section 4. In Section 5 we prove a direct
and a strong converse result of type A for the Phillips operators with explicit
constants.
2. Basic results
In this section we collect some elementary and basic results which will be used
throughout this paper. First we list some identities for the basis functions 𝑠𝑛,𝑘
which follow directly from their definition. For the sake of simplicity in the
302 Commutativity, direct and strong converse results...
notation we set 𝑠𝑛,𝑘(𝑥) = 0 for 𝑘 < 0. The identities we shall need are:
∞
𝑘=0
𝑠𝑛,𝑘(𝑥) = 1,(4)
∞
0
𝑠𝑛,𝑘(𝑡)𝑑𝑡 =1
𝑛,(5)
𝑠′
𝑛,𝑘(𝑥) = 𝑛[𝑠𝑛,𝑘−1(𝑥)−𝑠𝑛,𝑘 (𝑥)],(6)
𝜑(𝑥)2𝑠′
𝑛,𝑘(𝑥) = (𝑘−𝑛𝑥)𝑠𝑛,𝑘 (𝑥),(7)
𝜑(𝑥)4𝑠′′
𝑛,𝑘 = [(𝑘−𝑛𝑥)2−𝑘]𝑠𝑛,𝑘(𝑥),(8)
𝜑(𝑥)2𝑠𝑛,𝑘(𝑥)𝑠𝑛,𝑘+1 (𝑡) = 𝑠𝑛,𝑘+1 (𝑥)𝜑(𝑡)2𝑠𝑛,𝑘(𝑡).(9)
We will also need the second moment of the original Sz´asz-Mirakjan operators
𝑆𝑛given by
(10) 𝑆𝑛((𝑡−𝑥)2, 𝑥) = 𝑥
𝑛.
In our first lemma we state an explicit formula for the moments and the im-
ages of the Phillips operators for monomials. Note that there exists a formula
for the image of the monomials by the Phillips operators given in [2, p. 1504],
where the coefficients are given in terms of a recursion formula. By com-
paring the moments of the Phillips operators with the functions 𝐻𝑛,𝑚 in [9,
Lemma 4.10], case 𝑐= 0, one can see that apart from a factor (−1)𝑚the func-
tions 𝐻𝑛,𝑚 are exactly the moments of the Phillips operators. In [9] a recursion
formula and other representations are proved but now we are able to give an
explicit formula. The same connection holds true for the functions 𝐻𝑛−1,𝑚
in [3, Lemma 6.4] and the moments of the genuine Bernstein-Durrmeyer op-
erators as well as for the functions 𝐻𝑛+1,𝑚 in [9, Lemma 4.10], case 𝑐= 1 for
the genuine Baskakov-Durrmeyer operators.
Throughout this paper we denote by 𝑒𝜈(𝑡) = 𝑡𝜈,𝜈∈ℕ0, the 𝜈-th monomial
and define 𝑓𝜇,𝑥(𝑡) = (𝑡−𝑥)𝜇,𝜇∈ℕ0.
Lemma 2.1. For the images of the operators
𝑆𝑛for the monomials we
have
𝑆𝑛(𝑒0, 𝑥) = 1,
𝑆𝑛(𝑒𝜈, 𝑥) =
𝜈
𝑗=1 𝜈−1
𝑗−1𝜈!
𝑗!𝑛𝑗−𝜈𝑥𝑗, 𝜈 ∈ℕ.
Margareta Heilmann and Gancho Tachev 303
The moments of the operators
𝑆𝑛are given by
𝑆𝑛(𝑓0,𝑥, 𝑥) = 1,
𝑆𝑛(𝑓1,𝑥, 𝑥) = 0,
𝑆𝑛(𝑓𝜇,𝑥, 𝑥) =
[𝜇
2]
𝑗=1 𝜇−𝑗−1
𝑗−1𝜇!
𝑗!𝑛𝑗−𝜇𝑥𝑗, 𝜇 ≥2,
where 𝜇
2denotes the integer part of 𝜇
2.
Proof. The identity
𝑆𝑛(𝑒0, 𝑥) = 1 follows directly from (4) and (5). As
𝑆𝑛
interpolates at 0 we can use (3) and the image of 𝑆𝑛for the monomials given
in [10, Satz 4.1] to calculate for 𝜈∈ℕ
𝑥
0
𝑆𝑛(𝜈𝑒𝜈−1, 𝑢)𝑑𝑢 =𝜈
𝜈
𝑗=1 𝜈−1
𝑗−1(𝜈−1)!
(𝑗−1)! 𝑛𝑗−𝜈⋅1
𝑗𝑥𝑗
=𝑥
0
(
𝑆𝑛(𝑒𝜈, 𝑢)′𝑑𝑢
=
𝑆𝑛(𝑒𝜈, 𝑥).
For calculating the moments we follow the lines of [10, (4.6) in Korollar 4.4].
By using the binomial formula, the image of the monomials for
𝑆𝑛, appropriate
transformations for the summation indices and interchanging the order of
summation we get
𝑆𝑛(𝑓𝜇,𝑥(𝑡), 𝑥) =
𝜇
𝜈=0 𝜇
𝜈(−𝑥)𝜇−𝜈
𝑆𝑛(𝑒𝜈, 𝑥)
= (−𝑥)𝜇+
𝜇
𝜈=1 𝜇
𝜈(−𝑥)𝜇−𝜈
𝜈
𝑗=1 𝜈−1
𝑗−1𝜈!
𝑗!𝑛𝑗−𝜈𝑥𝑗
304 Commutativity, direct and strong converse results...
= (−𝑥)𝜇+
𝜇
𝜈=1𝜇
𝜈(−1)𝜇−𝜈
𝜇
𝑗=𝜇−𝜈+1𝜈−1
𝑗+𝜈−𝜇−1𝜈!
(𝑗+𝜈−𝜇)! 𝑛𝑗−𝜇𝑥𝑗
= (−𝑥)𝜇+
𝜇
𝑗=1
𝑥𝑗𝑛𝑗−𝜇
𝜇
𝜈=𝜇−𝑗+1
𝜇
𝜈(−1)𝜇−𝜈𝜈−1
𝑗+𝜈−𝜇−1𝜈!
(𝑗+𝜈−𝜇)!
= (−𝑥)𝜇+
𝜇−1
𝑗=1
𝑥𝑗𝑛𝑗−𝜇
𝑗−1
𝜈=0 𝜇
𝜈+𝜇−𝑗+ 1
×(−1)𝑗−1−𝜈𝜈+𝜇−𝑗
𝜈(𝜈+𝜇−𝑗+ 1)!
(𝜈+ 1)!
= (−𝑥)𝜇+
𝜇−1
𝑗=1
𝑥𝑗𝑛𝑗−𝜇(−1)𝑗−1𝜇!
(𝑗−1)!
×1
𝜇−𝑗
𝑗−1
𝜈=0
(−1)𝜈𝑗−1
𝜈𝜈+𝜇−𝑗
𝜈+ 1
=
[𝜇
2]
𝑗=1 𝜇−𝑗−1
𝑗−1𝜇!
𝑗!𝑛𝑗−𝜇𝑥𝑗.
For the last equality we have used [8, (3.48)]. □
3. Commutativity results
In contrast with the Sz´asz-Mirakjan operators, the Phillips operators have the
very nice property of commutativity
𝑆𝑚(
𝑆𝑛𝑓) =
𝑆𝑛(
𝑆𝑚𝑓). Another impor-
tant property is the commutativity with an appropriate differential operator.
This is a crucial step to establish a strong converse result of type A in the
terminology of [4].
In our first theorem we prove a nice identity for
𝑆𝑚(
𝑆𝑛𝑓). An analo-
gous result was proved by Abel and Ivan in [1] for the operators 𝑆𝑛. From
Theorem 3.1 the commutativity then follows as a corollary as well as a nice
representation for iterates of
𝑆𝑛.
Theorem 3.1. For all 𝑓∈𝐶𝛼[0,∞),𝑚 > 𝛼,𝑛 > 𝛼,𝑚𝑛
𝑚+𝑛> 𝛼 we have
(11)
𝑆𝑚(
𝑆𝑛𝑓) =
𝑆𝑚𝑛
𝑚+𝑛.
Margareta Heilmann and Gancho Tachev 305
Proof. As
𝑆𝑛(𝑓, 0) = 𝑓(0), the proposition is equivalent to
𝑚∞
𝑗=1
𝑠𝑚,𝑗 (𝑥)∞
0𝑠𝑚,𝑗−1(𝑡)𝑛∞
𝑘=1
𝑠𝑛,𝑘(𝑡)∞
0
𝑠𝑛,𝑘−1(𝑦)𝑓(𝑦)𝑑𝑦𝑑𝑡
+𝑓(0)
𝑚∞
𝑗=1
𝑠𝑚,𝑗 (𝑥)∞
0
𝑠𝑚,𝑗−1(𝑡)𝑒−𝑛𝑡 𝑑𝑡 +𝑒−𝑚𝑥
=𝑚𝑛
𝑚+𝑛
∞
𝑙=1
𝑠𝑚𝑛
𝑛+𝑚,𝑙(𝑥)∞
0
𝑠𝑚𝑛
𝑛+𝑚,𝑙−1(𝑦)𝑓(𝑦)𝑑𝑦 +𝑒−𝑚𝑛
𝑛+𝑚𝑥𝑓(0).
(12)
Since
𝑚∞
𝑗=1
𝑠𝑚,𝑗 (𝑥)∞
0
𝑠𝑚,𝑗−1(𝑡)𝑒−𝑛𝑡𝑑𝑡 =𝑚∞
𝑗=1
𝑠𝑚,𝑗 (𝑥)𝑚𝑗−1
(𝑛+𝑚)𝑗
=𝑒−𝑚𝑥 ∞
𝑗=1 𝑚2
𝑛+𝑚𝑥𝑗
𝑗!
=𝑒−𝑚𝑥 𝑒𝑚2
𝑛+𝑚𝑥−1,
we get that (12) is equivalent to
𝑚𝑛 ∞
0
𝑓(𝑦)∞
𝑗=1
∞
𝑘=1
𝑠𝑚,𝑗 (𝑥)𝑠𝑛,𝑘−1(𝑦)∞
0
𝑠𝑚,𝑗−1(𝑡)𝑠𝑛,𝑘 (𝑡)𝑑𝑡
𝑑𝑦
=𝑚𝑛
𝑚+𝑛
∞
𝑙=1
𝑠𝑚𝑛
𝑛+𝑚,𝑙(𝑥)∞
0
𝑠𝑚𝑛
𝑛+𝑚,𝑙−1(𝑦)𝑓(𝑦)𝑑𝑦.
(13)
We denote the left-hand side of (13) by ∞
0𝑓(𝑦)𝑇𝑚,𝑛(𝑥, 𝑦)𝑑𝑦. From
∞
0
𝑠𝑚,𝑗−1(𝑡)𝑠𝑛,𝑘 (𝑡)𝑑𝑡 =𝑚𝑗−1𝑛𝑘
(𝑚+𝑛)𝑗+𝑘⋅(𝑘+𝑗−1)!
(𝑗−1)!𝑘!
we get
𝑇𝑚,𝑛(𝑥, 𝑦) = 𝑚𝑛 ∞
𝑗=1
∞
𝑘=1
𝑠𝑚,𝑗 (𝑥)𝑠𝑛,𝑘−1(𝑦)𝑚𝑗−1𝑛𝑘
(𝑚+𝑛)𝑗+𝑘⋅(𝑘+𝑗−1)!
(𝑗−1)!𝑘!
=𝑚𝑛 𝑒−𝑚𝑥−𝑛𝑦 ∞
𝑗=1
𝑚2𝑗−1𝑥𝑗
(𝑚+𝑛)𝑗𝑗!(𝑗−1)!
×∞
𝑘=1
𝑛2𝑘−1𝑦𝑘−1
(𝑚+𝑛)𝑘𝑘!(𝑘−1)! ⋅(𝑘+𝑗−1)!.
(14)
306 Commutativity, direct and strong converse results...
For the inner sum of the right-hand side of (14) we can write
∞
𝑘=1
𝑛2𝑘−1𝑦𝑘−1
(𝑚+𝑛)𝑘𝑘!(𝑘−1)! ⋅(𝑘+𝑗−1)! = 𝑛
𝑚+𝑛
∞
𝑘=0
(𝑛2
𝑚+𝑛𝑦)𝑘
𝑘!(𝑘+ 1)! ⋅(𝑘+𝑗)!
=: 𝑇(𝑦).
From
𝑧𝑗𝑒𝑎𝑧 =∞
𝑘=0
𝑎𝑘𝑧𝑘+𝑗
𝑘!
and the Leibniz formula we obtain
𝑑𝑗−1
𝑑𝑧𝑗−1𝑧𝑗𝑒𝑎𝑧 =∞
𝑘=0
(𝑗+𝑘)!𝑎𝑘𝑧𝑘+1
𝑘!(𝑘+ 1)!
=
𝑗−1
𝑙=0 𝑗−1
𝑙𝑗!
(𝑙+ 1)!𝑧𝑙+1𝑎𝑙𝑒𝑎𝑧.
Substituting 𝑧= 1 and 𝑎=𝑛2
𝑚+𝑛𝑦in the above equation we derive
𝑇(𝑦) = 𝑛
𝑚+𝑛𝑒𝑛2
𝑚+𝑛𝑦
𝑗−1
𝑙=0 𝑗−1
𝑙𝑗!
(𝑙+ 1)! 𝑛2
𝑚+𝑛𝑦𝑙
.
Inserting this expression in (14) we get
𝑇𝑚,𝑛(𝑥, 𝑦) = 𝑛2𝑚
𝑚+𝑛𝑒−𝑚𝑥𝑒−𝑛𝑦 𝑒𝑛2
𝑚+𝑛𝑦∞
𝑗=1
𝑚2𝑗−1𝑥𝑗
(𝑚+𝑛)𝑗𝑗!(𝑗−1)!
×
𝑗−1
𝑙=0 𝑗−1
𝑙𝑗!
(𝑙+ 1)! 𝑛2
𝑚+𝑛𝑦𝑙
=𝑛2𝑚
𝑚+𝑛𝑒−𝑚𝑥𝑒−𝑛𝑦 𝑒𝑛2
𝑚+𝑛𝑦∞
𝑙=0
1
(𝑙+ 1)!𝑙!𝑛2
𝑚+𝑛𝑦𝑙
×∞
𝑗=𝑙+1
𝑚2𝑗−1𝑥𝑗
(𝑚+𝑛)𝑗⋅1
(𝑗−1−𝑙)!.
Since
∞
𝑗=𝑙+1
𝑚2𝑗−1𝑥𝑗
(𝑚+𝑛)𝑗⋅1
(𝑗−1−𝑙)! =𝑚2𝑙+1𝑥𝑙+1
(𝑚+𝑛)𝑙+1
∞
𝑗=0 𝑚2𝑥
𝑚+𝑛𝑗
𝑗!
=𝑚2𝑙+1𝑥𝑙+1
(𝑚+𝑛)𝑙+1 𝑒𝑚2
𝑚+𝑛𝑥,
Margareta Heilmann and Gancho Tachev 307
we get
𝑇𝑚,𝑛(𝑥, 𝑦) = 𝑚𝑛
𝑚+𝑛𝑒−𝑚𝑛
𝑚+𝑛𝑥𝑒−𝑚𝑛
𝑚+𝑛𝑦∞
𝑙=0 𝑚𝑛
𝑚+𝑛𝑥𝑙+1
(𝑙+ 1)! 𝑚𝑛
𝑚+𝑛𝑦𝑙
(𝑙)!
=𝑚𝑛
𝑚+𝑛
∞
𝑙=1
𝑠𝑚𝑛
𝑛+𝑚,𝑙(𝑥)𝑠𝑚𝑛
𝑛+𝑚,𝑙−1(𝑦).
So, in view of (13), we have proved our proposition. □
Corollary 3.1. For all 𝑓∈𝐶𝛼[0,∞),𝑚 > 𝛼,𝑛 > 𝛼,𝑚𝑛
𝑚+𝑛> 𝛼 we have
(15)
𝑆𝑚(
𝑆𝑛𝑓) =
𝑆𝑛(
𝑆𝑚𝑓)
and for 𝑙∈ℕ,𝑚
𝑙> 𝛼
(16)
𝑆𝑙
𝑚=
𝑆𝑚
𝑙.
We point out that (16) is essential for the derivation of a strong converse
result of type A.
As an appropriate differential operator we will use
(17)
𝐷2𝑓:= 𝜑2𝐷2𝑓
where 𝐷denotes the ordinary differentiation of a function with respect to its
variable.
Our second commutativity result is
Theorem 3.2. Let 𝑓∈𝐶𝛼[0,∞)and 𝑓′, 𝑓 ′′ ∈𝐶𝛼[0,∞). Then the oper-
ators
𝑆𝑛,𝑛 > 𝛼, and
𝐷2commute, namely
(18) (
𝐷2∘
𝑆𝑛)𝑓= (
𝑆𝑛∘
𝐷2)𝑓.
Proof. On using (6) we compute the first derivative of
𝑆𝑛𝑓.
𝐷(
𝑆𝑛(𝑓, 𝑥)) = −𝑛𝑒−𝑛𝑥𝑓(0) + 𝑛∞
𝑘=1
𝑛[𝑠𝑛,𝑘−1(𝑥)−𝑠𝑛,𝑘(𝑥)]
𝑠′
𝑛,𝑘(𝑥)∞
0
𝑠𝑛,𝑘−1(𝑡)𝑓(𝑡)𝑑𝑡
=−𝑛𝑒−𝑛𝑥𝑓(0) + 𝑛∞
𝑘=0
𝑛𝑠𝑛,𝑘(𝑥)∞
0
𝑠𝑛,𝑘(𝑡)𝑓(𝑡)𝑑𝑡
−𝑛∞
𝑘=1
𝑛𝑠𝑛,𝑘(𝑥)∞
0
𝑠𝑛,𝑘−1(𝑡)𝑓(𝑡)𝑑𝑡
308 Commutativity, direct and strong converse results...
=−𝑛𝑒−𝑛𝑥𝑓(0) + 𝑛2𝑒−𝑛𝑥 ∞
0
𝑒−𝑛𝑡𝑓(𝑡)𝑑𝑡
−𝑛∞
𝑘=1
𝑠𝑛,𝑘(𝑥)∞
0
𝑛[𝑠𝑛,𝑘−1(𝑡)−𝑠𝑛,𝑘(𝑡)]
𝑠′
𝑛,𝑘(𝑡)
𝑓(𝑡)𝑑𝑡
=𝑛𝑒−𝑛𝑥
−𝑓(0) +
−𝑒−𝑛𝑡𝑓(𝑡)𝑑𝑡∣∞
0
=𝑓(0)
+∞
0
𝑒−𝑛𝑡𝑓′(𝑡)𝑑𝑡
−𝑛∞
𝑘=1
𝑠𝑛,𝑘(𝑥)
𝑠𝑛,𝑘(𝑡)𝑓(𝑡)∣∞
0
=0 as 𝑘≥1
−∞
0
𝑠𝑛,𝑘(𝑡)𝑓′(𝑡)𝑑𝑡
=𝑛∞
𝑘=0
𝑠𝑛,𝑘(𝑥)∞
0
𝑠𝑛,𝑘(𝑡)𝑓′(𝑡)𝑑𝑡
=𝑆𝑛(𝑓′, 𝑥).
Thus we proved that
(19) 𝐷(
𝑆𝑛(𝑓, 𝑥)) = 𝑆𝑛(𝑓′, 𝑥).
Now from [10, (3.1)] we get for the second derivative
𝐷2(
𝑆𝑛(𝑓, 𝑥)) = 𝐷(𝑆𝑛(𝑓′, 𝑥))
=𝑛∞
𝑘=0
𝑠𝑛,𝑘(𝑥)∞
0
𝑠𝑛,𝑘+1(𝑡)𝑓′′(𝑡)𝑑𝑡.
Finally, on using (9) and observing that (
𝐷2𝑓)(0) = 0 we get
𝐷2(
𝑆𝑛(𝑓, 𝑥) = 𝜑(𝑥)2𝐷(𝑆𝑛(𝑓′, 𝑥))
=𝑛∞
𝑘=1
𝑠𝑛,𝑘(𝑥)∞
0
𝑠𝑛,𝑘−1(𝑡)(
𝐷2𝑓)(𝑡)𝑑𝑡
=
𝑆𝑛(
𝐷2𝑓, 𝑥).
Theorem 3.2 is proved. □
As an immediate consequence of Theorem 3.2 we get the following corollary
which improves significantly the constant in the result of [6, Lemma 5].
Corollary 3.2. For 𝑓∈𝑊2
∞(𝜑)there holds
∥
𝐷2(
𝑆𝑛𝑓)∥ ≤ ∥
𝐷2𝑓∥.
Margareta Heilmann and Gancho Tachev 309
Remark 3.1. For sufficiently smooth functions we get from Theorem 3.2
via induction
(
𝐷2)𝑙∘
𝑆𝑛=
𝑆𝑛∘(
𝐷2)𝑙.
It is easy to verify that
(
𝐷2)𝑙=
𝐷2𝑙:= 𝐷𝑙−1𝜑2𝑙𝐷𝑙+1.
Such an identity is not true in the case of the corresponding appropriate
differential operators for the genuine Bernstein-Durrmeyer and Baskakov-
Durrmeyer operators.
4. Strong Voronovskaja type theorem
In order to derive a strong converse inequality of type A we need an appro-
priate strong Voronovskaja type result. We formulate our results in terms of
a positive constant 𝑐which will be choosen later to get a good constant in the
strong converse inequality.
Theorem 4.1. Let 𝑔∈𝐶𝐵[0,∞),𝜑2𝑔′′′, 𝜑3𝑔′′′ ∈𝐶𝐵[0,∞)and 𝑛 > 0.
Then
𝑆𝑛𝑔−𝑔−1
𝑛𝜑2𝑔′′
≤√6
2⋅1
𝑛max 4
3√1 + 2𝑐⋅1
√𝑛∥𝜑3𝑔′′′∥,1 + 2𝑐
𝑐⋅1
𝑛∥𝜑2𝑔′′′∥,
where 𝑐denotes an arbitrary positive constant.
Proof. We apply the operators
𝑆𝑛to the Taylor expansion of 𝑔
𝑔(𝑡) = 𝑔(𝑥) + 𝑔′(𝑥)(𝑡−𝑥) + 1
2𝑔′′(𝑥)(𝑡−𝑥)2+1
2𝑡
𝑥
𝑔′′′(𝑢)(𝑡−𝑢)2𝑑𝑢
and use Lemma 2.1 to derive
(20)
𝑆𝑛(𝑔, 𝑥)−𝑔(𝑥)−1
𝑛(
𝐷2𝑔)(𝑥)≤1
2
𝑆𝑛𝑡
𝑥
𝑔′′′(𝑢)(𝑡−𝑢)2𝑑𝑢, 𝑥.
Case 1: 𝑥≥1
𝑐𝑛 . We have
(21)
𝑆𝑛𝑡
𝑥
𝑔′′′(𝑢)(𝑡−𝑢)2𝑑𝑢, 𝑥≤ ∥𝜑3𝑔′′′∥
𝑆𝑛𝑡
𝑥
(𝑡−𝑢)2
𝜑(𝑢)3𝑑𝑢, 𝑥.
310 Commutativity, direct and strong converse results...
As ∣𝑡−𝑢∣
𝑢≤∣𝑡−𝑥∣
𝑥we now observe that
𝑆𝑛𝑡
𝑥
(𝑡−𝑢)2
𝜑(𝑢)3𝑑𝑢, 𝑥
=∞
𝑘=1
𝑠𝑛,𝑘(𝑥)𝑛∞
0
𝑠𝑛,𝑘−1(𝑡)𝑡
𝑥
(𝑡−𝑢)2
𝑢3
2
𝑑𝑢𝑑𝑡 +𝑒−𝑛𝑥 𝑥
0
𝑢1
2𝑑𝑢
≤∞
𝑘=1
𝑠𝑛,𝑘(𝑥)𝑛∞
0
𝑠𝑛,𝑘−1(𝑡)∣𝑡−𝑥∣3
2
𝑥3
2𝑡
𝑥∣𝑡−𝑢∣1
2𝑑𝑢𝑑𝑡 +2
3𝑒−𝑛𝑥𝑥3
2
≤2
3⋅1
𝑥3
2
𝑆𝑛∣𝑡−𝑥∣3, 𝑥.
(22)
Now using the Cauchy-Schwarz inequality we get the estimate
𝑆𝑛∣𝑡−𝑥∣3, 𝑥≤
𝑆𝑛((𝑡−𝑥)2, 𝑥)⋅
𝑆𝑛((𝑡−𝑥)4, 𝑥)
=2𝑥
𝑛⋅12𝑥
𝑛2𝑥+2
𝑛,
(23)
where we have used Lemma 2.1 for the moments. For 𝑥≥1
𝑐𝑛 (22) and (23)
imply
𝑆𝑛𝑡
𝑥
(𝑡−𝑢)2
𝜑(𝑢)3𝑑𝑢, 𝑥≤4√6
3√1 + 2𝑐 𝑛−3/2.
Thus for the case 𝑥≥1
𝑐𝑛 we have
(24)
𝑆𝑛𝑡
𝑥
𝑔′′′(𝑢)(𝑡−𝑢)2𝑑𝑢, 𝑥≤4√6
3√1 + 2𝑐𝑛−3/2∥𝜑3𝑔′′′∥.
Case 2: 𝑥≤1
𝑐𝑛 . We have
𝜑(𝑥)2
𝑆𝑛𝑡
𝑥
𝑔′′′(𝑢)(𝑡−𝑢)2𝑑𝑢, 𝑥
≤ ∥𝜑2𝑔′′′∥𝜑(𝑥)2
𝑆𝑛𝑡
𝑥
(𝑡−𝑢)2
𝜑(𝑢)2𝑑𝑢, 𝑥.
(25)
Proceeding in a similar way as in Case 1 we get
𝜑(𝑥)2
𝑆𝑛𝑡
𝑥
(𝑡−𝑢)2
𝜑(𝑢)2𝑑𝑢, 𝑥≤
𝑆𝑛∣𝑡−𝑥∣𝑡
𝑥∣𝑡−𝑢∣𝑑𝑢, 𝑥
≤1
2
𝑆𝑛∣𝑡−𝑥∣3, 𝑥
≤𝜑(𝑥)2√61 + 2𝑐
𝑐𝑛−2.
Margareta Heilmann and Gancho Tachev 311
Thus we obtained for the case 𝑥≤1
𝑐𝑛
(26)
𝑆𝑛𝑡
𝑥
𝑔′′′(𝑢)(𝑡−𝑢)2𝑑𝑢, 𝑥≤√61 + 2𝑐
𝑐𝑛−2∥𝜑2𝑔′′′∥.
Substitution of (24) and (26) into (20) proves the proposition. □
5. Direct and strong converse result
Our direct and converse results will be formulated in terms of the following
K-functional
(27) 𝐾2
𝜑(𝑓, 𝛿2) := inf {∥𝑓−𝑔∥+𝛿2∥
𝐷2𝑔∥:𝑔∈𝑊2
∞(𝜑)}.
The corresponding second order Ditzian-Totik modulus of smoothness is given
by
(28) 𝜔2
𝜑(𝑓, 𝛿) = sup
0<ℎ≤𝛿∥Δ2
ℎ𝜑𝑓∥,
where
Δ2
ℎ𝜑(𝑥)𝑓(𝑥) = 𝑓(𝑥+ℎ𝜑(𝑥))−2𝑓(𝑥)+𝑓(𝑥−ℎ𝜑(𝑥)),if 𝑥±ℎ𝜑(𝑥)∈[0,∞),
0,otherwise.
It is known that 𝐾2
𝜑(𝑓, 𝛿2) and 𝜔2
𝜑(𝑓, 𝛿) are equivalent (see [5, Theorem 2.1.1]),
i.e., there exists an absolute constant 𝐶 > 0 and 𝛿0such that
𝐶−1𝜔2
𝜑(𝑓, 𝛿)≤𝐾2
𝜑(𝑓, 𝛿2)≤𝐶𝜔2
𝜑(𝑓, 𝛿),0< 𝛿 ≤𝛿0.
Our main results are
Theorem 5.1. For every 𝑓∈𝐶𝐵[0,∞)and 𝑛 > 0there holds
∥
𝑆𝑛𝑓−𝑓∥ ≤ 2𝐾2
𝜑𝑓, 1
𝑛.
Theorem 5.2. For every 𝑓∈𝐶𝐵[0,∞)and 𝑛 > 0the following inequal-
ity holds true
(29) 𝐾2
𝜑𝑓, 1
𝑛≤92.16 ∥
𝑆𝑛𝑓−𝑓∥.
As a consequence of Theorems 5.1 and 5.2 we get the following equivalence
result.
312 Commutativity, direct and strong converse results...
Corollary 5.1. For 𝑓∈𝐶𝐵[0,∞),𝑛 > 0we have the following equiva-
lences
1
2∥
𝑆𝑛𝑓−𝑓∥ ≤ 𝐾2
𝜑𝑓, 1
𝑛≤92.16 ∥
𝑆𝑛𝑓−𝑓∥,
𝐶1∥
𝑆𝑛𝑓−𝑓∥ ≤ 𝜔2
𝜑𝑓, 1
√𝑛≤𝐶2∥˜
𝑆𝑛𝑓−𝑓∥,
where 𝐶1, 𝐶2>0are absolute constants.
Proof of Theorem 5.1. Let 𝑔∈𝑊2
∞(𝜑) be arbitrary and 𝑥be fixed. From
the Taylor expansion
𝑔(𝑡)−𝑔(𝑥) = 𝑔′(𝑥)(𝑡−𝑥) + 𝑡
𝑥
𝑔′′(𝑠)(𝑡−𝑠)𝑑𝑠, 𝑡 ∈[0,∞),
we see that
(30) ∣˜
𝑆𝑛(𝑔, 𝑥)−𝑔(𝑥)∣ ≤ ˜
𝑆𝑛𝑡
𝑥
𝑔′′(𝑢)(𝑡−𝑢)𝑑𝑢, 𝑥.
It is known (see [5, (9.6.1)]) that
𝜑(𝑥)2𝑡
𝑥
𝑔′′(𝑢)(𝑡−𝑢)𝑑𝑢≤ ∣𝑡−𝑥∣𝑡
𝑥
𝜑(𝑢)2𝑔′′(𝑢)𝑑𝑢
≤ ∥𝜑2𝑔′′∥(𝑡−𝑥)2.
From (30) and the second moment
𝑆𝑛((𝑡−𝑥)2, 𝑥) = 2𝑥
𝑛(see Lemma 2.1) we
get
∣˜
𝑆𝑛(𝑔, 𝑥)−𝑔(𝑥)∣ ≤ 2
𝑛∥𝜑2𝑔′′∥.
This gives
∣˜
𝑆𝑛(𝑓, 𝑥)−𝑓(𝑥)∣ ≤ ∣ ˜
𝑆𝑛(𝑓−𝑔)(𝑥)∣+∣𝑔(𝑥)−𝑓(𝑥)∣+∣˜
𝑆𝑛(𝑔, 𝑥)−𝑔(𝑥)∣
≤2(∥𝑓−𝑔∥+1
𝑛∥𝜑2𝑔′′∥).
Taking the infimum of the right-hand side term over all 𝑔∈𝑊2
∞(𝜑) we obtain
the statement of the theorem. □
For the proof of Theorem 5.2 we will need three further estimates which are
proved in the next lemmas. Note that Lemma 5.1 improves upon the constant
in [6, Lemma 6] and Lemma 5.2 gives an explicit value for the constant in [6,
Lemma 7].
Lemma 5.1. For 𝑓∈𝐶𝐵[0,∞),𝑛 > 0we have
∥
𝐷2(
𝑆𝑛𝑓)∥ ≤ 2𝑛∥𝑓∥.
Margareta Heilmann and Gancho Tachev 313
Proof. From (8) we have
(31) 𝜑(𝑥)2
𝐷2
𝑆𝑛(𝑓, 𝑥)≤ ∥𝑓∥∞
𝑘=0
(𝑘−𝑛𝑥)2𝑠𝑛,𝑘(𝑥) + ∞
𝑘=1
𝑘𝑠𝑛,𝑘 (𝑥).
For the second moment of the Sz´asz-Mirakjan operator 𝑆𝑛we make use of
(10). As 𝑆𝑛preserves linear functions we derive from (31)
𝐷2
𝑆𝑛(𝑓, 𝑥)≤2𝑛∥𝑓∥.□
Lemma 5.2. For every 𝑔∈𝑊2
∞(𝜑)and 𝑛 > 0we have
∥𝜑3𝐷3(
𝑆𝑛𝑔)∥ ≤ 1.47√𝑛∥
𝐷2𝑔∥.
Proof. From (19) and [10, (3.1)] we have
𝜑(𝑥)4𝐷3(
𝑆𝑛(𝑔, 𝑥)) = 𝜑(𝑥)4𝐷2(𝑆𝑛(𝑔′, 𝑥))
=𝜑(𝑥)4𝑛∞
𝑘=0
𝑠′
𝑛,𝑘(𝑥)∞
0
𝑠𝑛,𝑘+1(𝑡)𝐷2𝑔(𝑡)𝑑𝑡
=𝑛∞
𝑘=0
(𝑘−𝑛𝑥)𝑠𝑛,𝑘+1(𝑥)∞
0
𝑠𝑛,𝑘(𝑡)(
𝐷2𝑔(𝑡))𝑑𝑡 ,
(32)
where we have used (7) and (9). On using (5) we get
𝜑(𝑥)4𝐷3(
𝑆𝑛(𝑔, 𝑥))≤ ∥
𝐷2𝑔∥∞
𝑘=0
(∣𝑘+ 1 −𝑛𝑥∣+ 1)𝑠𝑛,𝑘+1(𝑥)
=∥
𝐷2𝑔∥∞
𝑘=0
(∣𝑘−𝑛𝑥∣+1)𝑠𝑛,𝑘 (𝑥)−𝑒−𝑛𝑥(1 +𝑛𝑥)
≤∥
𝐷2𝑔∥𝑛𝑆𝑛((𝑡−𝑥)2, 𝑥)+1−𝑒−𝑛𝑥(1+𝑛𝑥)
=∥
𝐷2𝑔∥𝜑(𝑥)√𝑛+ 1 −𝑒−𝑛𝑥(1 + 𝑛𝑥).
(33)
The maximum of 1−𝑒−𝑛𝑥 (1+𝑛𝑥)
√𝑥is attained for 𝑛𝑥 ∈[3.2,3.25], hence we have
the estimate
1−𝑒−𝑛𝑥(1 + 𝑛𝑥)
√𝑛𝑥 ≤1−𝑒−3.25(1 + 3.2)
√3.2<0.47.
Inserting this bound into (33) we get
𝜑(𝑥)3𝐷3(
𝑆𝑛(𝑔, 𝑥))≤1.47√𝑛∥
𝐷2𝑔∥.□
314 Commutativity, direct and strong converse results...
Lemma 5.3. For every 𝑔∈𝑊2
∞(𝜑)we have
∥𝜑2𝐷3(
𝑆𝑛𝑔)∥ ≤ 2𝑛∥
𝐷2𝑔∥.
Proof. Making again use of (19) and [10, (3.1)], combined with (6), we get
𝜑(𝑥)2𝐷3(
𝑆𝑛(𝑔, 𝑥)) = 𝜑(𝑥)2𝐷2(𝑆𝑛(𝑔′, 𝑥))
=𝜑(𝑥)2𝑛∞
𝑘=0
𝑠′
𝑛,𝑘(𝑥)∞
0
𝑠𝑛,𝑘+1(𝑡)𝑔′′(𝑡)𝑑𝑡
=𝜑(𝑥)2𝑛∞
𝑘=0
𝑛[𝑠𝑛,𝑘−1(𝑥)−𝑠𝑛,𝑘(𝑥)]∞
0
𝑛
𝑘+1 𝑠𝑛,𝑘(𝑡)
𝐷2𝑔(𝑡)𝑑𝑡.
Applying (5) we deduce the result as follows:
∣𝜑(𝑥)2𝐷3((
𝑆𝑛(𝑔, 𝑥))∣ ≤ ∥
𝐷2𝑔∥𝑛∞
𝑘=1
𝑛𝑥𝑠𝑛,𝑘−1(𝑥)
𝑘+ 1 +∞
𝑘=0
𝑛𝑥𝑠𝑛,𝑘(𝑥)
𝑘+ 1
≤ ∥
𝐷2𝑔∥2𝑛∞
𝑘=0
𝑠𝑛,𝑘(𝑥)
= 2𝑛∥
𝐷2𝑔∥.
□
Proof of Theorem 5.2. Since
(34) 𝐾2
𝜑𝑓, 1
𝑛≤ ∥
𝑆𝑛𝑓−𝑓∥+1
𝑛∥
𝐷2(
𝑆𝑛𝑓)∥,
we have to estimate 1
𝑛∥
𝐷2(
𝑆𝑛𝑓)∥.
Let 𝑁∈ℕ. From Lemma 5.1 we obtain the estimate
1
𝑛∥
𝐷2(
𝑆𝑛𝑓)∥ ≤ 1
𝑛∥
𝐷2[
𝑆𝑛(𝑓−
𝑆𝑁
𝑛𝑓)]∥+1
𝑛∥
𝐷2[
𝑆𝑛(
𝑆𝑁
𝑛𝑓)]∥
≤2𝑁∥𝑓−
𝑆𝑛𝑓∥+1
𝑛∥
𝐷2(
𝑆𝑁+1
𝑛𝑓)∥.
(35)
We now apply the strong Voronovskaja–type result of Theorem 4.1 to the
function 𝑔=
𝑆𝑁+1
𝑛𝑓to obtain
1
𝑛∥
𝐷2(
𝑆𝑁+1
𝑛𝑓)∥
≤ ∥
𝑆𝑁+2
𝑛𝑓−
𝑆𝑁+1
𝑛𝑓∥+∥
𝑆𝑁+2
𝑛𝑓−
𝑆𝑁+1
𝑛𝑓−1
𝑛
𝐷2(
𝑆𝑁+1
𝑛𝑓)∥
≤ ∥𝑓−
𝑆𝑛𝑓∥
+√6
2𝑛max 4√1+2𝑐
3√𝑛∥𝜑3(
𝑆𝑁+1
𝑛𝑓)′′′∥,1+2𝑐
𝑐⋅1
𝑛∥𝜑2(
𝑆𝑁+1
𝑛𝑓)′′′∥.
(36)
Margareta Heilmann and Gancho Tachev 315
From Corollary 3.1, Lemma 5.2 and Lemma 5.3 we obtain the estimates
∥𝜑3(
𝑆𝑁+1
𝑛𝑓)′′′∥=∥𝜑3(
𝑆𝑛
𝑁
𝑆𝑛𝑓)′′′∥ ≤ 1.47 𝑛
𝑁∥
𝐷2(
𝑆𝑛𝑓)∥
∥𝜑2(
𝑆𝑁+1
𝑛𝑓)′′′∥=∥𝜑2(
𝑆𝑛
𝑁
𝑆𝑛𝑓)′′′∥ ≤ 2𝑛
𝑁∥
𝐷2(
𝑆𝑛𝑓)∥.
The latter estimates together with (35) and (36) imply
1
𝑛∥
𝐷2(
𝑆𝑛𝑓)∥ ≤ (2𝑁+ 1)∥𝑓−
𝑆𝑛𝑓∥
+√6
2𝑛∥
𝐷2(
𝑆𝑛𝑓)∥max 4⋅1.47
3⋅√1 + 2𝑐
√𝑁,21 + 2𝑐
𝑐⋅1
𝑁.
(37)
In order to get a good constant we now choose 𝑁= 16 and 𝑐=625
9604 . With
this choice we have
max 4⋅1.47
3⋅√1 + 2𝑐
√𝑁,21 + 2𝑐
𝑐⋅1
𝑁=9√2⋅67
200 .
Substituting this into (37) leads to
(38) 1
𝑛∥
𝐷2(
𝑆𝑛𝑓)∥ ≤ 33∥𝑓−
𝑆𝑛𝑓∥+1
𝑛
9√2⋅6⋅67
400 ∥
𝐷2(
𝑆𝑛𝑓)∥.
Note that 9√2⋅6⋅67
400 <1, so (38) is equivalent to
1
𝑛∥
𝐷2(
𝑆𝑛𝑓)∥ ≤ 6600
200 −9√3⋅67∥𝑓−
𝑆𝑛𝑓∥.
Together with (34) we end up with the estimate
𝐾2
𝜑𝑓, 1
𝑛≤6800 −9√3⋅67
200 −9√3⋅67 ∥
𝑆𝑛𝑓−𝑓∥ ≤ 92.16∥
𝑆𝑛𝑓−𝑓∥.
Theorem 5.2 is proved. □
316 Commutativity, direct and strong converse results...
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Margareta Heilmann and Gancho Tachev 317
Received December 5, 2011
Margareta Heilmann
Faculty of Mathematics and Natural Sciences
University of Wuppertal
Gaußstraße 20
D-42119 Wuppertal
GERMANY
E-mail: heilmann@math.uni-wuppertal.de
Gancho Tachev
University of Architecture
1 Hr. Smirnenski Blvd.
1046 Sofia
BULGARIA
E-mail: gtt fte@uacg.bg
