The second order modulus revisited:
remarks, applications, problems
Abstract.
Several questions concerning the second order modulus of smoothness are addressed in this note. The central part is a refined analysis of a construction of certain smooth functions by Zhuk and its application to several problems in approximation theory, such as degree of approximation and the preservation of global smoothness. Lower bounds for some optimal constants introduced by Sendov are given as well. We also investigate an alternative approach using quadratic splines studied by Sendov.
Key words and phrases:
second order modulus, degree of approximation, global smoothness preservation, Bernstein operators.2005 Mathematics Subject Classification:
41A25, 41A36, 41A44.1. Introduction
The present note could have likewise been called
“Note on a paper by
Zhuk” or “Note on
a paper by Sendov”, which was also a
preliminary title of this work. The point about this somewhat unusual
introductory remark is that both Sendov and Zhuk have recently dealt (again)
with certain natural questions concerning the classical second order modulus
of continuity (denoted by
Let us first introduce some notation. For a compact interval
and
By
The problem discussed here essentially originated in a paper G. Freud [13]. In 1959 he proved a certain assertion which Brudnyĭ later generalized to the following theorem of utmost importance in approximation theory.
Theorem 1.1 (Brudnyĭ, [7, Proposition 2]).
Let
(1) | ||||
(2) |
Here
It is of interest to have information on the magnitude of the constants
At several stages of this note, Bernstein polynomials over an interval
(3) |
As is well known, the polynomials
2. Further estimates for Zhuk’s functions
Let us first recall Zhuk’s approach to constructing his smoothing functions
For
by
Here
Then Zhuk defined the second order Steklov means
and showed the following
In the next lemma we show that there is a pointwise refinement of Lemma 2.1
Lemma 2.2.
Let
Proof.
We first rewrite
Let
Hence
Since
we have
Let
In this case,
This implies
Here we have
Since
For the remaining two differences we have, for,
Lemma 2.3.
For a compact interval
-
(i)
If
denotes the linear function interpolating af and , then -
(ii)
If
is the best approximation to by elements of , then
Proof of Lemma 2.2
(cont’d): Hence for
Here
giving the desired inequality for
As can be seen from the example below (see Section 4 in particular), it is
sometimes convenient to have estimates for lower order derivatives of
Lemma 2.4.
Proof.
The second inequality is a simple consequence of
which implies
The first inequality is obtained as follows. Write
Here,
where
Likewise,
Hence,
which implies
(differentiation under the integral, mean value theorem).
In order to estimate
Then
Now assume that
Then
Due to the linearity of
As we know from Lemma 2.3
on the interval
Hence it follows that
The remaining case, namely
can be treated analogously to the second one. ∎
Corollary 2.5.
As an immediate consequence of Lemma 2.4 one has the simpler inequalities
Corollary 2.6.
However, for assertions in which small constants are of interest in the final statement, it is not advisable to use the latter inequalities.
3. Lower bounds for
It was shown by Sendov [28, p. 198] that the constant
Definition 3.1.
We denote by
Very little is known about the funciton
Theorem 3.2.
Proof.
Let
and
Now consider the Bernstein polynomials
Choosing
It was shown in [2] (see Remark (ii) following Theorem 9 there) that the inequality
can hold only if
We consider the case
Theorem 3.3.
Proof.
A query published in the proceedings of the 1982 Edmonton Conference on approximation theory (see [19] and Example 4.13 (ii) below) contains the information that it was known then that
(4) |
The same fact was also observed by Păltănea in 1990, see [26, Theorem 3.2]. This means that
cannot hold for any constant
Now let
and
Choose
From (4), we know that
Remark 3.4.
The lower bound
for some natural
The numerical experiments then carried our failed to produce such a function. This experience, together with the then-known inequality.
led us to publish the query in the Alberta conference proceedings of 1982 mentioned earlier.
4. Applications
In this section we give a collection of applications of the inequalities in Section 2. At several stages we critically discuss the power of the general estimates derived here by comparing them with results obtained in some special situations.
4.1. General Operators
In the following lemma we show that functions in
Lemma 4.1.
For each
-
(i)
and
-
(ii)
Proof.
For
where
The case
For
Here
Hence
For the second derivative one has
where
∎
This implies
Using Lemma 4.1
and the results from Section 2
, we present next a partial
generalization of another theorem of Brudnyĭ (see [7, Theorem 9]),
which is more appropriate for application purposes than earlier
contributions by other authors. As far as earlier work is concerned,
particularly for the case of linear operators, that of Freud [13, Main theorem] and
Stečkin [32, Theorem 5], must be mentioned. For this
so-called smoothing technique, see also [12, Theorems
2.2 through 2.4]. While we restrict ourselves here to the case of
Theorem 4.2.
Let
-
a)
-
b)
-
c)
Then for all
Proof.
Corollary 4.3.
In many cases one has
Remark 4.4.
The constants
Remark 4.5.
-
(i)
If
is linear, then condition a) of Theorem 4.2 is automatically fulfilled with . -
(ii)
Typical examples of non-linear operators
satisfying the assumptions of Theorem 4.2 are those of the from , where , and the linear operator and are fixed. -
(iii)
Instances of linear operators
satisfying the assumptions of Theorem 4.2 are given by, e.g., , where is a point-evaluation functional and is some linear operator.
4.1.1. 4.2.Examples (non-linear case)
Global Smoothness Preservation
As a first application of Theorem 4.2
(or Corollary 4.3,
inequality concerning the preservation of global smoothness by the classical
Bernstein operators in terms of the second order modulus of smoothness. The
same inequality was derived in [10, Prop 3.5],
however, as
an application of a different general result.
The situation here is
i.e.,
Putting
While this is already better than a recent result by Adell and de la Cal
[1], for the same special cases improvements are available. If we
define Lipschitz classes with respect to
then the latter inequality shows that
The statement was recently improved by Ding-Xuan Zhou [34] who proved
This was also shown independently by I. Gavrea [16] for the cases
and showed that for this modulus one has
4.2.2.Modulus of the Remainder
A question related to that of the previous example is the magnitude of the
modulus of the remainder in the approximation by linear operators; see, e.g.,
[4] for earlier work in this direction. Here we consider the
case
Assuming further that
holds, we find
It thus follows that
For
If
4.2.3.Landau-type inequalities involving Moduli of Smoothness
Landau-type inequalities involving moduli of smoothness can be used in order to give more compact upper bounds in direct estimates; see the proof of Corollary 2.7 in [20] for an example. Below is an improved version of Lemma 2.6 in [20]. It also improves a recent result by Gavrea and Rasa [17].
We consider here the space
Furthermore, for all
The next step is to use Landau’s inequality. Indeed, one has (see [25], [3], [9], 71???? nu este pe original)
(5) |
This means that Theorem 4.2
can be applied with
This is an improvement of Lemma 2.6 in [20] and also of formula (4) in [17].
Gavrea and Rasa also gave a certain improvement of (5), namely
(6) |
which enabled them to improve a result from [9] (see Theorem 2.1 there). Combining their improvement with the above Lemma 2.1 we give a refinement of Theorem 2.3 in [9].
Theorem 4.6.
Let
Note how the upper bound of Theorem 4.6
simplifies for positive
operators
Proof.
We apply Theorem 4.2
with
The inequality of Theorem 4.6 is then an immediate consequence of Corollary 4.3 ∎
Remark 4.7.
It is of advantage to use the quantity
Applications of an inequality of the type given in Theorem 4.6 can be found in [9], for example.
4.1.2. 4.3.Approximation by Bounded Linear Operators
?? 4.3 este si pe original??? Another immediate consequence of Theorem 4.2 is the following
Corollary 4.8.
Let
-
(i)
for all -
(ii)
for all
Then for all
Remark 4.9.
For the case
4.2. 4.4.Approximation by Positive Linear Operators
In this section we give a pointwise inequality for the degree of approximation
by positive linear operators defined on
Lemma 4.10.
Let
We now apply Theorem 4.2
for positive linear operators with
Theorem 4.11.
If
Simpler inequalities hold if
Corollary 4.12.
Let the assumptions of Theorem 4.11 be satisfied.
-
(i)
If
, then for each we have -
(ii)
If
then
Example 4.13.
(Bernstein operators).
-
(i)
The representation
is well-known. Choosing in Corollary 4.12 (ii) givesThis estimate can also be directly derived from Zhuk’s paper referred to before. It should be compared to a recent result by Păltănea [26] who showed
Comparing the quantities (cf. Corollary 4.12 (ii))
shows that
i.e., for small values of
the constant in front of arising from Zhuk’s approach is better than Păltănea’s. -
(ii)
In Problem n.2 of [24] (see also [18]) the question was raised (again) as to the best possible value of the constant
in an estimate of the form(with
independent of and ). This question had been motivated by Sikkema’s striking result concerning the first order modulus (see [31]) and by related observations made in [22]. If we put in Corollary 4.12 (ii), the general inequality given there shows that is one possible value.Păltănea [26] proved the better result
. Using Păltănea’s method, in [5] it was recently shown that it is also possible to choose . However, the latter constant is probably not optimal.Another partial result along these lines was recently obtained in [23], in which the following was proved: Let
. Then there is a constant so that for all one hasThis result seems to indicate that our conjecture from [19], namely that the optimal value of
equals , is correct. However, an answer to the original problem is not yet available.
Remark 4.14.
An interesting different approach to derive inequalities as in Example 4.13
(again for the special case of Bernstein operators) was taken by
Gasharov (see [14], [15]). Instead of starting from
a general inequality like that in Corollary 4.12
(ii),
Gasharov uses his Steklov means
The second term
For the remaining two terms it is essential in his approach not
to use the common upper bound
These inequalities are worse than the estimates given earlier. Nonetheless, we
feel that his approach might be useful in obtaining better constants. We have
tried without success to carry such an approach over to the Steklov means
5. The quadratic splines of Sendov
In order to define Zhuk’s functions
Theorem 5.1.
Let
such that
(1)
(2)
Remark 5.2.
In [28] the author claims that the inequalities of Theorem 5.1
are true for all
It is the main objective of this appendix to prove that the constant 9/8
figuring in Theorem 5.1
can be replaced by 1. We feel that such an assertion
is in perfect harmony with our earlier observations
The quadratic spline
The analytic representation of
for
However,
Recalling the definition of a 2nd degree Bernstein polynomial over an interval
as well as (one-sided derivatives taken at
This observation yields an immediate proof of the second part of Theorem 5.1
Indeed, letting
we have
Recalling further that

Our next aim is to show that the constant 9/8 figuring in Theorem 5.1
can be
replaced by 1. To this end, it seems to be instructive to sketch the graph of
a typical spline
In Figure 1, the graph of
Theorem 5.3.


Case I:
Since
Case II:
We first look at the larger interval
The same argument shows that
Now observe that, by construction, over the interval
in cases 1 and 2:
in cases 3 and 4:
Analogous inequalities hold on
Case III:
The statement of the next lemma parallels that of Lemma 2.4
It shows that
lower order derivatives of
Lemma 5.4.
Let
Proof.
Case I:
Case II:
The case
In order to see that
∎
As we mentioned in Remark 5.2
, we were unable to verify Sendov’s Theorem 5.1
for all
Problem 5.5.
Let
-
(i)
-
(ii)
-
(iii)
-
(iv)
Acknowledgements.
The authors gratefully acknowledge Claudia Cottin, Rita Hülsbusch, Eva Müller-Faust, John Sevy, Hans-Jörg Wenz and Xin-long Zhou for their technical help in preparing this note, for their critical remarks on earlier versions, or for several attempts to solve 5.5.
References
- [1] J.A. Adell, J. de la Cal, Preservation of moduli of continuity for Bernstein-type operators, Manuscript 1993. https://doi.org/10.1007/978-1-4615-2494-6_1
- [2] G.A. Anastassiou, C. Cottin, H.H., Gonska, Global smoothness of approximating functions, Analysis, 11 (1991), pp. 43–57. https://doi.org/10.1524/anly.1991.11.1.43
-
[3]
I.B. Bashmakova, On approximation by Hermite
splines (Russian), In:
Numerical Methods in Boundary Value Problems of Mathematical Physics (Meshvyz. Temat. Sb. Tr.), 5-7. Leningrad: Leningradskii Inzhemerno-Stroitel’nyi Institut,1985. - [4] W.R. Bloom, D. Elliot, The modulus of continuity of the remainder in the approximation of Lipschitz functions, J. Approx. Theory, 31 (1981), pp. 59–66. https://doi.org/10.1016/0021-9045(81)90030-7
- [5] C. Badea, I. Badea, H. Gonska, Improved estimates on simultaneous approximation by Bernstein operators II, Manuscript 1993.
- [6] A. Boos, Jia-ding Cao, H. Gonska, Approximation by Boolean sums of positive linear operators V: on the constants in DeVore-Gopengauz-type inequalities. To appear in Calcolo. Temporary reference: Schriftenreihe des Instituts für Angewandte Informatik SIAI-EBS-4-93, European Business School, 1993. https://doi.org/10.1007/BF02575858
-
[7]
Ju.A.Brudnyi, Approximation of functions of
variables by quasi-polynomials (Russian), Izv. Akad. Nauk SSSR, Ser. Mat., 34 (1970), pp. 564–583. English translation: Math. USSR-Izvestija 4 (1970, n.3), pp. 568–586. https://doi.org/10.1070/IM1970v004n03ABEH000922 - [8] H. Burkill, Cesàro-Perron almost periodic functions, Proc. London Math. Soc (Series 3), 2 (1952), pp. 150–174. https://doi.org/10.1112/plms/s3-2.1.150
- [9] Jia-ding Cao, H. Gonska, Approximation by Boolean sums of positive linear operators, Rend. Mat. 6 (1986), 525-546.
- [10] C. Cottin, H. Gonska, Simultaneous approximation and global smoothness preservation, Rend. Circ. Mat. Palermo (2) Suppl. 33 (1993), pp. 259–279.
- [11] R.A. Devore, The Approximation of Continuous Functions by Positive Linear Operators, Berlin: Springer 1972. https://doi.org/10.1007/BFb0059493
- [12] R.A. Devore, Degree of approximation, in ?? ?? Approximation Theory II?? ?? . (Proc. Int. Symp. Austin 1976; ed. by G.G. Lorentz et al.), pp. 117–161. New York: Academic Press 1976.
- [13] G. Freud, Su i procedimenti lineari d’approssimazione, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur. 26 (1959), pp. 641–643.
- [14] N.G. Gasharov, On estimates for the approximation by linear operators preserving linear functions (Russian), In: ”Approximation of Functions by Special Classes of Operators”. 28035, Vologda, 1987.
- [15] N.G. Gasharov, Written comunication, January 1992.
- [16] I. Gavrea, About a conjecture, Studia Univ. Babeş-Bolyai Math. 38 (1993).
- [17] I. Gavrea, I. Raşa, Remarks on some quantitative Korovkin-type results, Manuscript 1993.
- [18] H.H. Gonska, Query in ?? ?? Unsolved Problems?? ?? ,In: ?? ?? Constructive Function Theory’81?? ?? .(Proc.Int.Conf.Varna 1981; ed. by Bl. Sendov et. al.), pp. 597–598, Sofia: Publishing House of the Bulgarian Acadmy of Sciences 1983.
- [19] H.H. Gonska, Two problems on best constants in direct estimates, In: Problem Section of Proc. Section Edmonton Conf. Approximation Theory (Edmonton, Alta., 1982; ed. by Z. Ditzian et al., 194, Providence, RI: Amer. Math. Soc. 1983.
- [20] H.H. Gonska, Quantitative Korovkin-type theorems on simultaneous approximation, Math. Z., 186 (1984), pp. 419–433. https://doi.org/10.1007/BF01174895
-
[21]
H.H. Gonska, Degree of approximation by lacunary
interpolators:
interpolation, Rocky Mountain J. Math. 19 (1989), pp. 157–171. https://doi.org/10.1216/RMJ-1989-19-1-157 - [22] H.H. Gonska, J. Meier, On approximation by Bernstein type operators: best constants, Studia Sci. Math. Hungar., 22 (1987), pp. 287–297.
- [23] H.H. Gonska, Ding-xuan Zhou, On an extremal problem concerning Bernstein operators, To appear in Serdica. Temporary reference: Schriftenreihe des Instituts für Angewandte Informatik SIAI-EBS-10-94, European Business School 1994.
- [24] H.H. Gonska, Xin-long Zhou, Polynomial approximation with side conditions: recent results and open problems, In:?? ?? Proc. First International Colloquium on Numerical Analysis (Plovdiv 1992; ed. by D.Bainov and V. Covachev), 61-71. Zeist/The Netherlands: VPS International Science Publishers 1993. https://doi.org/10.1515/9783112314111-007
- [25] D.S. Mitrinovic, Analytic Inequalities, Berlin et al: Springer 1970. https://doi.org/10.1007/978-3-642-99970-3
- [26] R. Păltănea, On the estimate of the pointwise approximation of functions by linear positive functionals, Studia Univ. Babes-Bolyai 53 (1990), no.1, pp. 11–24.
- [27] L.L. Schumaker, Spline Functions: Basic Theory, New York: John Wiley & Sons 1981.
- [28] Bl. Sendov, On a theorem of Ju. Brudnyi, Math. Balkanica (N.S.), 1 (1987), no.1, pp. 106–111.
- [29] Bl. Sendov, V.A. Popov, Averaged Moduli of Smoothness (Bulgarian), Sofia: Publishing House of the Bulgarian Academy of Sciences 1983.
- [30] Bl. Sendov, V.A. Popov, The Averaged Moduli of Smoothness, New York: John Wiley & Sons 1988.
- [31] P.C. Sikkema, Der Wert einiger Konstanten in der Theorie der Approximation mit Bernstein-Polynomen, Numer. Math.,3 (1961), pp. 107–116. https://doi.org/10.1007/BF01386008
- [32] S.B. Stečkin, The approximation of periodic functions by Féjer sums (Russian), Trudy Mat. Inst. Steklov 62 (1961), 48-60, English translation: Amer. Math. Soc. Transl., (2) 28 (1963), pp. 269–282. https://doi.org/10.1090/trans2/028/14
-
[33]
H. Withney, On functions with bounded
-th differences, J. Math. Pures et Appl., 36 (1957), pp. 67–95. - [34] Ding-xuan Zhou, On a problem of Gonska. To appear in Resultate Math.
- [35] V.V. Zhuk, Functions of the Lip 1 class and S.N. Bernstein’s polynomials (Russian), Vestnik Leningrad.Univ. Math Mekh. Astronom. 1989, vyp.1, pp. 25–30.