In the present note we modify a linear positive Markov process of discrete type by using so called multiplicative calculus. In this framework, a convergence property and the error of approximation are established. In the final part some numerical examples are delivered.
Authors
O. Agratini
Babes-Bolyai University, Faculty of Mathematics and Computer Science, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
H. Karsli
Bolu Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, Golkoy Bolu, Turkey
Keywords
Linear positive operator; non-Newtonian calculus; modulus of multiplicative smoothness.
References
See the expanding block below.
Paper coordinates
O. Agratini, Modifying an approximation process using non-Newtonian calculus, Stud. Univ. Babes-Bolyai Math. 65 (2020) no. 2, 291–301, doi: 10.24193/subbmath.2020.2.10
Modifying an approximation process using
non-Newtonian Calculus
Octavian Agratini
Babeş-Bolyai University,
Faculty of Mathematics and Computer Science,
Str. Kogălniceanu,
1, 400084 Cluj-Napoca, Romania
and
Tiberiu Popoviciu Institute of Numerical
Analysis, Romanian Academy,
Str. Fântânele, 57, 400320 Cluj-Napoca,
Romania
agratini@math.ubbcluj.roHarun Karsli
Bolu Abant Izzet Baysal University
Faculty of Science and Arts,
Department of Mathematics,
14030 Golkoy Bolu, Turkey
karsli_h@ibu.edu.tr
Abstract.
In the present note we modify a linear positive
Markov process of discrete type by using so called multiplicative calculus.
In this framework, a convergence property and the error of approximation are
established.
In the final part some numerical examples are delivered.
Key words and phrases:
Linear positive operator, non-Newtonian calculus, modulus of multiplicative smoothness.
1991 Mathematics Subject Classification:
41A36, 47S30.
1. Introduction
The study of the linear methods of approximation, which are given by
sequences of linear and positive operators, has become a firmly rooted part
of an old area of mathematical research called Approximation Theory. It has
a great potential for applications to a wide variety of issues.
The starting point of this note is a general approximation process of
discrete type which acts on the real valued functions defined on a compact
interval . Since a linear substitution maps any compact
interval into , we will only consider functions defined on . Each operator of the class to which we refer, uses an
equidistant network with a flexible step of the form , where is a strictly decreasing
sequence of real numbers with the property
(1.1)
The operators we are referring to are designed as follows
(1.2)
where the function is
continuous for each . The
condition (1.2) guarantees that is indeed a network on the
compact .
Typically, the operators described by (1.2) satisfy the condition of
reproducing constants. Being linear operators, this property is involved in
achieving the following identity
(1.3)
Throughout the paper, we consider this as a working hypothesis. Note that
such operators are called Markov operators.
At this point we refer to non-Newtonian calculus also called as multiplicative calculus. In the 1970s, Michael Grossman and Robert Katz
[5] have developed this type of calculation moving the roles of
subtraction and addition to division and multiplication. See also Dick
Stanley’s paper [8]. This type of calculus was also called geometric calculus in order to emphasize that changes in function arguments
are measured by differences, while changes in values are measured in ratios.
Recently, Bashirov et al. [1] have given the complete mathematical
description of multiplicative calculus. In the last decade there have been
extensions of this notion in different directions of mathematics, even if it
has a relatively restrictive area of applications than the classical
calculus of Newton and Leibniz covering only positive quantities. Of the
area where this type of approach has proven efficacy, we mention the theory
of economic growth, see, for example, [4].
In the present paper our aim is to bring up multiplicative calculus to the
attention of researchers in the branch of positive approximation processes.
We could find no references that treat any kind of multiplicative calculus
to the above mentioned directions.
In the next two sections we present basic elements of multiplicative
calculus and new results as regards the positive approximation processes. A
particular case is treated at the end of the paper.
2. Preliminaries
We introduce the second central moment of , ,
operators, i.e.,
. Taking into account Bohman-Korovkin criterion,
since (1.3) is fulfilled,
in order the sequence to become
an approximation process on the space , it is necessary and
sufficient to take place the following relation
(2.1)
We also consider that this identity is achieved.
Set . Also, stands for all strictly
positive real valued functions defined on and
We collected some information about multiplicative calculus. Here, the
symbol represents the difference in non-Newtonian geometric
calculus which means the division in the classical calculus.
Consequently, means , provided that makes sense.
In non-Newtonian geometric calculus, the multiplicative absolute value (or
modulus) of an element be a number such that
Owing to the definition of multiplicative absolute value, the multiplicative
distance between two elements is given by
From the definition of the multiplicative absolute value, it is obvious that
for all
For a closed interval , denoting , we present the following
Definition 2.1.
A function is said to tend to the limit
as tends to , if, corresponding to any arbitrary chosen
number , there exists a positive number such
that
for all values of for which
Here
and
We use the notation or , .
Definition 2.2.
A function is said to be multiplicative
continuous at , if
holds.
In other words, a function is said to be
multiplicative continuous at , if, corresponding to any arbitrary
chosen number , there exists a positive number
such that
for all values of for which
Similar to the classic modulus of smoothness, can be defined the modulus of
multiplicative smoothness.
Definition 2.3.
Let be an interval and . The modulus of multiplicative smoothness of is denoted by and is defined as follows
ii) if , then ,
consequently is a non-decreasing
function.
We associate to the operators defined by (1.2) with the fulfillment of
hypotheses (1.3) and (2.1), the following operators
(2.3)
for each function . This new class of operators loses the
linearity property.
We also notice that it keeps the constants. Indeed, by virtue of property (1.3), if , , then , .
Further on, our goal is to highlight approximation properties of the
sequence .
3. Results
Theorem 3.1
Let and the operators , , be defined by (2.3). The following
relation
(3.1)
holds at each point of multiplicative continuity of .
Proof. Let be arbitrarily fixed. In
order to prove the theorem we have to show
holds true at each point of multiplicative continuity of
If is a constant function then one has
and hence
holds true at every point This proves (3.1) for
constant functions.
Now, we assume that is not a constant function. Since
the multiplicative absolute value is always greater than or equal to 1, it is
sufficient only to show that the inequality
(3.2)
is valid for , being a certain rank. By using (2.3) and (1.3) we can write
(3.3)
Since , in
accordance with Definition 2.2, there exists a positive number such that
(3.4)
for all values of for which
(3.5)
We split up the set as follows
Returning at (3.2) we break down the product as follows
The first product can be increased in the following way
If , then we obtain exactly the inequality (3.2). Otherwise , choosing , we obtain the same
inequality from (3.2) with which does
not alter the statement.
In this moment, the proof of (3.2) is completed, consequently (3.1)
takes place.
Next, we establish an upper bound of the error of approximation by using the
modulus of multiplicative smoothness.
Theorem 3.2
Let and the operators , , be defined by (2.3). For large
enough, the following relation
(3.11)
holds at each point of multiplicative continuity of .
The constant is defined at (3.9).
Proof. We use the identity (3.3) and the
decomposition of that product according to the relation (3). Based
both on the definition of , see (2.2), and the inequalities set out in (3.10) that are valid for any and sufficiently large, we can write
Choosing , we obtain the desired result.
4. A special case
In this section, we give a particular example of operators satisfying the
assumptions employed in the previous sections.
By choosing
where is the Bernstein basis, then we obtain a special
case of the operators (2.3), namely multiplicative (geometric) version
of the celebrated Bernstein operators. More precisely, is
given by
(4.1)
As a consequence of Theorem 3.1 for functions and for
multiplicative (geometric) Bernstein operators
we have the following direct estimate:
Corollary 4.1
Let be a function. Let the operators , , be defined by (4.1). The
following relation
holds at each point of multiplicative continuity of .
4.2. Graphical and Numerical Representations
In the recent period, many operators have been investigated
that generalize the classical approximation operators and
the theoretical approach is usually accompanied by
illustrations of convergence properties of particular functions.
The included graphics are realized using software programs.
Among such papers, we randomly quote
[3], [7], [2], [6],
the last three appeared in the years 2019-2020.
Following this line,
we give some graphical and numerical examples to illustrate the
approximation results for multiplicative (geometric) Bernstein operators
obtained in the present paper.
We note that in the Figures 1, 2 and 3, the graph with the red line belongs to
the original function, the graph with the green line to the operators with
, and finally the graph consisting of blue line to the operators with
.
Example 4.2.
Let us consider the function
and we take its corresponding multiplicative Bernstein
operator
(4.1), that one
has for and for
Figure 1. Approximation of by multiplicative Bernstein
operators (4.1), for and .
In the following tables, by using the Wolfram Mathematica 11 Program, we
compute the error of approximation numerically at certain points for and
,
,
and finally
.
Example 4.3.
Let us consider the function and we
take its corresponding multiplicative Bernstein operator (4.1), that one has for and for
Figure 2. Approximation of by multiplicative Bernstein
operators (4.1), for and .
In the following tables, we compute the error of approximation numerically
at certain points for and
,
,
and finally
Example 4.4.
Let us consider the function
and we
take its corresponding multiplicative Bernstein operator (4.1), that one has for and for
Figure 3. Approximation of by multiplicative Bernstein
operators (4.1), for and .
In the following tables, we compute the error of approximation numerically
at certain points for and
,
,
and finally
Remark 4.5.
Unlike error evaluation for linear and positive operators, from (3.11)
we cannot deduce the convergence property of the sequence to . This note should be regarded as a pioneering
activity in order to introduce multiplicative calculus in the field promoted
by Korovkin type theory.
References
[1]
Bashirov, A.E., Kurpinar, E.M., Özyapici, A.,
Multiplicative calculus and its applications,
J. Math. Anal. Appl., 337(2008), 36-48.
[2]
Cai, Q.-Bo, Zhou, G., Li, J.,
Statistical approximation properties of -Bernstein
operators based on -integers,
Open Mathematics, 17(2019), Issue 1, 487-498.
[3]
Cătinaş, T.,
Some classes of surfaces generated by Nielson
and Marshall type operators on the triangle with one curved side,
Stud. Univ. Babeş-Bolyai Math., 61(2016), no. 3,
305-314.
[4]
Filip, D.A., Piatecki, C.,
A non-newtonian examination of the theory of exogenous
economic growth,
Mathematica Aeterna, Vol. 4(2014), no. 2, 101-117.
[5]
Grossman, M., Katz, R.,
Non-Newtonian Calculus,
Lee Press, Pigeon Cove, MA, 1972.
[6]
Gupta, V., Agrawal, G.,
Approximation for link Ismail-May operators,
Annal of Functional Analysis,
https://doi.org/10.1007/s43034-019-00051-y, online: 13 January 2020.
[7]
Mursaleen, M., Rahman, S., Ansari, K.J.,
Approximation by Jakimovski-Leviatan-Stancu-Durrmeyer type
operators,
Filomat, 33(2019), no. 6, 1517-1530.
[8]
Stanley, D.,
A multiplicative calculus,
Primus:
Problems, Resources, and Issues in Mathematics undergraduate Studies, Vol.
9(1999), no. 4, 310-326.
[1] Bashirov, A.E., Kurpinar, E.M., Ozyapici, A., Multiplicative calculus and its applications, J. Math. Anal. Appl., 337(2008), 36-48.
[2] Cai, Q.-Bo, Zhou, G., Li, J., Statistical approximation properties of λ-Bernstein operators based on q-integers, Open Math., 17(2019), no. 1, 487-498
[3] Catina¸s, T., Some classes of surfaces generated by Nielson and Marshall type operators on the triangle with one curved side, Stud. Univ. Babes-Bolyai Math., 61(2016), no. 3, 305-314.
[4] Filip, D.A., Piatecki, C., A non-newtonian examination of the theory of exogenous economic growth, Mathematica Aeterna, 4(2014), no. 2, 101-117.
[5] Grossman, M., Katz, R., Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA, 1972.
[6] Gupta, V., Agrawal, G., Approximation for link Ismail-May operators, Ann. Funct. Anal., https://doi.org/10.1007/s43034-019-00051-y, online: 13 January 2020.
[7] Mursaleen, M., Rahman, S., Ansari, K.J., Approximation by Jakimovski-LeviatanStancu-Durrmeyer type operators, Filomat, 33(2019), no. 6, 1517-1530.
[8] Stanley, D., A multiplicative calculus, Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 9(1999), no. 4, 310-326.
