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Approximation in Weighted Space with Generalized Max-Product
Type Favard-Szász-Mirakyan-Durrmeyer Operators
Nasan ALSATTUF, Sevilay KIRCI SERENBAY
Harran University, Institute of Science, Mathematics, Şanlıurfa, Türkiye
DOI : https://doi.org/10.51583/IJLTEMAS.2024.131025
Received: 02 November 2024; Accepted: 12 November 2024; Published: 18 November 2024
Abstract: In this paper, we explore the uniform approximation of functions using Generalized Favard- Szász-Mirakyan -
Durrmeyer operators of the max-product type with specific exponential weighted spaces. We analyze the approximation rate with
an appropriate continuity modulus.
Keywords: Durrmeyer Operators, Exponential Weighted Spaces, Maximum product operators, Approximation rate.
I. Introduction
Let be a function defined on . The Favard- Szász-Mirakyan operators
applied to are given by
The approximation properties of Favard- Szász-Mirakyan operator have been studied by many authors; a few examples
[2],[6],[13] and [14].
ve The Durrmeyer-type modification, as defined by Mazhar and Totik [12], is given by,
,
where
and
The linear structure is not always preserved or sufficient in approximation operators. Furthermore, standard algebraic operations
such as addition and subtraction may be inadequate. To address these issues, maximum-product type approximation operators
were introduced, utilizing the maximum algebraic operation. These operators are nonlinear and positive, thereby expanding the
range of tools available in approximation theory. There is extensive research in this field [1-5],[8-11].
The max-product type Favard-Szász-Mirakyan operator is defined as follows:
where and
is considered bounded on
(Bede et al. [4])
Many modifications of the max-product type Favard-Szász-Mirakyan operator have been studied. The Durrmeyer generalization
of these operators has been investigated in [3] and [4] for continuous and bounded functions defined on the interval [0,1] and a
different generalization has also been explored in [5].
In this paper, we demonstrate that the operators
can be utilized for uniform approximation with the weight
, where
. We also examine the rate of convergence of these operators. determine the rate of convergence of these operators to
the identity operator.
Any sequence of positive linear operators can be used for the uniform approximation of functions across a wide range of weights
defined by
, which are associated with certain operators ( [7] and [8]).
be a function defined over a non-compact interval . This function is continuous and strictly monotonic. The
interval corresponds precisely to . For , we denote the space of continuous functions as follows:
This space can be endowed with the norm
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
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.
The modulus of continuity
is given for every
and as follows
where the supremum is taken for all and such that
.
For and
we get the usual modulus of continuity , (see, [8]).
II. Auxiliary Results
In this section, we define generalized Favard-Szász-Mirakyan-Durrmeyer operators and present some supplementary results to
investigate the approximation properties of these operators.
Definition 1
, an integrable function and
.
the operator is referred to as max-product type Favard-Szász-Mirakyan -Durrmeyer operator, where
).
Definition 2
, an integrable function and
,
the operator defined by the above equality is referred to as the generalized max-product type Favard-Szász-Mirakyan-Durrmeyer
Operators, where
.
Lemma 1 For and
being a positive, increasing, and unbounded real sequence, we consider the following intervals for
and
, .
The intervals are non-empty, disjoint, and collectively they span the positive half line.
Indeed
Lemma 2 If
then,
.
Proof Let us denote
We get
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,
,
as opposed to
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.
Lemma 3 For and
being a positive, increasing, and unbounded real sequence, we have,
Proof For
and from lemma 2,
Let
, from lemma 2, for we have
and
and
,
we have
Using the inequality
we get ,
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.
Remark 1 We have
for and
. Indeed
from Lemma 3 and
,
Remark 2 For
, for every
for this reason
,
.
Lemma 4 For all and , the following inequality holds:
.
Proof
and
.
If then
and
because it is ,
If then
and
because it is ,
If then
and if
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It remains to evaluate the maximum of
, for
.
Let us denote
, we get
for every integer
we get
by taking
, and so on. Let us denote
We deduce that if
then
1} for we obtain
Lemma 5 For and , nℕ, there exists a
such that the following inequality holds:
Proof We have,
and
for ,
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,
For and we have
,
and for
we have
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Lemma 6 For , and
we have
.
Proof We have
where
from lemma 5,
.
Approximation in Weighted Space with Generalized Max-Product Type Favard-Szász-Mirakyan-Durrmeyer Operators
Theorem 1 For
and
,
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,
.
Proof
we get,
and
according to this
using remark 1 and lemma 6 also taking the value
) we obtain
which proves the theorem.
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INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
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www.ijltemas.in Page 221
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