/*======================================================================
Core CSS for LaTeXML documents converted to (X)HTML */
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.ltx_page_header,
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.ltx_page_header *[rel~="prev"],
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.ltx_page_footer *[rel~="up"] { display:block; text-align:center; }
.ltx_page_header *[rel~="next"],
.ltx_page_footer *[rel~="next"] { float:right; }
/* What was I trying for here; need more selective rule!
.ltx_page_header .ltx_ref,
.ltx_page_footer .ltx_ref {
margin:0 1em; }
*/
.ltx_page_header li {
padding:0.1em 0.2em 0.1em 1em;}
/* Main content */
.ltx_page_content { clear:both; }
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.ltx_page_logo a { font-variant: small-caps; }
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/* if shown */
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/* If ref got turned into span, it's "this section"*/
.ltx_page_navbar li span.ltx_ref { white-space:normal; overflow:visible; }
/* Ought to be easily removable/overridable? */
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Document Structure; Titles & Frontmatter */
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at let all other styles override it (with more specific rules)*/
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/* Hack to simulate run-in! put class="ltx_runin" on a title or tag
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.ltx_runin + .ltx_para,
.ltx_runin + .ltx_para p,
.ltx_runin + p { display:inline; }
.ltx_outdent { margin-left: -2em; }
/* .ltx_chapter_title, etc should be in ltx-article.css etc.
*/
.ltx_page_main { margin:0px; padding:1em 3em 1em 2em; }
.ltx_tocentry { list-style-type:none; }
/* support for common author block layouts.*/
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with pop-up affiliation, etc. */
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background:white; text-align:left;
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display:block; }
/*======================================================================
Para level */
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td.ltx_subfigure,
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/*======================================================================
Blocks, Lists, Floats */
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.ltx_td.ltx_align_left, .ltx_th.ltx_align_left,
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table.ltx_eqn_align tr.ltx_equation td.ltx_align_left + td.ltx_align_right,
table.ltx_eqn_align tr.ltx_equation td.ltx_align_left + td.ltx_align_center,
table.ltx_eqn_align tr.ltx_equation td.ltx_align_center + td.ltx_align_right,
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.ltx_item .ltx_tag + .ltx_para,
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/* NOTE: Need to try harder to get runin appearance? */
dl.ltx_description dt { margin-right:0.5em; float:left;
font-weight:bold; font-size:95%; }
dl.ltx_description dd { margin-left:5em; }
dl.ltx_description dl.ltx_description dd { margin-left:3em; }
/* Theorems */
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/* Bibliographies */
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/* Indices */
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background-color:transparent;
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text-align:right; }
/*
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max-width:0em; text-align:right; }
*/
.ltx_parbox {text-indent:0em; }
/* NOTE that it is CRITICAL to put position:relative outside & absolute inside!!
I wish I understood why!
Outer box establishes resulting size, neutralizes any outer positioning, etc;
inner establishes position of stuff to be rotated */
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position:relative; bottom:0pt;left:0pt;
overflow:visible; }
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Columns */
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Borders and such */
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/*======================================================================
Misc */
/* .ltx_verbatim*/
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/*======================================================================
Meta stuff, footnotes */
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/*right:5%; */
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/*======================================================================
SVG (pgf/tikz ?) basics */
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.ltx_svg_fog foreignObject > p { margin:0; padding:0; display:block; }
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/*======================================================================
Low-level Basics */
/* Note that LaTeX(ML)'s font model doesn't map quite exactly to CSS's */
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/* dingbats should be converted to unicode? */
/* Math font families handled within math: script, symbol, fraktur, blackboard ? */
/* Font Series => font-weight */
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/* Font Shapes => font-style or font-variant */
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font-style:normal;
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font-variant-numeric: oldstyle-nums; }
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/*
.ltx_font_mathscript { ? }
*/
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.ltx_red { color:red; }
/*.ltx_centering { text-align:center; margin:auto; }*/
/*.ltx_inline-block.ltx_centering,*/
/* Hmm.... is this right in general? */
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/* Dubious stuff */
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display:inline-block;
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.ltx_theorem .ltx_title + .ltx_p { margin-top:0em; }
Păltănea’s operators: old and new results
Păltănea’s operators: old and new results
Abstract.
This note concerns the study of an approximation linear positive process
introduced by R. Păltănea in 2008.
Considering the impact of this class of operators that depends on two
parameters, in a distinct section we realize a brief radiograph of the main
properties highlighted over time in various papers.
Our contribution materialized in the definition and study of the
approximation properties of King variant of these operators.
Mathematics Subject Classification:
41A36, 41A35, 41A25.
Keywords and phrases:
linear approximation process, Durrmeyer type operators,
modulus of smoothness, King type operator.
1. Introduction
This note falls under the field of Approximation Theory, more precisely
it aims at the study of linear and positive approximation processes.
It is known that Szász operators, along with the many discrete
and integral generalizations obtained over time, play a significant
role in this domain.
Our present study concerns a class of Szász-Durrmeyer type operators
introduced by Păltănea in 2008 [9].
They depend on two parameters and are described as follows.
For , and ,
|
|
|
(1) |
where
|
|
|
|
|
|
|
|
(2) |
and is a locally integrable function for which
formula (1) is well defined for all .
Păltănea operators preserve affine functions and make a link
between the Phillips operators and the classical Szász operators.
In the particular case and ,
the above operators become Phillips operators [11]
|
|
|
In the limit case (see Theorem 2)
one obtains the classical Szász operators
defined by
|
|
|
(3) |
The extension in the Durrmeyer sense of operators was achieved
by Mazhar and Totik [7] in 1985.
Unlike Păltănea operators, this extension does not reproduce
affine functions.
Our goal is twofold: to collect some known properties of Păltănea’s
operators in a brief synthesis as well as to construct a version
of King type by investigating its utility and its approximation properties.
Wanting to create a self-contained presentation, all the notions used
are described explicitly.
As much as possible, we have kept the original notations used in the
papers cited.
2. An eclectic collection of known results
We did not propose an exhaustive presentation of the results obtained
over time, but scoring of the most significant properties of this
approximation process.
Păltănea studied this class of operators in the papers
[9], [10], the following main properties being proved.
Set , the space of functions
which are Riemann integrable on each compact interval of
and for which exist certain numbers , such that
, .
For and , denote by the subspace
of of those functions which satisfy the above inequality
with .
One has
In [9, Theorem 2.1] it was showed that
exists for any , , .
An approximation property in the particular case
will be read as follows.
Theorem 1.
([9, Theorem 3.4]).
For any function and any number ,
there exists such that exists for
and for any compact set
we have
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The limit of the functions has been established when
tends to infinity.
Theorem 2.
([10, Theorem 4]).
For any , any and any , there is
such that exists for all and
we have
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where the Szász operator is defined in (3).
Also it was proved that preserves convexity of higher order
and it has the property of simultaneous approximation on the compact
sets
[10, Theorems 6, 9].
Further, we will highlight some papers based on Păltănea
operators and which bear his name in the title.
We recall that in [5] the authors gave a generalization of Szász operators
based on Appell polynomials.
Let
be an analytic function in the disc
, , and .
The Appell polynomials , ,
are defined by the generating function
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(4) |
For , Verma and Gupta [14] proposed the
Jakimovski-Leviatan-Păltănea operators defined as
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(5) |
where is defined at (1) and
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(6) |
being described at (4).
To establish the rate of convergence, the authors used the moduli
of smoothness of the first and second order which give direct information
about the smoothness of .
We recall their definitions.
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(7) |
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(8) |
where , the space of continuous and bounded real
valued functions defined on .
The following result was proved.
Theorem 3.
([14, Theorem 2]).
For , we have
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where
Considering the space
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being a constant depending on , a Voronovskaja type asymptotic
formula was also obtained.
Theorem 4.
([14, Theorem 3]).
For any function
such that , are continuous and belong to ,
we have
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Goyal and Agrawal [3] defined and studied
the Bézier variant of the operators
, .
Set ,
where is fixed.
For and ,
the Jakimovski-Leviatan-Păltănea-Bézier operator is of the form
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where
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see
[3, Eq. (1.2)].
Clearly,
defined by (6), .
For , turns out to be
defined by (5).
A substantial result is the establishment of the rate of convergence
for functions having a derivative of bounded variation.
Let , ,
be the class of all functions defined on having a derivative
of bounded variation on every bounded subinterval of
and any function of this class enjoys the property
, .
Theorem 5.
([3, Theorem 5]).
Let , and let
be the total variation of on .
For every and sufficiently large , we have
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where and the function is defined by
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Motivated by the above mentioned construction, in [8] the authors
introduced the Bézier-Păltănea operators based on Gould-Hopper
polynomials.
For this new generalization of Păltănea operators,
the authors obtained both the quantitative Voronovskaja type
theorem in terms of Ditzian-Totik modulus of smoothness and the rate
of pointwise convergence for the functions having a derivative of bounded
variation.
In the final part we mention a recent result obtained by Gupta and
Agrawal [4].
They proposed a hybrid integral type operator containing both Szász as well
as Baskakov bases in summation.
More precisely, in (1) they replaced ,
, , with , where
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being a constant belonging to the interval .
In the above stands for rising factorial, also called
Pochhammer function.
We recall is taken to be 1.
For these new operators, the notation
was used.
Among the results obtained we mention a Grüss-Voronovskaja type theorem.
Setting
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(9) |
where ,
the following statement was proved.
Theorem 6.
([4, Theorem 3.2]).
Let belong to .
For any we have
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3. On the King variant of the operators
Set , , .
Two decades ago, King [6] had the idea to modify the Bernstein
operators such that to reproduce the monomials and .
Consequently, the modified operators enjoy the property of keeping
the functions
as fixed points, for any real constants and .
From approximation theory point of view the construction is useful.
In spite of the fact that the new operators have the degree of exactness
null, the maximum rate of convergence is smaller.
Over time this technique was applied to many linear approximation
processes, becoming known as the King method.
We propose to apply it to Păltănea operators (1).
It is known that
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(10) |
see [10, Eq. (2.1)].
Considering
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(11) |
where ,
we define the operators
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(12) |
.
Remarks.
(i) By using a bivariate kernel, we can write (12) in a more compact
form, as follows
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where
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In the above represents Dirac delta function for which
(ii) For any we can easily deduce that the operators
are non-expansive, this means
.
The proof uses the identities
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(13) |
Relations (10) and (11) involve the identities
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(14) |
Since any compact interval is isomorphic
to , arbitrarily fixed, in our approach will use only this interval.
Theorem 7.
For any , any function
and any number ,
there exists such that for ,
is well-defined and
we have
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Let be arbitrarily fixed.
The relation
uniformly on takes place.
Based on (14), the proof of the above theorem follows exactly
the same line as the proof of Theorem 3.4 from [9],
so we omit it.
For a positive linear operator its second central
moment defined by
where
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(15) |
plays a crucial role when estimating its local rate of convergence.
The identities (10) and (14) imply
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(16) |
It turns out that the second central moment of their King type
variant (12)
is smaller than the second central moment of the
Păltănea operators
(1) on the whole interval .
Lemma 1.
(ii) The inequality
is valid, for each .
Proof.
(i) Let . The first inequality follows from the observation
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since .
Furthermore, we have
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which proves the second inequality.
(ii) By (16), this statement is a consequence of the
previous result.
∎
By virtue of the classical results regarding the local rate of convergence
established by Shisha and Mond [12], the relations (10),
(14) and (16)
guarantee
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for any , where is defined at (7).
Remark.
Since associated with a function is an increasing
function, Lemma 1 (ii) demonstrates that the upper bound for
the absolute error of is smaller than
that for .
The evaluation of the rate of convergence can be carried out in weighted
spaces, for example in defined at (9)
and endowed with the usual norm
,
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where , .
Theorem 8.
Let be defined by (12).
For every ,
converges to in norm, i.e.,
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(17) |
Proof.
It is known that is a Korovkin set in ,
see, e.g.,
[1, Proposition 4.2.5.-(6)].
Taking in view identities (14), it remains for us to prove
(17) only for .
Applying two times Lemma 1 (i), we obtain the estimate
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for all , from which we deduce
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Thus, we got what we proposed, consequently (17) takes place.
∎
To obtain the following new result we need an inequality that we present
in what follows.
Any discrete or integral linear positive operator of summation
type satisfies the classical inequality
where is given at (15).
Because the operators contain as the first term
a quantity not included in the sum, for a self-contained presentation,
we prove the relation
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(18) |
The proof is based on Cauchy–Schwarz inequality both for integrals and for series,
and it runs as follows.
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see (13).
Further, we define
and
.
We get
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and the proof of (18) is completed.
A less frequently used tool to approximate signals is the so called
Steklov mean.
The benefit of special function is that continuous functions can be
approximated by smoother functions.
For , the Stelov mean of second order and step
is defined by
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(19) |
and verifies the inequalities
and if
exist,
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(21) |
In the above stands for the sup-norm,
, .
The key of the proofs of these relations consists in rewriting the definitions
(7) and (8) as follows
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where .
The proofs of (20) and (21) can be found in
[2, Eqs. (5.2)-(5.4)].
Remark.
For the full information of the reader, in accordance with
The Great Soviet Encyclopedia, 3rd Edition (1969-1978),
we mention that the initial form of this type of function was introduced
in 1907 by Vladimir Steklov (Stekloff) [13] by the equality
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where is so small that the interval lies in the domain
of the definition of the locally integrable function .
Theorem 9.
Let be defined by (11).
For every and , the following inequality
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holds.
Proof.
Let be arbitrarily fixed.
For , our relation is obvious.
Let .
Applying the Steklov mean given at (19), we can write
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(22) |
Using the fact that the operators are non-expansive and taking in view
(20), we obtain
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Further, using successively Taylor’s expansion, the identity
and relations (18), (21) we get
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At this point we choose
and returning at (3) we assemble the established increases.
Our statement is fully motivated.
∎
References
-
[1]
F. Altomare, M. Campiti,
Korovkin-type Approximation Theory and its Applications,
de Gruyter Studies in Mathematics, Vol. 17, Walter de Gruyter,
Berlin, 1994.
-
[2]
A. Aral, V. Gupta,
(p,q)-Type Beta functions of second kind,
Adv. Oper. Theory, 1(2016), no. 1, 134-146.
-
[3]
M. Goyal, P.N. Agrawal,
Bézier variant of the Jakimovski-Leviatan-Păltănea
operators based on Appell polynomials,
Ann. Univ. Ferrara, 63(2017), 289–302.
-
[4]
V. Gupta, P.M. Agrawal,
Approximation by modified Păltănea operators,
Publications de l’Institute Mathematique, 107(121)(2020), 137–164.
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[5]
A. Jakimovski, D. Leviatan,
Generalized Szász operators for the approximation in the infinite
interval,
Mathematica (Cluj), 11(1969), 97–103.
-
[6]
J.P. King,
Positive linear operators which preserve ,
Acta Math. Hungar., 99(2003), No. 3, 203–208.
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[7]
S.M. Mazhar, V. Totik,
Approximation by modified Szász operators,
Acta Sci. Math. (Szeged), 49(1985), 257–269.
-
[8]
M. Mursaleen, S. Rahman, K.J. Ansari,
On the approximation by Bézier-Păltănea operators based
on Gould-Hopper polynomials,
Math. Commun., 24(2019), No. 2, 147–164.
-
[9]
R. Păltănea,
Modified Szász-Mirakjan operators of integral form,
Carpathian J. Math., 24(2008), No. 3, 378–385.
-
[10]
R. Păltănea,
Simultaneous approximation by a class of Szász-Mirakjan operators,
J. Applied Functional Analysis, 9(2014), Nos. 3-4, 356–368.
-
[11]
R.S. Phillips,
An inversion formula for Laplace transforms and semi-groups
of linear operators,
Annals of Mathematics, 59(1954), No. 2, 325–356.
-
[12]
O. Shisha, B. Mond,
The degree of convergence of linear positive operators,
Proc. Nat. Acad. Sci. USA, 60(1968), 1196–1200.
-
[13]
V. Steklov,
Sur les expressions asymptotiques de certaines fonctions,
définies par les équations différentielles linéaires du
second ordre, et leurs applications au problème du développement d’une
fonction arbitraire en séries procédant suivant les-dites fonctions,
Communications de la Société Mathématique de Kharkow,
2-ée série, 10(1907), 97-199.
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[14]
D.K. Verma, V. Gupta,
Approximation for Jakimovski-Leviatan-Păltănea operators,
Ann. Univ. Ferrara, 61(2015), No. 2, 367–380.
