/*======================================================================
Core CSS for LaTeXML documents converted to (X)HTML */
/* Generic Page layout */
.ltx_page_header,
.ltx_page_footer { font-size:0.8em; }
.ltx_page_header *[rel~="prev"],
.ltx_page_footer *[rel~="prev"] { float:left; }
.ltx_page_header *[rel~="up"],
.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; }
.ltx_page_header { border-bottom:1px solid; margin-bottom:5px; }
.ltx_page_footer { clear:both; border-top:1px solid; margin-top:5px; }
.ltx_page_header:after,
.ltx_page_footer:after,
.ltx_page_content:after {
content:"."; display:block; height:0; clear:both; visibility:hidden; }
.ltx_page_footer:before {
content:"."; display:block; height:0; clear:both; visibility:hidden; }
.ltx_page_logo { font-size:80%; margin-top: 5px; clear:both; float:right; }
.ltx_page_logo a { font-variant: small-caps; }
.ltx_page_logo img { vertical-align:-3px; }
/* if shown */
.ltx_page_navbar li { white-space:nowrap; display:block; overflow:hidden; }
/* 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? */
.ltx_pagination.ltx_role_newpage { height:2em; }
/*======================================================================
Document Structure; Titles & Frontmatter */
/* undo bold here to remove the browser's native h# styling,
at let all other styles override it (with more specific rules)*/
.ltx_title { font-size:100%; font-weight:normal; }
/* Hack to simulate run-in! put class="ltx_runin" on a title or tag
for it to run-into the following text. */
.ltx_runin { display:inline; }
.ltx_runin:after { content:" "; }
.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.*/
/* add class ltx_authors_1line to get authors in single line
with pop-up affiliation, etc. */
.ltx_authors_1line .ltx_creator,
.ltx_authors_1line .ltx_author_before,
.ltx_authors_1line .ltx_author_after { display:inline;}
.ltx_authors_1line .ltx_author_notes { display:inline-block; }
.ltx_authors_1line .ltx_author_notes:before { content:"*"; color:blue;}
.ltx_authors_1line .ltx_author_notes span { display:none; }
.ltx_authors_1line .ltx_author_notes:hover span {
display:block; position:absolute; z-index:10;
background:white; text-align:left;
border: 1px solid black; border-radius: 0 5px 5px 5px; box-shadow: 5px 5px 10px gray; }
/* add class=ltx_authors_multiline to get authors & affliations on separate lines*/
.ltx_authors_multiline .ltx_creator,
.ltx_authors_multiline .ltx_author_before,
.ltx_authors_multiline .ltx_author_after,
.ltx_authors_multiline .ltx_author_notes,
.ltx_authors_multiline .ltx_author_notes .ltx_contact {
display:block; }
/*======================================================================
Para level */
.ltx_float {
margin: 1ex 3em 1ex 3em; }
td.ltx_subfigure,
td.ltx_subtable,
td.ltx_subfloat { width:50%; }
/* theorems, figure, tables, floats captions.. */
/*======================================================================
Blocks, Lists, Floats */
.ltx_p,
.ltx_quote,
.ltx_block,
.ltx_para {
display: block; }
/* alignment within blocks */
.ltx_align_left { text-align:left; }
.ltx_align_right { text-align:right; }
.ltx_align_center { text-align:center; }
.ltx_align_justify { text-align:justify; }
.ltx_align_top { vertical-align:top; }
.ltx_align_bottom { vertical-align:bottom; }
.ltx_align_middle { vertical-align:middle; }
.ltx_align_baseline { vertical-align:baseline; }
.ltx_align_floatleft { float:left; }
.ltx_align_floatright { float:right; }
.ltx_td.ltx_align_left, .ltx_th.ltx_align_left,
.ltx_td.ltx_align_right, .ltx_th.ltx_align_right,
.ltx_td.ltx_align_center, .ltx_th.ltx_align_center { white-space:nowrap; }
.ltx_td.ltx_align_left.ltx_wrap, .ltx_th.ltx_align_left.ltx_wrap,
.ltx_td.ltx_align_right.ltx_wrap, .ltx_th.ltx_align_right.ltx_wrap,
.ltx_td.ltx_align_center.ltx_wrap, .ltx_th.ltx_align_center.ltx_wrap,
.ltx_td.ltx_align_justify, .ltx_th.ltx_align_justify { white-space:normal; }
.ltx_tabular .ltx_tabular { width:100%; }
.ltx_inline-block { display:inline-block; }
/* equations in non-aligned mode (not normally used) */
.ltx_eqn_div { display:block; width:95%; text-align:center; }
/* equations in aligned mode (aligning tags, etc as well as equations) */
.ltx_eqn_table { display:table; width:100%; border-collapse:collapse; }
.ltx_eqn_row { display:table-row; }
.ltx_eqn_cell { display:table-cell; width:auto; }
/* Padding between column pairs in ams align */
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,
table.ltx_eqn_align tr.ltx_equation td.ltx_align_center + td.ltx_align_center { padding-left:3em; }
table.ltx_eqn_eqnarray tr.ltx_eqn_lefteqn + tr td.ltx_align_right { min-width:2em; }
.ltx_eqn_eqno { max-width:0em; overflow:visible; white-space: nowrap; }
.ltx_eqn_eqno.ltx_align_right .ltx_tag { float:right; }
.ltx_eqn_center_padleft,
.ltx_eqn_center_padright { width:50%; min-width:2em;}
.ltx_eqn_left_padleft,
.ltx_eqn_right_padright { min-width:2em; }
.ltx_eqn_left_padright,
.ltx_eqn_right_padleft { width:100%; }
/* Various lists */
.ltx_itemize,
.ltx_enumerate,
.ltx_description {
display:block; }
.ltx_itemize .ltx_item,
.ltx_enumerate .ltx_item {
display: list-item; }
/* Position the tag to look like a normal item bullet. */
li.ltx_item > .ltx_tag {
display:inline-block; margin-left:-1.5em; min-width:1.5em;
text-align:right; }
.ltx_item .ltx_tag + .ltx_para,
.ltx_item .ltx_tag + .ltx_para .ltx_p { display:inline; }
/* 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 */
.ltx_theorem {margin:1em 0em 1em 0em; }
.ltx_title_theorem { font-size:100%; }
/* Bibliographies */
.ltx_bibliography dt { margin-right:0.5em; float:left; }
.ltx_bibliography dd { margin-left:3em; }
/*.ltx_biblist { list-style-type:none; }*/
.ltx_bibitem { list-style-type:none; }
.ltx_bibitem .ltx_tag { font-weight:bold; margin-left:-2em; width:3em; }
/*.bibitem-tag + div { display:inline; }*/
.ltx_bib_title { font-style:italic; }
.ltx_bib_article .bib-title { font-style:normal !important; }
.ltx_bib_journal { font-style:italic; }
.ltx_bib_volume { font-weight:bold; }
/* Indices */
.ltx_indexlist li { list-style-type:none; }
.ltx_indexlist { margin-left:1em; padding-left:1em;}
/* Listings */
.ltx_listing {
display:block;
margin: 1ex 3em 1ex 0em;
overflow-x:auto;
text-align: left; }
.ltx_float .ltx_listing {
margin: 0; }
.ltx_listingline { white-space:nowrap; min-height:1em; }
.ltx_lst_numbers_left .ltx_listingline .ltx_tag {
background-color:transparent;
margin-left:-3em; width:2.5em;
position:absolute;
text-align:right; }
.ltx_lst_numbers_right .ltx_listingline .ltx_tag {
background-color:transparent;
width:2.5em;
position:absolute; right:3em;
text-align:right; }
/*
position:absolute; left:0em;
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 */
.ltx_transformed_outer {
position:relative; bottom:0pt;left:0pt;
overflow:visible; }
.ltx_transformed_inner {
display:block;
position:absolute;bottom:0pt;left:0pt; }
.ltx_transformed_inner > .ltx_p {text-indent:0em; margin:0; padding:0; }
/* If simulating a table (html5), try to get rowspan to work...sorta? */
span.ltx_rowspan { position:absolute; top:0; bottom:0; }
/* by default, p doesn't indent */
.ltx_p { text-indent:0em; white-space:normal; }
/* explicit control of indentation (on ltx_para) */
.ltx_indent > .ltx_p:first-child { text-indent:2em!important; }
.ltx_noindent > .ltx_p:first-child { text-indent:0em!important; }
/*======================================================================
Columns */
.ltx_page_column1 {
width:44%; float:left; } /* IE uses % of wrong container*/
.ltx_page_column2 {
width:44%; float:right; }
.ltx_page_columns > .ltx_page_column1 {
width:48%; float:left; }
.ltx_page_columns > .ltx_page_column2 {
width:48%; float:right; }
.ltx_page_columns:after {
content:"."; display:block; height:0; clear:both; visibility:hidden; }
/*======================================================================
Borders and such */
.ltx_tabular { display:inline-table; border-collapse:collapse; }
.ltx_tabular.ltx_centering { display:table; }
.ltx_thead,
.ltx_tfoot,
.ltx_tbody { display:table-row-group; }
.ltx_tr { display:table-row; }
.ltx_td,
.ltx_th { display:table-cell; }
.ltx_framed { border:1px solid black;}
.ltx_tabular .ltx_td,
.ltx_tabular .ltx_th { padding:0.1em 0.5em; }
/* regular lines */
.ltx_border_t { border-top:1px solid black; }
.ltx_border_r { border-right:1px solid black; }
.ltx_border_b { border-bottom:1px solid black; }
.ltx_border_l { border-left:1px solid black; }
/* double lines */
.ltx_border_tt { border-top:3px double black; }
.ltx_border_rr { border-right:3px double black; }
.ltx_border_bb { border-bottom:3px double black; }
.ltx_border_ll { border-left:3px double black; }
/* Light lines */
.ltx_border_T { border-top:1px solid gray; }
.ltx_border_R { border-right:1px solid gray; }
.ltx_border_B { border-bottom:1px solid gray; }
.ltx_border_L { border-left:1px solid gray; }
/* Framing */
.ltx_framed_rectangle { border-style:solid; border-width:1px; }
.ltx_framed_top { border-top-style:solid; border-top-width:1px; }
.ltx_framed_left { border-left-style:solid; border-left-width:1px; }
.ltx_framed_right { border-right-style:solid; border-right-width:1px; }
.ltx_framed_bottom,
.ltx_framed_underline { border-bottom-style:solid; border-bottom-width:1px; }
.ltx_framed_topbottom { border-top-style:solid; border-top-width:1px;
border-bottom-style:solid; border-bottom-width:1px; }
.ltx_framed_leftright { border-left-style:solid; border-left-width:1px;
border-right-style:solid; border-right-width:1px; }
/*======================================================================
Misc */
/* .ltx_verbatim*/
.ltx_verbatim { text-align:left; }
/*======================================================================
Meta stuff, footnotes */
.ltx_note_content { display:none; }
/*right:5%; */
.ltx_note_content {
max-width: 70%; font-size:90%; left:15%;
text-align:left;
background-color: white;
padding: 0.5em 1em 0.5em 1.5em;
border: 1px solid black; border-radius: 0 5px 5px 5px; box-shadow: 5px 5px 10px gray; }
.ltx_note_mark { color:blue; }
.ltx_note_type { font-weight: bold; }
.ltx_note { display:inline-block; text-indent:0; } /* So we establish containing block */
.ltx_note_content .ltx_note_mark { position:absolute; left:0.2em; top:-0.1em; }
.ltx_note:hover .ltx_note_content,
.ltx_note .ltx_note_content:hover {
display:block; position:absolute; z-index:10; }
.ltx_ERROR { color:red; }
.ltx_rdf { display:none; }
.ltx_missing { color:red;}
.ltx_nounicode { color:red; }
/*======================================================================
SVG (pgf/tikz ?) basics */
/* Stuff appearing in svg:foreignObject */
.ltx_svg_fog foreignObject { margin:0; padding:0; overflow:visible; }
.ltx_svg_fog foreignObject > p { margin:0; padding:0; display:block; }
/*.ltx_svg_fog foreignObject > p { margin:0; padding:0; display:block; white-space:nowrap; }*/
/*======================================================================
Low-level Basics */
/* Note that LaTeX(ML)'s font model doesn't map quite exactly to CSS's */
/* Font Families => font-family */
.ltx_font_serif { font-family: serif; }
.ltx_font_sansserif { font-family: sans-serif; }
.ltx_font_typewriter { font-family: monospace; }
/* dingbats should be converted to unicode? */
/* Math font families handled within math: script, symbol, fraktur, blackboard ? */
/* Font Series => font-weight */
.ltx_font_bold { font-weight: bold; }
.ltx_font_medium { font-weight: normal; }
/* Font Shapes => font-style or font-variant */
.ltx_font_italic { font-style: italic; font-variant:normal; }
.ltx_font_upright { font-style: normal; font-variant:normal; }
.ltx_font_slanted { font-style: oblique; font-variant:normal; }
.ltx_font_smallcaps { font-variant: small-caps; font-style:normal; }
.ltx_font_oldstyle { font-variant: oldstyle-nums; /* experimental css3 ? Doesn't seem to work!*/
font-style:normal;
-moz-font-feature-settings: "onum";
-ms-font-feature-settings: "onum";
-webkit-font-feature-settings: "onum";
font-variant-numeric: oldstyle-nums; }
.ltx_font_mathcaligraphic { font-family: "Lucida Calligraphy", "Zapf Chancery","URW Chancery L"; }
/*
.ltx_font_mathscript { ? }
*/
cite { font-style: normal; }
.ltx_red { color:red; }
/*.ltx_centering { text-align:center; margin:auto; }*/
/*.ltx_inline-block.ltx_centering,*/
/* Hmm.... is this right in general? */
.ltx_centering { display:block; margin:auto; text-align:center; }
/* Dubious stuff */
.ltx_hflipped {
display:inline-block;
-moz-transform: scaleX(-1);
-o-transform: scaleX(-1);
-webkit-transform: scaleX(-1);
transform: scaleX(-1);
filter: FlipH;
-ms-fliter: "FlipH"; }
.ltx_vflipped {
display:inline-block;
-moz-transform: scaleY(-1);
-o-transform: scaleY(-1);
-webkit-transform: scaleY(-1);
transform: scaleY(-1);
filter: FlipV;
-ms-fliter: "FlipV"; }
/* .ltx_phantom handled in xslt */
/* A somewhat amsart-like style */
.ltx_title_document {
text-transform:uppercase; font-size: 150%; text-align:center; }
.ltx_title_section {
font-family:serif; font-weight:normal; font-variant:small-caps; font-size: 120%;
text-align:center; }
.ltx_authors { text-align:center; }
.ltx_author_before,.ltx_author_after,
.ltx_role_author .ltx_personname {
text-transform:uppercase; }
.ltx_abstract { margin-bottom:1em; }
.ltx_title_abstract { font-family:serif; font-weight:normal; font-variant:small-caps;}
.ltx_title_keywords,
.ltx_title_classification { display:inline; font-size:100%; font-weight:normal; font-style:italic; }
.ltx_role_dedicatory { font-size:100%; font-style:italic; text-align:center; margin:1em;}
.ltx_figure .ltx_tag,
.ltx_table .ltx_tag {
font-family:serif; font-variant:small-caps;}
.ltx_title_subsection,
.ltx_title_subsubsection {
font-family:serif; font-weight:bold;}
.ltx_title_subsection:after,
.ltx_title_subsubsection:after,
.ltx_title_paragraph:after { content:"."; }
/* abstract, subsection, subsubsection and paragraph titles are runin */
.ltx_abstract .ltx_title,
.ltx_abstract .ltx_title + .ltx_p { display:inline; }
.ltx_subsection .ltx_title,
.ltx_subsection .ltx_title + .ltx_para,
.ltx_subsection .ltx_title + .ltx_para > .ltx_p { display:inline; }
.ltx_subsubsection .ltx_title,
.ltx_subsubsection .ltx_title + .ltx_para,
.ltx_subsubsection .ltx_title + .ltx_para > .ltx_p { display:inline; }
.ltx_paragraph .ltx_title,
.ltx_paragraph .ltx_title + .ltx_para,
.ltx_paragraph .ltx_title + .ltx_para > .ltx_p { display:inline; }
.ltx_title +.ltx_para > .ltx_p {text-indent:2em; }
.ltx_itemize,
.ltx_enumerate,
.ltx_description
{ margin-left: clamp(1em, 3%, 2em); }
.ltx_title_document { font-size:170%; text-align:center; margin:0.5em 0 0.5em 0; }
.ltx_authors,
.ltx_role_author { text-align:center; margin:0.5em 0 0.5em 0; }
.ltx_role_author .ltx_personname { font-size: 120%; }
.ltx_date { text-align:center; font-size: 120%; margin:0.5em 0 0.5em 0; }
.ltx_subtitle { text-align:center; font-size: 120%; padding-left:0.2em; margin-left:-0.5em; }
.ltx_title_abstract { text-align:center; font-size: 100%; font-weight:bold; }
.ltx_abstract { margin-left:4em; margin-right:4em; }
.ltx_title_acknowledgements,
.ltx_title_keywords,
.ltx_title_classification {
text-align:left; font-size: 100%; font-weight:bold; margin:0.5 0 0 0; }
.ltx_appendix,
.ltx_section,
.ltx_subsection,
.ltx_subsubsection { margin-top:1.5em; }
.ltx_paragraph,
.ltx_subparagraph { margin-top:1.0em; }
.ltx_title_appendix,
.ltx_title_section,
.ltx_title_bibliography { font-size:140%; font-weight:bold; margin-bottom:1em; }
.ltx_title_subsection { font-size:120%; font-weight:bold; margin-bottom:1em; }
.ltx_title_subsubsection { font-size:100%; font-weight:bold; margin-bottom:1em; }
/* Paragraph & Subparagraph titles should be runin! */
.ltx_title_paragraph { font-size:100%; font-weight:bold; display:inline;
margin-right:1em; }
.ltx_paragraph .ltx_title,
.ltx_paragraph .ltx_title + .ltx_para,
.ltx_paragraph .ltx_title + .ltx_para > .ltx_p { display:inline; }
.ltx_title_subparagraph { font-size:100%; font-weight:bold; display:inline;
margin-left:2em; margin-right:1em; }
.ltx_subparagraph .ltx_title,
.ltx_subparagraph .ltx_title + .ltx_para,
.ltx_subparagraph .ltx_title + .ltx_para > .ltx_p { display:inline; }
.ltx_figure { text-align:center; margin:auto; margin:0.5em; }
.ltx_table { text-align:center; margin:auto; margin:0.5em; }
/* experimental: flex model, to center short captions, justify long ones */
.ltx_table,
.ltx_figure { display:flex; flex-direction:column; align-items:center; }
.ltx_table .ltx_caption,
.ltx_figure .ltx_caption { text-align:justify; }
/* first p in para gets indented */
.ltx_para > .ltx_p:first-child { text-indent:2em; }
/* except the initial in a section */
section > .ltx_title +.ltx_para > .ltx_p,
section > .ltx_title +.ltx_date +.ltx_para > .ltx_p {text-indent:0em; }
.ltx_title_abstract + .ltx_p {text-indent:2em; }
.ltx_itemize,
.ltx_enumerate { margin-left:1em; }
.ltx_theorem { margin-top:0.5em; margin-bottom:0.5em; }
.ltx_theorem .ltx_title { margin-bottom:0.1em; }
.ltx_theorem .ltx_title + .ltx_para,
.ltx_theorem .ltx_title + .ltx_para .ltx_p,
.ltx_theorem .ltx_title + .ltx_p { margin-top:0em; }
From uniform to statistical convergence of binomial type operators
From uniform to statistical convergence of binomial
type operators
Octavian Agratini
Babeş-Bolyai University
Faculty of Mathematics and Computer Science
Str. Kogălniceanu, 1
400084 Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis
Romanian Academy
Str. Fântânele, 57
400320 Cluj-Napoca, Romania
agratini@math.ubbcluj.ro
Abstract.
Sequences of binomial operators introduced by using umbral
calculus are investigated from the point of view of statistical
convergence.
This approach is based on a detailed presentation of delta
operators and their associated basic polynomials.
Bernstein-Sheffer linear positive operators are analyzed
and some particular cases are highlighted:
Cheney-Sharma operators, Stancu operators, Lupaş operators.
Keywords and phrases:
Statistical convergence, binomial sequence, linear positive operator,
umbral calculus,
Bernstein-Sheffer operator, Pincherle derivative.
2010 Mathematics Subject Classification:
05A40, 41A36, 47A58.
1. Introduction
Let be a sequence of linear positive operators
acting on the space of all real-valued and continuous
functions defined on the interval , equipped with the norm
of the uniform convergence, namely
.
Bohman-Korovkin’s theorem asserts:
if the operators , , map
into itself such that
|
|
|
(1.1) |
then, one has
|
|
|
(1.2) |
In the above represents the monomial of -th degree,
and
, .
A current subject in Approximation Theory is the approximation
of continuous functions by using the statistical convergence,
the first research of this topic being done by Gadjiev and Orhan
[5].
This approach models and improves the technique of signals
approximation in different function spaces.
On the other hand, sequences of polynomials of binomial type
have been the subject of many mathematical studies, drawing
to light their role in Approximation Theory.
Practically, the theory of the approximation operators
of binomial type is based on the technique of the
umbral calculus.
In its modern form, this is a strong tool for calculations with
polynomials representing a successful combination between
the finite differences calculus and certain chapters of
Probability Theory.
The topic discussed in this chapter is at the confluence
of the two concepts mentioned above, statistical convergence
and binomial type operators, from the point of view of the
approximation of some function classes.
The material is structured in three sections.
First of all we recall the variant of Bohman-Korovkin theorem
via statistical convergence and we present elementary facts
about polynomial sequences of binomial type.
Further on, we deal with delta operators and their basic polynomials.
In the last section we will analyze the approximation
properties of some binomial operators in terms of the statistical
convergence.
We mention that at the first sight this work seems disproportionate,
dominated by a lot of notions introduced and results already
achieved.
The goal was to be self contained paper.
It will be seen that for a clear understanding of the last
paragraph, it was necessary to structure the article
in this way.
2. Preliminaries
The concept of statistical convergence was first defined by
Steinhaus [13] and Fast [4].
It is based on the notion of the asymptotic density of subsets
of .
The density of denoted by
is given by
|
|
|
where stands for the characteristic function of the set .
Clearly,
.
A sequence of real numbers is said to be
statistically convergent to a real number , if, for every
,
the limit being denoted by
.
It is known that any convergent sequence is statistically convergent
but the converse of this statement is not true.
Even though this notion was introduced in 1951, its application
to the study of sequences of positive linear operators was attempted only
in 2002.
We refer to the
A.D. Gadjiev and C. Orhan [5] result,
which reads as follows.
Theorem 2.1.
If the sequence of positive linear operators
satisfies the condition
|
|
|
(2.1) |
then, one has
|
|
|
(2.2) |
As usual, stands for the space of all real-valued
bounded functions defined on , endowed with the sup-norm.
The identities (2.1) and (2.2) generalize respectively
relations (1.1), (1.2).
From this moment, the statistical convergence of positive
linear operators represented a new direction in the study
of so-called KAT-Korovkin-type Approximation Theory.
Set .
For any we denote by
the linear space of polynomials of degree no greater
than and by the set of all polynomials of
degree .
We also set
representing the commutative algebra of polynomials with
coefficients in , this symbol standing either for
the field or for the field .
A sequence
such that
for every
is called a polynomial sequence.
Definition 2.2.
A polynomial sequence
is called of binomial type if for any
the following identities hold
|
|
|
(2.3) |
The most common example of binomial sequence is
(the monomials).
Some nontrivial examples are given below.
a) The generalized factorial power with the step :
,
|
|
|
The Vandermonde formula, i.e.
|
|
|
guarantees that this is a binomial type sequence.
There are two particular cases:
for we obtain the lower-factorials which, usually,
are denoted by ;
for we obtain
the upper-factorials denoted by Pochhammer’s symbol
.
By convention we consider
b) Abel polynomials:
,
|
|
|
Rewriting the identity (2.3) for these polynomials we obtain
the Abel-Jensen (1902) combinatorial formula
|
|
|
c) Gould polynomials:
,
|
|
|
The space of all linear operators
will be denoted by .
Among these operators an important role will be played
by the shift operator, named .
For every , is defined by
|
|
|
An operator which switches with all shift
operators, that is
|
|
|
is called a shift-invariant operator and the set of these
operators is denoted by .
3. On delta operators
Definition 3.1.
An operator
is called delta operator if
and is a non zero constant.
Let denote the set of all delta operators.
For a better understanding we present some examples of delta operators.
In the following the
symbol stands for the identity operator on the space .
a) The derivative operator,
denoted by .
b) The operators used in calculus of divided differences.
Let be a fixed number belonging to the field .
We set
|
|
|
|
|
|
|
|
|
|
|
|
It is evident that
,
.
The properties of these operators
as well as their usefulness
can be found in [6].
c) Abel operator,
.
For any ,
.
Writing (symbolically) the Taylor’s series in the following
manner
|
|
|
(3.1) |
we can also get
.
d) Gould operator,
, .
Definition 3.2.
Let be a delta operator.
A polynomial sequence
is called the sequence of basic polynomials associated
to if
(i) for any ,
(ii) for any ,
(iii)
for any and .
Here are some examples.
The basic polynomials associated to the operators
, and , are respectively
, and
.
Also, we can easily prove that
respectively
is the sequence of basic polynomials associated to Abel operator
, respectively Gould operator .
The connection between delta operator and the binomial type
sequences is given by the following result
[9, Theorem 1].
Theorem 3.4.
Let be a sequence of polynomials.
It is a sequence of binomial type if and only if
it is a basic sequence for some delta operator.
The following statement generalizes the Taylor expansion theorem
to delta operators and their basic polynomials.
Theorem 3.5.
Let be a shift-invariant operator and let be a delta
operator with its basic sequence .
Then the following identity holds
|
|
|
(3.2) |
Let be a delta operator and let
be the ring of the formal power series in the variable over
the same field.
Here the product means the Cauchy product between two series.
Further, let
be the ring of shift-invariant operators, the product being
defined as usually:
for any
we have
,
, .
Then there exists an isomorphism from
onto such that
|
|
|
(3.3) |
This isomorphism allows us to conclude:
a shift-invariant operator is invertible if and only if
.
Since for every
we have
we deduce that any delta operator is not invertible.
Also, we can write
, where and
is a formal power series, to indicate that the operator
corresponds to the series under the isomorphism
defined by (3.3).
Theorem 3.7.
Let be a delta operator with
its sequence of basic polynomials.
Let and be the inverse formal power
series of .
Then
|
|
|
(3.4) |
where has the form
.
Another characterization of delta operators was included in
[9] without proof.
For this reason we prove the following statement.
Theorem 3.8.
is a delta operator if and only if
for some shift-invariant operator , where the inverse operator
exists.
Proof.
If in (3.3) we substitute by a delta operator , then
we get
and
.
Consequently, we can write
Denoting
by we have
and
,
thus is invertible, see the conclusion that emerges from (3.3).
So, can be written as .
Reciprocally, for every such that
is invertible, is a shift-invariant operator,
, and
|
|
|
thus .
Now we are ready to analyze some binomial operators
investigating their statistical convergence to the identity operator.
4. Classes of binomial operators
We consider a delta operator and its sequence of basic polynomials
,
under the assumption that
for every .
Also, according to Theorem 3.7 we shall keep the same
meaning of the functions and .
For every we consider
defined as follows
|
|
|
(4.1) |
They are called by P. Sablonnière [10]
Bernstein-Sheffer operators.
As D.D. Stancu and M.R. Occorsio motivated in [12],
these operators can be named Popoviciu operators.
T. Popoviciu [8] indicated the construction (4.1)
in front of the sum appearing the factor
from the identities
|
|
|
see (3.4).
If we choose it becomes obvious that
.
In the particular case , becomes
genuine Bernstein operator of degree .
An integral generalization of in Kantorovich sense
was introduced and studied in [1].
The operators , , are linear and reproduce
the constants.
Indeed, choosing in (2.3)
we obtain
.
The positivity of these operators are given by the sign of the
coefficients of the series
.
More precisely, in [8] and in [10] the authors
established the followings.
Lemma 4.1.
is a positive operator on for every
if and only if and for all .
Moreover, if satisfies the above conditions, then one has
|
|
|
(4.2) |
where
,
the sequence
being generated by
|
|
|
Theorem 4.2.
Let the operators , ,
be defined by (4.1) such that the hypothesis of Lemma 4.1
takes place.
If
,
then
|
|
|
(4.3) |
Proof.
We apply Theorem 2.1.
Based on algebraic operations with statistically convergent
sequences of real numbers, our hypothesis guarantees the identity (4.3).
For a profound documentation of operations with such sequences
[2, Theorem 3.1] can be consulted.
Further, choosing particular delta operators we reobtain
some classical linear positive operator of discrete type.
Example 4.3.
If with its basic sequence and assuming that
the parameter depends on ,
,
one obtains the Cheney-Sharma operators [3].
The corresponding operators , ,
are defined by the equation
|
|
|
Clearly .
To compute , , we follow the same path as
in [3, Section 3].
We can deduce:
if the sequence is statistically convergent to
zero, then (4.3) takes place.
Example 4.4.
If , ,
with its basic polynomials
becomes Stancu operator [11] denoted by
,
|
|
|
|
|
|
being a parameter which may depend on a natural number .
One has
|
|
|
in accordance with [11, Lemma 4.1].
If
and
,
then (4.3) takes place.
Indeed, it is enough to prove (2.1) for .
We get
|
|
|
|
|
|
|
|
and the conclusion follows.
At this moment we take a break in order to illustrate some
further properties of the binomial sequences.
Keeping the notations ,
,
, ,
we assume that the conditions of Lemma 4.1 are fulfilled.
In [7] A. Lupaş proved new inequalities between the
terms of the binomial sequences .
For any and one has
|
|
|
(4.4) |
In the above represents Pincherle derivative of .
The concept is detailed further.
Knowing that the operator ,
is called multiplication operator,
we recall that the Pincherle derivative of an operator
is defined by the formula
For example we get
, , , .
Example 4.5.
Following Lupaş we can define the operators
, ,
|
|
|
It is proved that
|
|
|
see (4.4).
Consequently, if
|
|
|
then the operators
, , satisfy identity (4.3).
Concluding remarks.
The paper reintroduces some linear positive operators of discrete
type by using umbral calculus.
Relative to these operators have been studied approximation
properties in Banach space .
The approach was based on Bohman-Korovkin theorem via statistical
convergence.
The usefulness of this type of convergence can be summarized
as follows:
the statistical convergence of a sequence is that the majority,
in a certain sense, of its elements converges and we are
not interested in what happens to the remaining elements.
The advantage of replacing the uniform convergence by statistical
convergence consists in the fact that the second convergence
is efficient in summing divergent sequences which may
have unbounded subsequences.
In short it is more lax.
References
-
[1]
O. Agratini,
On a certain class of approximation operators,
Pure Mathematics and Applications, 11(2000), No. 2, 119-127.
-
[2]
H. Albayrak, S. Pehlivan,
Statistical convergence and statistical continuity on locally
solid Riesz spaces,
Topology and its Applications, 159(2012), 1887-1893.
-
[3]
E.W. Cheney, A. Sharma,
On a generalization of Bernstein polynomials,
Rivista di Matematica della Università di Parma (2),
5(1964), 77-84.
-
[4]
H. Fast,
Sur le convergence statistique,
Colloquium Mathematicum, 2(1951), 241-244.
-
[5]
A.D. Gadjiev, C. Orhan,
Some approximation theorems via statistical convergence,
Rocky Mountain Journal of Mathematics, 32(2002), 129-138.
-
[6]
C. Jordan,
Calculus of Finite Differences,
Chelsea Publishing Company, New York, 1950.
-
[7]
A. Lupaş,
Approximation operators of binomial type,
New Developments in Approximation Theory (Dortmund, 1980),
175-198, International Series of Numerical Mathematics,
Vol. 132, Birkhäuser Verlag Basel/Switzerland, 1999.
-
[8]
T. Popoviciu,
Remarques sur les polynômes binomiaux,
Bulletin de la Société des Sciences de Cluj, Roumanie,
6(1931), 146-148
(also reproduced in Mathematica (Cluj), 6(1932), 8-10).
-
[9]
G.-C. Rota, D. Kahaner, A. Odlyzko,
On the Foundations of Combinatorial Theory,
VIII. Finite Operator Calculus,
Journal of Mathematical Analysis and Application,
42(1973), 685-760.
-
[10]
P. Sablonnière,
Positive Bernstein-Sheffer operators,
Journal of Approximation Theory,
83(1995), 330-341.
-
[11]
D.D. Stancu,
Approximation of functions by a new class of linear
polynomial operators,
Revue Roumaine des Mathématique Pures et Appliquées, 13(1968), No. 8, 1173-1194.
-
[12]
D.D. Stancu, M.R. Occorsion,
On approximation by binomial operators of Tiberiu Popoviciu type,
Revue d’Analyse Numérique et de Théorie de l’Approximation,
27(1998), No. 1, 167-181.
-
[13]
H. Steinhaus,
Sur la convergence ordinaire et la convergence asymptotique,
Colloquium Mathematicum, 2(1951), 73-74.
