{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:23:13Z","timestamp":1776784993857,"version":"3.51.2"},"reference-count":27,"publisher":"American Mathematical Society (AMS)","issue":"254","license":[{"start":{"date-parts":[[2006,11,3]],"date-time":"2006-11-03T00:00:00Z","timestamp":1162512000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"w Subscript lamda Baseline left-parenthesis x right-parenthesis colon equals left-parenthesis 1 minus x squared right-parenthesis Superscript lamda minus 1 slash 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>w<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bb\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>:=<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bb\n                                  \n                                <\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">w_{\\lambda }(x):=(1-x^2)^{\\lambda -1\/2}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P Subscript n Superscript left-parenthesis lamda right-parenthesis\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mi>\n                                \u03bb\n                                \n                              <\/mml:mi>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">P_n^{(\\lambda )}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be the ultraspherical polynomials with respect to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"w Subscript lamda Baseline left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>w<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bb\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">w_{\\lambda }(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Then we denote by\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E Subscript n plus 1 Superscript left-parenthesis lamda right-parenthesis\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mi>\n                                \u03bb\n                                \n                              <\/mml:mi>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">E_{n+1}^{(\\lambda )}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    the Stieltjes polynomials with respect to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"w Subscript lamda Baseline left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>w<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bb\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">w_{\\lambda }(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    satisfying\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartLayout 1st Row  integral Subscript negative 1 Superscript 1 Baseline w Subscript lamda Baseline left-parenthesis x right-parenthesis upper P Subscript n Superscript left-parenthesis lamda right-parenthesis Baseline left-parenthesis x right-parenthesis upper E Subscript n plus 1 Superscript left-parenthesis lamda right-parenthesis Baseline left-parenthesis x right-parenthesis x Superscript m d x StartLayout Enlarged left-brace 1st Row 1st Column equals 0 comma 2nd Column a m p semicolon 0 less-than-or-equal-to m greater-than n plus 1 comma 2nd Row 1st Column not-equals 0 comma 2nd Column a m p semicolon m equals n plus 1 period EndLayout EndLayout\">\n                        <mml:semantics>\n                          <mml:mtable columnalign=\"right center left\" rowspacing=\"3pt\" columnspacing=\"0 thickmathspace\" side=\"left\" displaystyle=\"true\">\n                            <mml:mtr>\n                              <mml:mtd>\n                                <mml:msubsup>\n                                  <mml:mo>\n                                    \u222b\n                                    \n                                  <\/mml:mo>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mo>\n                                      \u2212\n                                      \n                                    <\/mml:mo>\n                                    <mml:mn>1<\/mml:mn>\n                                  <\/mml:mrow>\n                                  <mml:mn>1<\/mml:mn>\n                                <\/mml:msubsup>\n                                <mml:msub>\n                                  <mml:mi>w<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>\n                                      \u03bb\n                                      \n                                    <\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:msubsup>\n                                  <mml:mi>P<\/mml:mi>\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                    <mml:mi>\n                                      \u03bb\n                                      \n                                    <\/mml:mi>\n                                    <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                  <\/mml:mrow>\n                                <\/mml:msubsup>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:msubsup>\n                                  <mml:mi>E<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:mn>1<\/mml:mn>\n                                  <\/mml:mrow>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                    <mml:mi>\n                                      \u03bb\n                                      \n                                    <\/mml:mi>\n                                    <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                  <\/mml:mrow>\n                                <\/mml:msubsup>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:msup>\n                                  <mml:mi>x<\/mml:mi>\n                                  <mml:mi>m<\/mml:mi>\n                                <\/mml:msup>\n                                <mml:mi>d<\/mml:mi>\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mrow>\n                                  <mml:mo>{<\/mml:mo>\n                                  <mml:mtable columnalign=\"left left\" rowspacing=\".2em\" columnspacing=\"1em\" displaystyle=\"false\">\n                                    <mml:mtr>\n                                      <mml:mtd>\n                                        <mml:mo>=<\/mml:mo>\n                                        <mml:mn>0<\/mml:mn>\n                                        <mml:mo>,<\/mml:mo>\n                                      <\/mml:mtd>\n                                      <mml:mtd>\n                                        <mml:mi>a<\/mml:mi>\n                                        <mml:mi>m<\/mml:mi>\n                                        <mml:mi>p<\/mml:mi>\n                                        <mml:mo>;<\/mml:mo>\n                                        <mml:mn>0<\/mml:mn>\n                                        <mml:mo>\n                                          \u2264\n                                          \n                                        <\/mml:mo>\n                                        <mml:mi>m<\/mml:mi>\n                                        <mml:mo>&gt;<\/mml:mo>\n                                        <mml:mi>n<\/mml:mi>\n                                        <mml:mo>+<\/mml:mo>\n                                        <mml:mn>1<\/mml:mn>\n                                        <mml:mo>,<\/mml:mo>\n                                      <\/mml:mtd>\n                                    <\/mml:mtr>\n                                    <mml:mtr>\n                                      <mml:mtd>\n                                        <mml:mo>\n                                          \u2260\n                                          \n                                        <\/mml:mo>\n                                        <mml:mn>0<\/mml:mn>\n                                        <mml:mo>,<\/mml:mo>\n                                      <\/mml:mtd>\n                                      <mml:mtd>\n                                        <mml:mi>a<\/mml:mi>\n                                        <mml:mi>m<\/mml:mi>\n                                        <mml:mi>p<\/mml:mi>\n                                        <mml:mo>;<\/mml:mo>\n                                        <mml:mi>m<\/mml:mi>\n                                        <mml:mo>=<\/mml:mo>\n                                        <mml:mi>n<\/mml:mi>\n                                        <mml:mo>+<\/mml:mo>\n                                        <mml:mn>1.<\/mml:mn>\n                                      <\/mml:mtd>\n                                    <\/mml:mtr>\n                                  <\/mml:mtable>\n                                  <mml:mo fence=\"true\" stretchy=\"true\" symmetric=\"true\"\/>\n                                <\/mml:mrow>\n                              <\/mml:mtd>\n                            <\/mml:mtr>\n                          <\/mml:mtable>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{eqnarray*} \\int _{-1}^1 w_{\\lambda }(x) P_n^{(\\lambda )}(x)E_{n+1}^{(\\lambda )}(x) x^m dx \\begin {cases} =0, &amp; 0 \\le m &gt; n+1,\\\\ \\neq 0, &amp; m=n+1. \\end{cases} \\end{eqnarray*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    In this paper, we show uniform convergence of the Hermite\u2013Fej\u00e9r interpolation polynomials\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Subscript n plus 1 Baseline left-bracket dot right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mo>\n                              \u22c5\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H_{n+1}[\\cdot ]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper H Subscript 2 n plus 1 Baseline left-bracket dot right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mo>\n                              \u22c5\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathcal H}_{2n+1}[\\cdot ]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    based on the zeros of the Stieltjes polynomials\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E Subscript n plus 1 Superscript left-parenthesis lamda right-parenthesis\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mi>\n                                \u03bb\n                                \n                              <\/mml:mi>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">E_{n+1}^{(\\lambda )}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and the product\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E Subscript n plus 1 Superscript left-parenthesis lamda right-parenthesis Baseline upper P Subscript n Superscript left-parenthesis lamda right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msubsup>\n                              <mml:mi>E<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>\n                                  \u03bb\n                                  \n                                <\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:msubsup>\n                              <mml:mi>P<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>\n                                  \u03bb\n                                  \n                                <\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">E_{n+1}^{(\\lambda )}P_n^{(\\lambda )}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0 less-than-or-equal-to lamda less-than-or-equal-to 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">0 \\le \\lambda \\le 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0 less-than-or-equal-to lamda less-than-or-equal-to 1 slash 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">0 \\le \\lambda \\le 1\/2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , respectively. To prove these results, we prove that the Lebesgue constants of Hermite\u2013Fej\u00e9r interpolation operators for the Stieltjes polynomials\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E Subscript n plus 1 Superscript left-parenthesis lamda right-parenthesis\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mi>\n                                \u03bb\n                                \n                              <\/mml:mi>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">E_{n+1}^{(\\lambda )}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and the product\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E Subscript n plus 1 Superscript left-parenthesis lamda right-parenthesis Baseline upper P Subscript n Superscript left-parenthesis lamda right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msubsup>\n                              <mml:mi>E<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>\n                                  \u03bb\n                                  \n                                <\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:msubsup>\n                              <mml:mi>P<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>\n                                  \u03bb\n                                  \n                                <\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">E_{n+1}^{(\\lambda )}P_n^{(\\lambda )}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are optimal, that is, the Lebesgue constants\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-vertical-bar upper H Subscript n plus 1 Baseline double-vertical-bar Subscript normal infinity Baseline left-parenthesis 0 less-than-or-equal-to lamda less-than-or-equal-to 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">\n                              \u2016\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:msub>\n                              <mml:mo fence=\"false\" stretchy=\"false\">\n                                \u2016\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"normal\">\n                                  \u221e\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\|H_{n+1}\\|_{\\infty }(0 \\le \\lambda \\le 1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-vertical-bar script upper H Subscript 2 n plus 1 Baseline double-vertical-bar Subscript normal infinity Baseline left-parenthesis 0 less-than-or-equal-to lamda less-than-or-equal-to 1 slash 2 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">\n                              \u2016\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:msub>\n                              <mml:mo fence=\"false\" stretchy=\"false\">\n                                \u2016\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"normal\">\n                                  \u221e\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\|{\\mathcal H}_{2n+1}\\|_{\\infty } (0 \\le \\lambda \\le 1\/2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    have optimal order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In the case of the Hermite\u2013Fej\u00e9r interpolation polynomials\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper H Subscript 2 n plus 1 Baseline left-bracket dot right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mo>\n                              \u22c5\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathcal H}_{2n+1}[\\cdot ]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1 slash 2 greater-than lamda less-than-or-equal-to 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">1\/2 &gt; \\lambda \\le 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , we prove weighted uniform convergence. Moreover, we give some convergence theorems of Hermite\u2013Fej\u00e9r and Hermite interpolation polynomials for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0 less-than-or-equal-to lamda less-than-or-equal-to 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">0 \\le \\lambda \\le 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in weighted\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Subscript p\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">L_p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    norms.\n                  <\/p>","DOI":"10.1090\/s0025-5718-05-01795-3","type":"journal-article","created":{"date-parts":[[2006,2,15]],"date-time":"2006-02-15T11:05:20Z","timestamp":1140001520000},"page":"743-766","source":"Crossref","is-referenced-by-count":4,"title":["Hermite and Hermite\u2013Fej\u00e9r interpolation for Stieltjes polynomials"],"prefix":"10.1090","volume":"75","author":[{"given":"H.","family":"Jung","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,11,3]]},"reference":[{"key":"1","first-page":"A336--A338","article-title":"Int\u00e9gration num\u00e9rique, abscisse de Kronrod-Patterson et polyn\u00f4mes de Szeg\u0151s","volume":"270","author":"Barrucand, Pierre","year":"1970","journal-title":"C. 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