{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:56:23Z","timestamp":1776804983411,"version":"3.51.2"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"318","license":[{"start":{"date-parts":[[2019,11,27]],"date-time":"2019-11-27T00:00:00Z","timestamp":1574812800000},"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                    In this paper we characterize extremal polynomials and the best constants for the Szeg\u0151\u2013Markov\u2013Bernstein-type inequalities, associated with iterated weight functions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"rho Subscript k Baseline left-parenthesis x right-parenthesis equals upper A left-parenthesis x plus h right-parenthesis rho Subscript k minus 1 Baseline left-parenthesis x plus h right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03c1\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>k<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mspace width=\"negativethinmathspace\"\/>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mspace width=\"negativethinmathspace\"\/>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mi>h<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03c1\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>k<\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mi>h<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\rho _{k}\\left ( x\\right ) \\!=\\! A\\left ( x+h\\right ) \\rho _{k-1}\\left ( x+h\\right )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of any classical weight\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"rho 0 left-parenthesis x right-parenthesis equals rho left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03c1\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>\n                              \u03c1\n                              \n                            <\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\rho _{0}\\left ( x\\right ) = \\rho \\left ( x\\right )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of discrete variable\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x equals a plus i h\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mi>h<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x = a+ih<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , which is defined to be the solution of a difference boundary value problem of the Pearson type. It yields the effective way to compute numerical values of the best constants for all six basic discrete classical weights of the Charlier, Meixner, Kravchuk, Hahn\u00a0I, Hahn II, and Chebyshev kind. In addition, it enables us to establish the generic identities between the Lagrange barycentric coefficients and Christoffel numbers of Gauss quadratures for these classical discrete weight functions, which extends to the discrete case the recent results due to Wang et al. and the authors, published in [Math. Comp. 81 (2012) and 83 (2014), pp. 861\u2013877 and 2893\u20132914, respectively] and [Math. Comp. 86 (2017), pp. 2409\u20132427].\n                  <\/p>","DOI":"10.1090\/mcom\/3396","type":"journal-article","created":{"date-parts":[[2018,8,15]],"date-time":"2018-08-15T09:28:36Z","timestamp":1534325316000},"page":"1791-1804","source":"Crossref","is-referenced-by-count":6,"title":["Difference inequalities and barycentric identities for classical discrete iterated weights"],"prefix":"10.1090","volume":"88","author":[{"given":"Przemys\u0142aw","family":"Rutka","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ryszard","family":"Smarzewski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2018,11,27]]},"reference":[{"issue":"2-3","key":"1","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/S0096-3003(01)00070-4","article-title":"Extremal problems, inequalities, and classical orthogonal polynomials","volume":"128","author":"Agarwal, Ravi P.","year":"2002","journal-title":"Appl. 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