{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:49:53Z","timestamp":1776721793877,"version":"3.51.2"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"214","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Our main result shows that certain generalized convex functions on a real interval possess a unique best\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^{1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    approximation from the family of piecewise polynomial functions of fixed degree with varying knots. This result was anticipated by Kioustelidis in [\n                    <italic>\n                      Uniqueness of optimal piecewise polynomial\n                      <inline-formula content-type=\"math\/mathml\">\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L 1\">\n                          <mml:semantics>\n                            <mml:msub>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:annotation encoding=\"application\/x-tex\">L_{1}<\/mml:annotation>\n                          <\/mml:semantics>\n                        <\/mml:math>\n                      <\/inline-formula>\n                      approximations for generalized convex functions\n                    <\/italic>\n                    , from \u201cFunctional Analysis and Approximation\u201d, Internat. Ser. Numer. Math., vol. 60 (1981), 421\u2013432]; however the proof given there is nonconstructive and uses topological degree as the primary tool, in a fashion similar to the proof the comparable result for the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^{2}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    case in [J. Chow,\n                    <italic>\n                      On the uniqueness of best\n                      <inline-formula content-type=\"math\/mathml\">\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L 2 left-bracket 0 comma 1 right-bracket\">\n                          <mml:semantics>\n                            <mml:mrow>\n                              <mml:msub>\n                                <mml:mi>L<\/mml:mi>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mn>2<\/mml:mn>\n                                <\/mml:mrow>\n                              <\/mml:msub>\n                              <mml:mo stretchy=\"false\">[<\/mml:mo>\n                              <mml:mn>0<\/mml:mn>\n                              <mml:mo>,<\/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\">L_{2}[0,1]<\/mml:annotation>\n                          <\/mml:semantics>\n                        <\/mml:math>\n                      <\/inline-formula>\n                      approximation by piecewise polynomials with variable breakpoints\n                    <\/italic>\n                    , Math. Comp.\n                    <bold>39<\/bold>\n                    (1982), 571\u2013585.]. By contrast, the proof given here proceeds by demonstrating the global convergence of an algorithm to calculate a best approximation over the domain of all possible knot vectors. The proof uses the contraction mapping theorem to simultaneously establish convergence and uniqueness. This algorithm was suggested by Kioustelidis [\n                    <italic>\n                      Optimal segmented polynomial\n                      <inline-formula content-type=\"math\/mathml\">\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript p\">\n                          <mml:semantics>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>p<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:annotation encoding=\"application\/x-tex\">L^{p}<\/mml:annotation>\n                          <\/mml:semantics>\n                        <\/mml:math>\n                      <\/inline-formula>\n                      -approximation\n                    <\/italic>\n                    , Computing\n                    <bold>26<\/bold>\n                    (1981), 239\u2013246.]. In addition, an asymptotic uniqueness result and a nonuniqueness result are indicated, which analogize known results in the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^{2}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    case.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00709-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"647-660","source":"Crossref","is-referenced-by-count":4,"title":["Unicity in piecewise polynomial \ud835\udc3f\u00b9-approximation via an algorithm"],"prefix":"10.1090","volume":"65","author":[{"given":"R.","family":"Gayle","sequence":"first","affiliation":[]},{"given":"J.","family":"Wolfe","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","unstructured":"J. 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