{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:36:24Z","timestamp":1776785784439,"version":"3.51.2"},"reference-count":28,"publisher":"American Mathematical Society (AMS)","issue":"255","license":[{"start":{"date-parts":[[2007,5,3]],"date-time":"2007-05-03T00:00:00Z","timestamp":1178150400000},"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                    For any interpolatory ternary subdivision scheme with two-ring stencils for a regular triangular or quadrilateral mesh, we show that the critical H\u00f6lder smoothness exponent of its basis function cannot exceed\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"log Subscript 3 Baseline 11 left-parenthesis almost-equals 2.18266 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>log<\/mml:mi>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mn>11<\/mml:mn>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mo>\n                              \u2248\n                              \n                            <\/mml:mo>\n                            <mml:mn>2.18266<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\log _3 11 (\\approx 2.18266)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where the critical H\u00f6lder smoothness exponent of a function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f colon double-struck upper R squared right-arrow from bar double-struck upper R\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">\n                              \u21a6\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f : \\mathbb {R}^2\\mapsto \\mathbb {R}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is defined to be\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"nu Subscript normal infinity Baseline left-parenthesis f right-parenthesis colon-equal sup left-brace right-brace colon nu colon element-of element-of f of LipLip nu period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03bd\n                                \n                              <\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-REL\">\n                              <mml:mo>\u2254<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo movablelimits=\"true\" form=\"prefix\">sup<\/mml:mo>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:mi>\n                              \u03bd\n                              \n                            <\/mml:mi>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mi>Lip<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03bd\n                              \n                            <\/mml:mi>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\nu _\\infty (f) \\coloneq \\sup \\{ \\nu : f\\in \\operatorname {Lip} \\nu \\}.<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    On the other hand, for both regular triangular and quadrilateral meshes, we present several examples of interpolatory ternary subdivision schemes with two-ring stencils such that the critical H\u00f6lder smoothness exponents of their basis functions do achieve the optimal smoothness upper bound\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"log Subscript 3 Baseline 11\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>log<\/mml:mi>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mn>11<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\log _3 11<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Consequently, we obtain optimal smoothest\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">C^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    interpolatory ternary subdivision schemes with two-ring stencils for the regular triangular and quadrilateral meshes. Our computation and analysis of optimal multidimensional subdivision schemes are based on the projection method and the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script l Subscript p\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>\n                              \u2113\n                              \n                            <\/mml:mi>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\ell _p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -norm joint spectral radius.\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01821-7","type":"journal-article","created":{"date-parts":[[2006,5,24]],"date-time":"2006-05-24T14:43:01Z","timestamp":1148481781000},"page":"1287-1308","source":"Crossref","is-referenced-by-count":18,"title":["Optimal \ud835\udc36\u00b2 two-dimensional interpolatory ternary subdivision schemes with two-ring stencils"],"prefix":"10.1090","volume":"75","author":[{"given":"Bin","family":"Han","sequence":"first","affiliation":[]},{"given":"Rong-Qing","family":"Jia","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,5,3]]},"reference":[{"issue":"453","key":"1","doi-asserted-by":"publisher","first-page":"vi+186","DOI":"10.1090\/memo\/0453","article-title":"Stationary subdivision","volume":"93","author":"Cavaretta, Alfred S.","year":"1991","journal-title":"Mem. 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