{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T03:26:04Z","timestamp":1776828364718,"version":"3.51.2"},"reference-count":19,"publisher":"American Mathematical Society (AMS)","issue":"220","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=\"upper S Subscript beta Baseline colon equals StartSet z element-of double-struck upper C colon StartAbsoluteValue upper I m z EndAbsoluteValue greater-than beta EndSet\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>S<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03b2\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>:=<\/mml:mo>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">C<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>Im<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>\n                              \u03b2\n                              \n                            <\/mml:mi>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">S_{\\beta }:=\\{z\\in \\mathbb {C}:| \\operatorname {Im}z|&gt;\\beta \\}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be a strip in the complex plane. For fixed integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r greater-than-or-equal-to 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">r\\ge 0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Subscript normal infinity comma beta Superscript r\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>\n                                \u03b2\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                          <\/mml:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^r_{\\infty ,\\beta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    denote the class of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 pi\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2\\pi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -periodic functions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f\">\n                        <mml:semantics>\n                          <mml:mi>f<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , which are analytic in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S Subscript beta\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>\n                                \u03b2\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">S_{\\beta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and satisfy\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartAbsoluteValue f Superscript left-parenthesis r right-parenthesis Baseline left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-equal-to 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msup>\n                              <mml:mi>f<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>r<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\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\">|f^{(r)}(z)|\\le 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S Subscript beta\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>\n                                \u03b2\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">S_{\\beta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Denote by\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Subscript normal infinity comma beta Superscript r comma double-struck upper R\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>\n                                \u03b2\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>r<\/mml:mi>\n                              <mml:mo>,<\/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:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^{r,\\mathbb R}_{\\infty ,\\beta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    the subset of functions from\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Subscript normal infinity comma beta Superscript r\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>\n                                \u03b2\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                          <\/mml:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^r_{\\infty ,\\beta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    that are real-valued on the real axis. Given a function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f element-of upper H Subscript normal infinity comma beta Superscript r\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"normal\">\n                                  \u221e\n                                  \n                                <\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>\n                                  \u03b2\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>r<\/mml:mi>\n                            <\/mml:msubsup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f\\in H^r_{\\infty ,\\beta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , we try to recover\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f left-parenthesis zeta right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03b6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f(\\zeta )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    at a fixed point\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"zeta element-of double-struck upper R\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b6\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2208\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\">\\zeta \\in \\mathbb {R}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    by an algorithm\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A\">\n                        <mml:semantics>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">A<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    on the basis of the information\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper I f equals left-parenthesis a 0 left-parenthesis f right-parenthesis comma a 1 left-parenthesis f right-parenthesis comma ellipsis comma a Subscript n minus 1 Baseline left-parenthesis f right-parenthesis comma b 1 left-parenthesis f right-parenthesis comma ellipsis comma b Subscript n minus 1 Baseline left-parenthesis f right-parenthesis right-parenthesis comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\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:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\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:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2026\n                              \n                            <\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/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:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\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:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2026\n                              \n                            <\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/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:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*}If=(a_{0}(f),a_{1}(f),\\dots ,a_{n-1}(f),b_{1}(f),\\dots ,b_{n-1}(f)), \\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a Subscript j Baseline left-parenthesis f right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>j<\/mml:mi>\n                              <\/mml:mrow>\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>\n                          <mml:annotation encoding=\"application\/x-tex\">a_{j}(f)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b Subscript j Baseline left-parenthesis f right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>j<\/mml:mi>\n                              <\/mml:mrow>\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>\n                          <mml:annotation encoding=\"application\/x-tex\">b_{j}(f)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are the Fourier coefficients of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f\">\n                        <mml:semantics>\n                          <mml:mi>f<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We find the intrinsic error of recovery\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E left-parenthesis upper H Subscript normal infinity comma beta Superscript r Baseline comma upper I right-parenthesis colon equals inf Underscript upper A colon double-struck upper C Superscript 2 n minus 1 Baseline right-arrow double-struck upper C Endscripts sup Underscript f element-of upper H Subscript normal infinity comma beta Superscript r Baseline Endscripts StartAbsoluteValue f left-parenthesis zeta right-parenthesis minus upper A left-parenthesis upper I f right-parenthesis EndAbsoluteValue period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"normal\">\n                                  \u221e\n                                  \n                                <\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>\n                                  \u03b2\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>r<\/mml:mi>\n                            <\/mml:msubsup>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>:=<\/mml:mo>\n                            <mml:munder>\n                              <mml:mo lspace=\"0\" rspace=\"0\" movablelimits=\"true\">inf<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>A<\/mml:mi>\n                                <mml:mo>\n                                  :\n                                  \n                                <\/mml:mo>\n                                <mml:msup>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi mathvariant=\"double-struck\">C<\/mml:mi>\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>\n                                      \u2212\n                                      \n                                    <\/mml:mo>\n                                    <mml:mn>1<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mo stretchy=\"false\">\n                                  \u2192\n                                  \n                                <\/mml:mo>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"double-struck\">C<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:munder>\n                              <mml:mo movablelimits=\"true\" form=\"prefix\">sup<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>f<\/mml:mi>\n                                <mml:mo>\n                                  \u2208\n                                  \n                                <\/mml:mo>\n                                <mml:msubsup>\n                                  <mml:mi>H<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi mathvariant=\"normal\">\n                                      \u221e\n                                      \n                                    <\/mml:mi>\n                                    <mml:mo>,<\/mml:mo>\n                                    <mml:mi>\n                                      \u03b2\n                                      \n                                    <\/mml:mi>\n                                  <\/mml:mrow>\n                                  <mml:mi>r<\/mml:mi>\n                                <\/mml:msubsup>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03b6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*} E(H^r_{\\infty ,\\beta } ,I):= \\operatorname * {inf}_{A\\colon \\mathbb {C}^{2n-1}\\to \\mathbb {C}}\\,\\sup _{f\\in H^r_{\\infty ,\\beta } }|f(\\zeta )- A(If)|. \\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    Furthermore the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis 2 n minus 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2212\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\">(2n-1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -dimensional optimal information error, optimal sampling error and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -widths of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Subscript normal infinity comma beta Superscript r comma double-struck upper R\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>\n                                \u03b2\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>r<\/mml:mi>\n                              <mml:mo>,<\/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:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^{r,\\mathbb R}_{\\infty ,\\beta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C\">\n                        <mml:semantics>\n                          <mml:mi>C<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">C<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the space of continuous functions on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket 0 comma 2 pi right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[0,2\\pi ]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , are determined. The optimal sampling error turns out to be strictly greater than the optimal information error. Finally the same problems are investigated for the class\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Subscript p comma beta\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>\n                                \u03b2\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">H_{p,\\beta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , consisting of all\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 pi\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2\\pi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -periodic functions, which are analytic in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S Subscript beta\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>\n                                \u03b2\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">S_{\\beta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -integrable boundary values. In the case\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p equals 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p=2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    sampling fails to yield optimal information as well in odd as in even dimensions.\n                  <\/p>","DOI":"10.1090\/s0025-5718-97-00896-x","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1579-1592","source":"Crossref","is-referenced-by-count":7,"title":["Optimal information for approximating periodic analytic functions"],"prefix":"10.1090","volume":"66","author":[{"given":"K.","family":"Osipenko","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"K.","family":"Wilderotter","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1997]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"82","DOI":"10.2307\/1989993","article-title":"Maximal orders in rational cyclic algebras of composite degree","volume":"46","author":"Perlis, Sam","year":"1939","journal-title":"Trans. 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Ju.","year":"1976","journal-title":"Mat. Zametki","ISSN":"https:\/\/id.crossref.org\/issn\/0025-567X","issn-type":"print"},{"issue":"4","key":"11","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1070\/IM1995v045n01ABEH001635","article-title":"On \ud835\udc5b-widths, optimal quadrature formulas and optimal reconstruction of functions that are analytic in a strip","volume":"58","author":"Osipenko, K. Yu.","year":"1994","journal-title":"Izv. Ross. Akad. Nauk Ser. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/1607-0046","issn-type":"print"},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1006\/jath.1995.1072","article-title":"On \ud835\udc41-widths of holomorphic functions of several variables","volume":"82","author":"Osipenko, K. Yu.","year":"1995","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"key":"13","doi-asserted-by":"crossref","unstructured":"K. Yu. 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Mat. Zh.","ISSN":"https:\/\/id.crossref.org\/issn\/0041-6053","issn-type":"print"},{"issue":"2","key":"19","doi-asserted-by":"publisher","first-page":"304","DOI":"10.1006\/jath.1995.1080","article-title":"Optimal sampling of periodic analytic functions","volume":"82","author":"Wilderotter, Klaus","year":"1995","journal-title":"J. Approx. 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