{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T19:03:42Z","timestamp":1781895822261,"version":"3.54.5"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"290","license":[{"start":{"date-parts":[[2015,6,20]],"date-time":"2015-06-20T00:00:00Z","timestamp":1434758400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We prove the curse of dimensionality for multivariate integration of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C Superscript r\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mi>r<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">C^r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    functions: The number of needed function values to achieve an error\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03f5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\epsilon<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is larger than\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c Subscript r Baseline left-parenthesis 1 plus gamma right-parenthesis Superscript d\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mi>r<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>\n                              \u03b3\n                              \n                            <\/mml:mi>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mi>d<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">c_r (1+\\gamma )^d<\/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=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon less-than-or-equal-to epsilon 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03f5\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03f5\n                                \n                              <\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\epsilon \\le \\epsilon _0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c Subscript r Baseline comma gamma greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mi>r<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03b3\n                              \n                            <\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">c_r,\\gamma &gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The proofs are based on volume estimates for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">r=1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    together with smoothing by convolution. This allows us to obtain smooth fooling functions for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r greater-than 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">r&gt;1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-2014-02855-x","type":"journal-article","created":{"date-parts":[[2014,6,20]],"date-time":"2014-06-20T10:45:16Z","timestamp":1403261116000},"page":"2853-2863","source":"Crossref","is-referenced-by-count":31,"title":["The curse of dimensionality for numerical integration of smooth functions"],"prefix":"10.1090","volume":"83","author":[{"given":"A.","family":"Hinrichs","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"E.","family":"Novak","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.","family":"Ullrich","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"H.","family":"Wo\u017aniakowski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2014,6,20]]},"reference":[{"key":"1","series-title":"National Bureau of Standards Applied Mathematics Series, No. 55","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","author":"Abramowitz, Milton","year":"1964"},{"issue":"4","key":"2","first-page":"3","article-title":"Approximate computation of multiple integrals","volume":"1959","author":"Bahvalov, N. 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F.","year":"1988","ISBN":"https:\/\/2.zoppoz.workers.dev:443\/https\/id.crossref.org\/isbn\/0126975450"},{"issue":"5","key":"10","doi-asserted-by":"publisher","first-page":"638","DOI":"10.1016\/S0885-064X(03)00069-4","article-title":"Multivariate integration in \ud835\udc36^{\u221e}([0,1]^{\ud835\udc51}) is not strongly tractable","volume":"19","author":"Wojtaszczyk, Jakub Onufry","year":"2003","journal-title":"J. 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