{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T15:52:54Z","timestamp":1774626774409,"version":"3.50.1"},"reference-count":9,"publisher":"American Mathematical Society (AMS)","issue":"287","license":[{"start":{"date-parts":[[2014,11,20]],"date-time":"2014-11-20T00:00:00Z","timestamp":1416441600000},"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>We study multivariate integration for a weighted Korobov space of periodic infinitely many times differentiable functions for which the Fourier coefficients decay exponentially fast. The weights are defined in terms of two non-decreasing sequences <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold a equals left-brace a Subscript i Baseline right-brace\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"bold\">a<\/mml:mi>\n      <\/mml:mrow>\n      <mml:mo>=<\/mml:mo>\n      <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n      <mml:msub>\n        <mml:mi>a<\/mml:mi>\n        <mml:mi>i<\/mml:mi>\n      <\/mml:msub>\n      <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathbf {a}=\\{a_i\\}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold b equals left-brace b Subscript i Baseline right-brace\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"bold\">b<\/mml:mi>\n      <\/mml:mrow>\n      <mml:mo>=<\/mml:mo>\n      <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n      <mml:msub>\n        <mml:mi>b<\/mml:mi>\n        <mml:mi>i<\/mml:mi>\n      <\/mml:msub>\n      <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathbf {b}=\\{b_i\\}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> of numbers no less than one and a parameter <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"omega element-of left-parenthesis 0 comma 1 right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>\u03c9<\/mml:mi>\n      <mml:mo>\u2208<\/mml:mo>\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\">\\omega \\in (0,1)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. Let <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"e left-parenthesis n comma s right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>e<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>n<\/mml:mi>\n      <mml:mo>,<\/mml:mo>\n      <mml:mi>s<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">e(n,s)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> be the minimal worst-case error of all algorithms that use <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> function values in the <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s\">\n  <mml:semantics>\n    <mml:mi>s<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">s<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-variate case. We would like to check conditions on <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold a\">\n  <mml:semantics>\n    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n      <mml:mi mathvariant=\"bold\">a<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathbf {a}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold b\">\n  <mml:semantics>\n    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n      <mml:mi mathvariant=\"bold\">b<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathbf {b}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"omega\">\n  <mml:semantics>\n    <mml:mi>\u03c9<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">\\omega<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> such that <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"e left-parenthesis n comma s right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>e<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>n<\/mml:mi>\n      <mml:mo>,<\/mml:mo>\n      <mml:mi>s<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">e(n,s)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> decays exponentially fast, i.e., for some <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q element-of left-parenthesis 0 comma 1 right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>q<\/mml:mi>\n      <mml:mo>\u2208<\/mml:mo>\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\">q\\in (0,1)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p greater-than 0\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>p<\/mml:mi>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mn>0<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">p&gt;0<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> we have <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"e left-parenthesis n comma s right-parenthesis equals script upper O left-parenthesis q Superscript n Super Superscript p Superscript Baseline right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>e<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>n<\/mml:mi>\n      <mml:mo>,<\/mml:mo>\n      <mml:mi>s<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo>=<\/mml:mo>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi>\n      <\/mml:mrow>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:msup>\n        <mml:mi>q<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mspace width=\"thinmathspace\"\/>\n          <mml:msup>\n            <mml:mi>n<\/mml:mi>\n            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n              <mml:mspace width=\"thinmathspace\"\/>\n              <mml:mi>p<\/mml:mi>\n            <\/mml:mrow>\n          <\/mml:msup>\n        <\/mml:mrow>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">e(n,s)=\\mathcal {O}(q^{\\,n^{\\,p}})<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> as <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> goes to infinity. The factor in the <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper O\">\n  <mml:semantics>\n    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n      <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathcal {O}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> notation may depend on <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s\">\n  <mml:semantics>\n    <mml:mi>s<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">s<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> in an arbitrary way. We prove that exponential convergence holds iff <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B colon equals sigma-summation Underscript i equals 1 Overscript normal infinity Endscripts 1 slash b Subscript i Baseline greater-than normal infinity\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>B<\/mml:mi>\n      <mml:mo>:=<\/mml:mo>\n      <mml:munderover>\n        <mml:mo>\u2211<\/mml:mo>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>i<\/mml:mi>\n          <mml:mo>=<\/mml:mo>\n          <mml:mn>1<\/mml:mn>\n        <\/mml:mrow>\n        <mml:mi mathvariant=\"normal\">\u221e<\/mml:mi>\n      <\/mml:munderover>\n      <mml:mn>1<\/mml:mn>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:msub>\n        <mml:mi>b<\/mml:mi>\n        <mml:mi>i<\/mml:mi>\n      <\/mml:msub>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mi mathvariant=\"normal\">\u221e<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">B:=\\sum _{i=1}^\\infty 1\/b_i&gt;\\infty<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> independently of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold a\">\n  <mml:semantics>\n    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n      <mml:mi mathvariant=\"bold\">a<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathbf {a}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"omega\">\n  <mml:semantics>\n    <mml:mi>\u03c9<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">\\omega<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. Furthermore, the largest <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> of exponential convergence is <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1 slash upper B\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>1<\/mml:mn>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mi>B<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">1\/B<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. We also study exponential convergence with weak, polynomial and strong polynomial tractability. This means that <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"e left-parenthesis n comma s right-parenthesis less-than-or-equal-to upper C left-parenthesis s right-parenthesis q Superscript n Super Superscript p\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>e<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>n<\/mml:mi>\n      <mml:mo>,<\/mml:mo>\n      <mml:mi>s<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo>\u2264<\/mml:mo>\n      <mml:mi>C<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>s<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mspace width=\"thinmathspace\"\/>\n      <mml:msup>\n        <mml:mi>q<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mspace width=\"thinmathspace\"\/>\n          <mml:msup>\n            <mml:mi>n<\/mml:mi>\n            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n              <mml:mspace width=\"thinmathspace\"\/>\n              <mml:mi>p<\/mml:mi>\n            <\/mml:mrow>\n          <\/mml:msup>\n        <\/mml:mrow>\n      <\/mml:msup>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">e(n,s)\\le C(s)\\,q^{\\,n^{\\,p}}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> for all <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> and <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s\">\n  <mml:semantics>\n    <mml:mi>s<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">s<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and with <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"log upper C left-parenthesis s right-parenthesis equals exp left-parenthesis o left-parenthesis s right-parenthesis right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>log<\/mml:mi>\n      <mml:mspace width=\"thinmathspace\"\/>\n      <mml:mi>C<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>s<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo>=<\/mml:mo>\n      <mml:mi>exp<\/mml:mi>\n      <mml:mo>\u2061<\/mml:mo>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>o<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>s<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\log \\,C(s)=\\exp (o(s))<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> for weak tractability, with a polynomial bound on <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"log upper C left-parenthesis s right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>log<\/mml:mi>\n      <mml:mspace width=\"thinmathspace\"\/>\n      <mml:mi>C<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>s<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\log \\,C(s)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> for polynomial tractability, and with uniformly bounded <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C left-parenthesis s right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>C<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>s<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">C(s)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> for strong polynomial tractability. We prove that the notions of weak, polynomial and strong polynomial tractability are equivalent, and hold iff <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B greater-than normal infinity\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>B<\/mml:mi>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mi mathvariant=\"normal\">\u221e<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">B&gt;\\infty<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a Subscript i\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>a<\/mml:mi>\n      <mml:mi>i<\/mml:mi>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">a_i<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> are exponentially growing with <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"i\">\n  <mml:semantics>\n    <mml:mi>i<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">i<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. We also prove that the largest (or the supremum of) <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> for exponential convergence with strong polynomial tractability belongs to <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket 1 slash left-parenthesis 2 upper B right-parenthesis comma 1 slash upper B right-bracket\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo stretchy=\"false\">[<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mn>2<\/mml:mn>\n      <mml:mi>B<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo>,<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mi>B<\/mml:mi>\n      <mml:mo stretchy=\"false\">]<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">[1\/(2B),1\/B]<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>.<\/p>","DOI":"10.1090\/s0025-5718-2013-02739-1","type":"journal-article","created":{"date-parts":[[2013,11,20]],"date-time":"2013-11-20T17:06:39Z","timestamp":1384967199000},"page":"1189-1206","source":"Crossref","is-referenced-by-count":38,"title":["Multivariate integration of infinitely many times differentiable functions in weighted Korobov spaces"],"prefix":"10.1090","volume":"83","author":[{"given":"Peter","family":"Kritzer","sequence":"first","affiliation":[]},{"given":"Friedrich","family":"Pillichshammer","sequence":"additional","affiliation":[]},{"given":"Henryk","family":"Wo\u017aniakowski","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2013,11,20]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"337","DOI":"10.2307\/1990404","article-title":"Theory of reproducing kernels","volume":"68","author":"Aronszajn, N.","year":"1950","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"2","series-title":"Mathematical Surveys and Monographs","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/178","volume-title":"Quadrature theory","volume":"178","author":"Brass, Helmut","year":"2011","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821853610"},{"issue":"274","key":"3","doi-asserted-by":"publisher","first-page":"905","DOI":"10.1090\/S0025-5718-2010-02433-0","article-title":"Exponential convergence and tractability of multivariate integration for Korobov spaces","volume":"80","author":"Dick, Josef","year":"2011","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"4","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511761188","volume-title":"Digital nets and sequences","author":"Dick, Josef","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521191593"},{"key":"5","series-title":"EMS Tracts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.4171\/026","volume-title":"Tractability of multivariate problems. Vol. 1: Linear information","volume":"6","author":"Novak, Erich","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9783037190265"},{"key":"6","series-title":"EMS Tracts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.4171\/084","volume-title":"Tractability of multivariate problems. Volume II: Standard information for functionals","volume":"12","author":"Novak, Erich","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9783037190845"},{"issue":"219","key":"7","doi-asserted-by":"publisher","first-page":"1119","DOI":"10.1090\/S0025-5718-97-00834-X","article-title":"An intractability result for multiple integration","volume":"66","author":"Sloan, I. H.","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","series-title":"Computer Science and Scientific Computing","isbn-type":"print","volume-title":"Information-based complexity","author":"Traub, J. F.","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0126975450"},{"key":"9","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1007\/978-1-4419-6594-3_17","article-title":"Towards a general error theory of the trapezoidal rule","author":"Waldvogel, J\u00f6rg","year":"2011"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2014-83-287\/S0025-5718-2013-02739-1\/S0025-5718-2013-02739-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-287\/S0025-5718-2013-02739-1\/S0025-5718-2013-02739-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T05:30:54Z","timestamp":1627623054000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-287\/S0025-5718-2013-02739-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,11,20]]},"references-count":9,"journal-issue":{"issue":"287","published-print":{"date-parts":[[2014,5]]}},"alternative-id":["S0025-5718-2013-02739-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2013-02739-1","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,11,20]]}}}