{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:16:59Z","timestamp":1776795419568,"version":"3.51.2"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"280","license":[{"start":{"date-parts":[[2013,3,20]],"date-time":"2013-03-20T00:00:00Z","timestamp":1363737600000},"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                    We obtain decay rates for singular values and eigenvalues of integral operators generated by square integrable kernels on the unit sphere in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper R Superscript m plus 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>m<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {R}^{m+1}<\/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=\"m greater-than-or-equal-to 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">m\\geq 2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , under assumptions on both, certain derivatives of the kernel and the integral operators generated by such derivatives. This type of problem is common in the literature but the assumptions are usually defined using standard differentiation in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper R Superscript m plus 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>m<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {R}^{m+1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In this paper, the assumptions are all defined via the Laplace-Beltrami derivative, a concept first investigated by Rudin in the early fifties and genuinely spherical in nature. The rates we present depend on both, the differentiability order used to define the smoothness conditions and the dimension\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . They are shown to be optimal.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2012-02595-6","type":"journal-article","created":{"date-parts":[[2012,3,21]],"date-time":"2012-03-21T01:28:22Z","timestamp":1332293302000},"page":"2303-2317","source":"Crossref","is-referenced-by-count":14,"title":["Eigenvalue decay of positive integral operators on the sphere"],"prefix":"10.1090","volume":"81","author":[{"given":"M.","family":"Castro","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"V.","family":"Menegatto","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2012,3,20]]},"reference":[{"key":"1","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-8137-3","volume-title":"Harmonic function theory","volume":"137","author":"Axler, Sheldon","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0387952187","edition":"2"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1016\/0022-1236(88)90112-7","article-title":"\ud835\udc60-numbers of integral operators with H\u00f6lder continuous kernels over metric compacta","volume":"81","author":"Carl, Bernd","year":"1988","journal-title":"J. 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