{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T18:43:50Z","timestamp":1776105830415,"version":"3.50.1"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"9","license":[{"start":{"date-parts":[[2007,3,23]],"date-time":"2007-03-23T00:00:00Z","timestamp":1174608000000},"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":["Proc. Amer. Math. Soc."],"abstract":"<p>\n                    In 1939, G. H. Hardy proved that, under certain conditions, the only functions satisfying\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"integral Subscript 0 Superscript 1 Baseline f left-parenthesis lamda Subscript m Baseline t right-parenthesis f left-parenthesis lamda Subscript n Baseline t right-parenthesis d t equals 0 comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msubsup>\n                              <mml:mo>\n                                \u222b\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03bb\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>m<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03bb\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*} \\int _{0}^{1}f(\\lambda _{m}t)f(\\lambda _{n}t)dt=0, \\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    where the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"lamda Subscript n\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\lambda _{n}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are the zeros 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                    , are the Bessel functions. We replace the above integral by the Jackson\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q\">\n                        <mml:semantics>\n                          <mml:mi>q<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -integral and give the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q\">\n                        <mml:semantics>\n                          <mml:mi>q<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -analogue of Hardy\u2019s result.\n                  <\/p>","DOI":"10.1090\/s0002-9939-06-08285-2","type":"journal-article","created":{"date-parts":[[2006,5,24]],"date-time":"2006-05-24T14:49:36Z","timestamp":1148482176000},"page":"2695-2701","source":"Crossref","is-referenced-by-count":11,"special_numbering":"567","title":["Functions \ud835\udc5e-orthogonal with respect to their own zeros"],"prefix":"10.1090","volume":"134","author":[{"given":"Luis","family":"Abreu","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,3,23]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"1197","DOI":"10.1090\/S0002-9939-04-07589-6","article-title":"A \ud835\udc5e-sampling theorem related to the \ud835\udc5e-Hankel transform","volume":"133","author":"Abreu, L. D.","year":"2005","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"issue":"3-4","key":"2","doi-asserted-by":"publisher","first-page":"310","DOI":"10.1007\/BF03322057","article-title":"A note on the analytic form of the Kramer sampling theorem","volume":"34","author":"Everitt, W. N.","year":"1998","journal-title":"Results Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0378-6218","issn-type":"print"},{"key":"3","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","volume-title":"Basic hypergeometric series","volume":"35","author":"Gasper, George","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0521350492"},{"key":"4","doi-asserted-by":"crossref","unstructured":"G. H. Hardy, Notes on special systems of orthogonal functions (II): On functions orthogonal with respect to their own zeros, J. Lond. Math. 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Ismail, Some properties of Jackson\u2019s third q-Bessel function, unpublished manuscript."},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1215\/S0012-7094-94-07617-5","article-title":"The quantum group of plane motions and the Hahn-Exton \ud835\udc5e-Bessel function","volume":"76","author":"Koelink, H. T.","year":"1994","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"690","DOI":"10.1006\/jmaa.1994.1327","article-title":"On the zeros of the Hahn-Exton \ud835\udc5e-Bessel function and associated \ud835\udc5e-Lommel polynomials","volume":"186","author":"Koelink, H. T.","year":"1994","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"445","DOI":"10.2307\/2154118","article-title":"On \ud835\udc5e-analogues of the Fourier and Hankel transforms","volume":"333","author":"Koornwinder, Tom H.","year":"1992","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"11","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1002\/sapm195938168","article-title":"A generalized sampling theorem","volume":"38","author":"Kramer, H. P.","year":"1959","journal-title":"J. Math. and Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0097-1421","issn-type":"print"},{"issue":"3","key":"12","doi-asserted-by":"publisher","first-page":"855","DOI":"10.2307\/2160480","article-title":"A \ud835\udc5e-analogue of the Wronskian and a second solution of the Hahn-Exton \ud835\udc5e-Bessel difference equation","volume":"120","author":"Swarttouw, R. F.","year":"1994","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"key":"13","unstructured":"J. M. Whittaker, Interpolatory function theory  (1935)."}],"container-title":["Proceedings of the American Mathematical Society"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/proc\/2006-134-09\/S0002-9939-06-08285-2\/S0002-9939-06-08285-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/proc\/2006-134-09\/S0002-9939-06-08285-2\/S0002-9939-06-08285-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T17:23:33Z","timestamp":1776101013000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/proc\/2006-134-09\/S0002-9939-06-08285-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,3,23]]},"references-count":13,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2006,9]]}},"alternative-id":["S0002-9939-06-08285-2"],"URL":"https:\/\/doi.org\/10.1090\/s0002-9939-06-08285-2","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6826","0002-9939"],"issn-type":[{"value":"1088-6826","type":"electronic"},{"value":"0002-9939","type":"print"}],"subject":[],"published":{"date-parts":[[2006,3,23]]}}}