{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:07:11Z","timestamp":1776830831341,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"322","license":[{"start":{"date-parts":[[2020,9,9]],"date-time":"2020-09-09T00:00:00Z","timestamp":1599609600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate more general high frequency solutions to PDEs. We present a superposition of Gaussian beams over an arbitrary bounded set of 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                    in phase space, and show that the tools recently developed in [Math. Comp. 82 (2013), pp. 919\u2013952] can be applied to obtain the propagation error of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k Superscript 1 minus StartFraction upper N Over 2 EndFraction minus StartFraction d minus m Over 4 EndFraction\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>\n                                \u2212\n                                \n                              <\/mml:mo>\n                              <mml:mfrac>\n                                <mml:mi>N<\/mml:mi>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mfrac>\n                              <mml:mo>\n                                \u2212\n                                \n                              <\/mml:mo>\n                              <mml:mfrac>\n                                <mml:mrow>\n                                  <mml:mi>d<\/mml:mi>\n                                  <mml:mo>\n                                    \u2212\n                                    \n                                  <\/mml:mo>\n                                  <mml:mi>m<\/mml:mi>\n                                <\/mml:mrow>\n                                <mml:mn>4<\/mml:mn>\n                              <\/mml:mfrac>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">k^{1- \\frac {N}{2}- \\frac {d-m}{4}}<\/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=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper 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                    is the order of beams and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the spatial dimension. Moreover, we study the sharpness of this estimate in examples.\n                  <\/p>","DOI":"10.1090\/mcom\/3462","type":"journal-article","created":{"date-parts":[[2019,7,24]],"date-time":"2019-07-24T09:35:37Z","timestamp":1563960937000},"page":"675-697","source":"Crossref","is-referenced-by-count":2,"title":["General superpositions of Gaussian beams and propagation errors"],"prefix":"10.1090","volume":"89","author":[{"given":"Hailiang","family":"Liu","sequence":"first","affiliation":[]},{"given":"James","family":"Ralston","sequence":"additional","affiliation":[]},{"given":"Peimeng","family":"Yin","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2019,9,9]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"973","DOI":"10.4310\/cms.2009.v7.n4.a9","article-title":"Gaussian beams summation for the wave equation in a convex domain","volume":"7","author":"Bougacha, Salma","year":"2009","journal-title":"Commun. 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"5","isbn-type":"print","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1090\/conm\/640\/12852","article-title":"Error estimates of the Bloch band-based Gaussian beam superposition for the Schr\u00f6dinger equation","author":"Liu, Hailiang","year":"2015","ISBN":"https:\/\/id.crossref.org\/isbn\/9781470409890"},{"key":"6","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/j.wavemoti.2017.05.004","article-title":"Error estimates for Gaussian beam methods applied to symmetric strictly hyperbolic systems","volume":"73","author":"Liu, Hailiang","year":"2017","journal-title":"Wave Motion","ISSN":"https:\/\/id.crossref.org\/issn\/0165-2125","issn-type":"print"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"428","DOI":"10.1137\/090761598","article-title":"Recovery of high frequency wave fields for the acoustic wave equation","volume":"8","author":"Liu, Hailiang","year":"2009","journal-title":"Multiscale Model. Simul.","ISSN":"https:\/\/id.crossref.org\/issn\/1540-3459","issn-type":"print"},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"622","DOI":"10.1137\/090756909","article-title":"Recovery of high frequency wave fields from phase space-based measurements","volume":"8","author":"Liu, Hailiang","year":"2009","journal-title":"Multiscale Model. Simul.","ISSN":"https:\/\/id.crossref.org\/issn\/1540-3459","issn-type":"print"},{"issue":"282","key":"9","doi-asserted-by":"publisher","first-page":"919","DOI":"10.1090\/S0025-5718-2012-02656-1","article-title":"Error estimates for Gaussian beam superpositions","volume":"82","author":"Liu, Hailiang","year":"2013","journal-title":"Math. 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Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/1539-6746","issn-type":"print"},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"105","DOI":"10.4310\/CMS.2013.v11.n1.a4","article-title":"Global geometrical optics method","volume":"11","author":"Zheng, Chunxiong","year":"2013","journal-title":"Commun. Math. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/1539-6746","issn-type":"print"},{"issue":"6","key":"13","doi-asserted-by":"publisher","first-page":"2905","DOI":"10.1137\/130935720","article-title":"Optimal error estimates for first-order Gaussian beam approximations to the Schr\u00f6dinger equation","volume":"52","author":"Zheng, Chunxiong","year":"2014","journal-title":"SIAM J. Numer. 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