{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:38:20Z","timestamp":1776836300627,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"256","license":[{"start":{"date-parts":[[2007,6,22]],"date-time":"2007-06-22T00:00:00Z","timestamp":1182470400000},"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 construct wavelets on general\n                    <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>\n                    -dimensional domains or manifolds via a domain decomposition technique, resulting in so-called composite wavelets. With this construction, wavelets with supports that extend to more than one patch are only continuous over the patch interfaces. Normally, this limited smoothness restricts the possibility for matrix compression, and with that the application of these wavelets in (adaptive) methods for solving operator equations. By modifying the scaling functions on the interval, and with that on the\n                    <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>\n                    -cube that serves as parameter domain, we obtain composite wavelets that have\n                    <italic>patchwise cancellation properties<\/italic>\n                    of any required order, meaning that the restriction of any wavelet to each patch is again a wavelet. This is also true when the wavelets are required to satisfy zeroth order homogeneous Dirichlet boundary conditions on (part of) the boundary. As a result, compression estimates now depend only on the patchwise smoothness of the wavelets that one may choose. Also taking stability into account, our composite wavelets have all the properties for the application to the (adaptive) solution of well-posed operator equations of orders\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 t\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>t<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2 t<\/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=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t element-of left-parenthesis negative one half comma three halves right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mfrac>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mfrac>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mfrac>\n                              <mml:mn>3<\/mml:mn>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mfrac>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">t \\in (-\\frac {1}{2},\\frac {3}{2})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01867-9","type":"journal-article","created":{"date-parts":[[2006,8,16]],"date-time":"2006-08-16T10:28:45Z","timestamp":1155724125000},"page":"1871-1889","source":"Crossref","is-referenced-by-count":22,"title":["Wavelets with patchwise cancellation properties"],"prefix":"10.1090","volume":"75","author":[{"given":"Helmut","family":"Harbrecht","sequence":"first","affiliation":[]},{"given":"Rob","family":"Stevenson","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,6,22]]},"reference":[{"key":"1","unstructured":"[BF01] S. 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Ph.D. Thesis, Technische Universit\u00e4t Chemnitz, Germany, 2001."},{"key":"16","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1002\/mana.200310171","article-title":"Biorthogonal wavelet bases for the boundary element method","volume":"269\/270","author":"Harbrecht, Helmut","year":"2004","journal-title":"Math. Nachr.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-584X","issn-type":"print"},{"key":"17","unstructured":"[KS04] A. Kunoth and J. Sahner. Wavelets on manifolds: An optimized construction. SFB 611 Preprint 163, Universit\u00e4t Bonn, July 2004. To appear in Math. Comp."},{"key":"18","doi-asserted-by":"crossref","unstructured":"[Sch98] R. Schneider. Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur L\u00f6sung gro\u00dfer vollbesetzter Gleigungssysteme. Habilitationsschrift, 1995. Advances in Numerical Mathematics. Teubner, Stuttgart, 1998.","DOI":"10.1007\/978-3-663-10851-1"},{"key":"19","unstructured":"[Ste04a] R.P. Stevenson. Composite wavelet bases with extended stability and cancellation properties. Technical Report 1304, Utrecht University, July 2004. To appear in in SIAM J. Numer. Anal."},{"issue":"5","key":"20","doi-asserted-by":"publisher","first-page":"1110","DOI":"10.1137\/S0036141002411520","article-title":"On the compressibility operators in wavelet coordinates","volume":"35","author":"Stevenson, Rob","year":"2004","journal-title":"SIAM J. Math. 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