{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:38:19Z","timestamp":1776836299363,"version":"3.51.2"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"255","license":[{"start":{"date-parts":[[2007,5,3]],"date-time":"2007-05-03T00:00:00Z","timestamp":1178150400000},"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>A key ingredient of the construction of biorthogonal wavelet bases for Sobolev spaces on manifolds, which is based on topological isomorphisms is the Hestenes extension operator. Here we firstly investigate whether this particular extension operator can be replaced by another extension operator. Our main theoretical result states that an important class of extension operators based on interpolating boundary values cannot be used in the construction setting required by Dahmen and Schneider. In the second part of this paper, we investigate and optimize the Hestenes extension operator. The results of the optimization process allow us to implement the construction of biorthogonal wavelets from Dahmen and Schneider. As an example, we illustrate a wavelet basis on the 2-sphere.<\/p>","DOI":"10.1090\/s0025-5718-06-01828-x","type":"journal-article","created":{"date-parts":[[2006,5,24]],"date-time":"2006-05-24T14:43:01Z","timestamp":1148481781000},"page":"1319-1349","source":"Crossref","is-referenced-by-count":22,"title":["Wavelets on manifolds: An optimized construction"],"prefix":"10.1090","volume":"75","author":[{"given":"Angela","family":"Kunoth","sequence":"first","affiliation":[]},{"given":"Jan","family":"Sahner","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,5,3]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"key":"2","unstructured":"[B] D. Bertsekas, Nonlinear Programming, Athena Scientific, 1999."},{"key":"3","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-6333-3","volume-title":"A practical guide to splines","volume":"27","author":"de Boor, Carl","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0387903569"},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4064\/sm-76-1-1-58","article-title":"Spline bases in classical function spaces on compact \ud835\udc36^{\u221e} manifolds. I","volume":"76","author":"Ciesielski, Z.","year":"1983","journal-title":"Studia Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0039-3223","issn-type":"print"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1006\/acha.1997.0242","article-title":"The wavelet element method. I. Construction and analysis","volume":"6","author":"Canuto, Claudio","year":"1999","journal-title":"Appl. Comput. Harmon. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1063-5203","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1006\/acha.1998.0247","article-title":"Biorthogonal spline wavelets on the interval\u2014stability and moment conditions","volume":"6","author":"Dahmen, Wolfgang","year":"1999","journal-title":"Appl. Comput. Harmon. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1063-5203","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1137\/S0036141098333451","article-title":"Wavelets on manifolds. I. Construction and domain decomposition","volume":"31","author":"Dahmen, Wolfgang","year":"1999","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"228","key":"8","doi-asserted-by":"publisher","first-page":"1533","DOI":"10.1090\/S0025-5718-99-01092-3","article-title":"Composite wavelet bases for operator equations","volume":"68","author":"Dahmen, Wolfgang","year":"1999","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3-4","key":"9","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1007\/BF03322055","article-title":"Wavelets with complementary boundary conditions\u2014function spaces on the cube","volume":"34","author":"Dahmen, Wolfgang","year":"1998","journal-title":"Results Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0378-6218","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1137\/S0036142997330949","article-title":"Element-by-element construction of wavelets satisfying stability and moment conditions","volume":"37","author":"Dahmen, Wolfgang","year":"1999","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"11","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1215\/S0012-7094-41-00812-8","article-title":"Extension of the range of a differentiable function","volume":"8","author":"Hestenes, M. R.","year":"1941","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"key":"12","doi-asserted-by":"crossref","unstructured":"[Ha] H. Harbrecht, Wavelet Galerkin Schemes for the Boundary Element Method in Three Dimensions, Dissertation, Technische Universit\u00e4t Chemnitz, 2001.","DOI":"10.1007\/978-3-0348-8276-7_14"},{"key":"13","unstructured":"[IGPM] A. Barinka, T. Bartsch, K. Urban, J. Vorloeper, The Multilevel Library, Institut f\u00fcr Geometrie und Praktische Mathematik, RWTH Aachen, 2001."},{"key":"14","unstructured":"[KS] A. Kunoth, J. Sahner, Wavelets on manifolds: An optimized construction (extended version), SFB 611 Preprint #163, Universit\u00e4t Bonn, July 2004, revised, April 2005, available at \\url{http:\/\/www.ins.uni-bonn.de\/ kunoth\/papers\/papers.html}"},{"key":"15","unstructured":"[S] J. Sahner, On the Optimized Construction of Wavelets on Manifolds, Diploma Thesis (in English), Universit\u00e4t Bonn, September 2003."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01828-X\/S0025-5718-06-01828-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01828-X\/S0025-5718-06-01828-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:36:25Z","timestamp":1776782185000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01828-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,5,3]]},"references-count":15,"journal-issue":{"issue":"255","published-print":{"date-parts":[[2006,7]]}},"alternative-id":["S0025-5718-06-01828-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-06-01828-x","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2006,5,3]]}}}