{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:50:01Z","timestamp":1776721801298,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"214","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n The aim of this paper is to investigate a multiresolution analysis of nested subspaces of trigonometric polynomials. The pair of scaling functions which span the sample spaces are fundamental functions for Hermite interpolation on a dyadic partition of nodes on the interval\n \n \n \n \n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n \n \u03c0\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n [0,2\\pi )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Two wavelet functions that generate the corresponding orthogonal complementary subspaces are constructed so as to possess the same fundamental interpolatory properties as the scaling functions. Together with the corresponding dual functions, these interpolatory properties of the scaling functions and wavelets are used to formulate the specific decomposition and reconstruction sequences. Consequently, this trigonometric multiresolution analysis allows a completely explicit algorithmic treatment.\n <\/p>","DOI":"10.1090\/s0025-5718-96-00719-3","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"683-722","source":"Crossref","is-referenced-by-count":31,"title":["Trigonometric wavelets for Hermite interpolation"],"prefix":"10.1090","volume":"65","author":[{"given":"Ewald","family":"Quak","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","isbn-type":"print","first-page":"217","article-title":"Wavelets with boundary conditions on the interval","author":"Auscher, Pascal","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0121745902"},{"key":"2","isbn-type":"print","first-page":"63","article-title":"Lacunary trigonometric interpolation on equidistant nodes","author":"Cavaretta, A. S., Jr.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/0122136500"},{"key":"3","series-title":"Wavelet Analysis and its Applications","isbn-type":"print","volume-title":"An introduction to wavelets","volume":"1","author":"Chui, Charles K.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0121745848"},{"issue":"2-3","key":"4","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1007\/BF01198002","article-title":"On trigonometric wavelets","volume":"9","author":"Chui, C. K.","year":"1993","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"903","DOI":"10.2307\/2153941","article-title":"On compactly supported spline wavelets and a duality principle","volume":"330","author":"Chui, Charles K.","year":"1992","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"6","series-title":"A Wiley-Interscience Publication","isbn-type":"print","volume-title":"Circulant matrices","author":"Davis, Philip J.","year":"1979","ISBN":"https:\/\/id.crossref.org\/isbn\/0471057711"},{"key":"7","doi-asserted-by":"crossref","unstructured":"[7] T. N. T. Goodman, Interpolatory Hermite spline wavelets, J. Approx. Theory 78 (1994), 174\u2013189.","DOI":"10.1006\/jath.1994.1071"},{"key":"8","doi-asserted-by":"crossref","unstructured":"[8] Y. W. Koh, S. L. Lee and H. H. Tan, Periodic orthogonal splines and wavelets, Applied and Computational Harmonic Analysis 2 (1995), 201\u2013218.","DOI":"10.1006\/acha.1995.1014"},{"key":"9","doi-asserted-by":"crossref","unstructured":"[9] R. A. Lorentz and A. A. Sahakian, Orthogonal trigonometric Schauder bases of optimal degree for \ud835\udc36(\ud835\udc3e), Journal of Fourier Analysis and Applications 1 (1994), 103\u2013112.","DOI":"10.1007\/s00041-001-4005-8"},{"issue":"2-3","key":"10","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1007\/BF01198009","article-title":"A note on orthonormal polynomial bases and wavelets","volume":"9","author":"Offin, D.","year":"1993","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"key":"11","doi-asserted-by":"crossref","unstructured":"[11] J. Prestin and E. Quak, Trigonometric interpolation and wavelet decompositions, Numerical Algorithms 9 (1995), 293\u2013317.","DOI":"10.1007\/BF02141593"},{"key":"12","unstructured":"[12] \\bysame, A duality principle for trigonometric wavelets, Wavelets, Images, and Surface Fitting (P. J. Laurent, A. Le M\u00e9haut\u00e9 and L. L. Schumaker, eds.), A K Peters, Boston, 1994, pp. (407\u2013418)."},{"key":"13","unstructured":"[13] \\bysame, Decay properties of trigonometric wavelets, Proceedings of the Cornelius Lanczos International Centenary Conference (J. D. Brown, M. T. Chu, D. C. Ellison and R. J. Plemmons, eds.), SIAM, Philadelphia, 1994, pp. (413\u2013415)."},{"issue":"3","key":"14","doi-asserted-by":"publisher","first-page":"384","DOI":"10.1070\/SM1992v072n02ABEH002143","article-title":"An orthogonal trigonometric basis","volume":"182","author":"Privalov, A. A.","year":"1991","journal-title":"Mat. Sb.","ISSN":"https:\/\/id.crossref.org\/issn\/0368-8666","issn-type":"print"},{"key":"15","series-title":"International Series of Monographs in Pure and Applied Mathematics","volume-title":"Theory of approximation of functions of a real variable","volume":"34","author":"Timan, A. 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