{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:21:44Z","timestamp":1776784904984,"version":"3.51.2"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"253","license":[{"start":{"date-parts":[[2006,6,23]],"date-time":"2006-06-23T00:00:00Z","timestamp":1151020800000},"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":"

\n Let\n \n \n \n \n f<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n f(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a smooth function whose derivative of a given order must be computed. The signal\n \n \n \n \n f<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n f(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is affected by two kinds of perturbation. The perturbation caused by the presence of the machine epsilon\n \n \n \n \n \n \u03f5\n \n <\/mml:mi>\n M<\/mml:mi>\n <\/mml:msub>\n \\epsilon _M<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the computer may be considered to be an extremely high-frequency noise of very small amplitude. The way to minimize its effect consists of choosing an appropriate value for the step size of the difference quotient. The second perturbation, caused by the presence of noise, requires first the signal to be treated in some way. It is the purpose of this work to construct a wavelet-based band-pass filter that deals with the two cited perturbations simultaneously. In effect our wavelet acts like a \u201csmoothed difference quotient\" whose stepsize is of the same order as that of the usual difference quotient. Moreover the wavelet effectively removes the noise and computes the derivative with an accuracy equal to the one obtained by the corresponding difference quotient in the absence of noise.\n <\/p>","DOI":"10.1090\/s0025-5718-05-01767-9","type":"journal-article","created":{"date-parts":[[2005,11,16]],"date-time":"2005-11-16T10:22:35Z","timestamp":1132136555000},"page":"259-280","source":"Crossref","is-referenced-by-count":3,"title":["Wavelet-based filters for accurate computation of derivatives"],"prefix":"10.1090","volume":"75","author":[{"given":"Maurice","family":"Hasson","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,6,23]]},"reference":[{"issue":"227","key":"1","doi-asserted-by":"publisher","first-page":"1121","DOI":"10.1090\/S0025-5718-99-01033-9","article-title":"For numerical differentiation, dimensionality can be a blessing!","volume":"68","author":"Anderssen, Robert S.","year":"1999","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"2","isbn-type":"print","volume-title":"An introduction to numerical analysis","author":"Atkinson, Kendall E.","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/0471624896","edition":"2"},{"key":"3","series-title":"Prentice Hall Information and System Sciences Series","isbn-type":"print","volume-title":"Detection of abrupt changes: theory and application","author":"Basseville, Mich\u00e8le","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0131267809"},{"key":"4","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3069-2","volume-title":"Advanced mathematical methods for scientists and engineers. I","author":"Bender, Carl M.","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0387989315"},{"issue":"10","key":"5","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1016\/0898-1221(91)90196-B","article-title":"Numerical differentiation of analytic functions using quadratures on the semicircle","volume":"22","author":"Cali\u00f2, Franca","year":"1991","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"key":"6","unstructured":"Fran\u00e7ois Chaplais and Sylvain Faure, Wavelets and differentiation, Internal Report, Centre Automatique et Syst\u00e8mes, \u00c9cole Nationale des Mines de Paris (2000), 1\u201314."},{"key":"7","series-title":"CBMS-NSF Regional Conference Series in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970104","volume-title":"Ten lectures on wavelets","volume":"61","author":"Daubechies, Ingrid","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712742"},{"key":"8","unstructured":"David L. Donoho, Private communication, 2003."},{"key":"9","unstructured":"Dan E. Dudgeon and Russell M. Mersereau, Multidimensional digital signal processing, Prentice-Hall, Englewood Cliffs, N.J., 1984."},{"key":"10","unstructured":"Sylvain Faure, Impl\u00e9mentation sous simulux de la d\u00e9rivation des signaux \u00e0 partir des coefficients d\u2019ondelettes, Internal Report, Centre Automatique et Syst\u00e8mes, \u00c9cole Nationale des Mines de Paris (1999), 1\u201312."},{"issue":"1","key":"11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0021-9045(03)00025-X","article-title":"Error formulas for divided difference expansions and numerical differentiation","volume":"122","author":"Floater, Michael S.","year":"2003","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"key":"12","isbn-type":"print","volume-title":"Introduction to partial differential equations","author":"Folland, Gerald B.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0691043612","edition":"2"},{"key":"13","series-title":"CBMS Regional Conference Series in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/cbms\/079","volume-title":"Littlewood-Paley theory and the study of function spaces","volume":"79","author":"Frazier, Michael","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0821807315"},{"issue":"6","key":"14","doi-asserted-by":"publisher","first-page":"512","DOI":"10.2307\/2695705","article-title":"Inverse problems light: numerical differentiation","volume":"108","author":"Hanke, Martin","year":"2001","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"},{"key":"15","series-title":"Lecture Notes in Statistics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-2222-4","volume-title":"Wavelets, approximation, and statistical applications","volume":"129","author":"H\u00e4rdle, Wolfgang","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0387984534"},{"key":"16","series-title":"Studies in Advanced Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1201\/9781420049985","volume-title":"A first course on wavelets","author":"Hern\u00e1ndez, Eugenio","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0849382742"},{"issue":"6","key":"17","doi-asserted-by":"publisher","first-page":"979","DOI":"10.1137\/0908079","article-title":"Numerical differentiation by high order interpolation","volume":"8","author":"Hoffman, Peter","year":"1987","journal-title":"SIAM J. Sci. Statist. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0196-5204","issn-type":"print"},{"issue":"1","key":"18","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1007\/s002110050111","article-title":"A variational method for numerical differentiation","volume":"70","author":"Knowles, Ian","year":"1995","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1-2","key":"19","first-page":"109","article-title":"On the numerical differentiation","volume":"26","author":"Lupa\u015f, Alexandru","year":"1997","journal-title":"Rev. Anal. Num\\'{e}r. Th\\'{e}or. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/1222-9024","issn-type":"print"},{"key":"20","first-page":"11","article-title":"Numerical differentiation of analytic functions","volume":"2","author":"McGregor, Malcolm T.","year":"1991","journal-title":"Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0353-8893","issn-type":"print"},{"key":"21","series-title":"Actualit\\'{e}s Math\\'{e}matiques. [Current Mathematical Topics]","isbn-type":"print","volume-title":"Ondelettes et op\\'{e}rateurs. I","author":"Meyer, Yves","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/2705661250"},{"issue":"5","key":"22","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0898-1221(98)00001-7","article-title":"Discrete mollification and automatic numerical differentiation","volume":"35","author":"Murio, D. A.","year":"1998","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"issue":"235","key":"23","doi-asserted-by":"publisher","first-page":"1131","DOI":"10.1090\/S0025-5718-01-01307-2","article-title":"On stable numerical differentiation","volume":"70","author":"Ramm, Alexander G.","year":"2001","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"24","series-title":"Van Nostrand Reinhold Mathematical Studies","volume-title":"Smoothing and approximation of functions","author":"Shapiro, Harold S.","year":"1969"},{"issue":"3","key":"25","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1007\/s10092-003-0074-1","article-title":"Generalized associated polynomials and their application in numerical differentiation and quadrature","volume":"40","author":"Skrzipek, M.-R.","year":"2003","journal-title":"Calcolo","ISSN":"https:\/\/id.crossref.org\/issn\/0008-0624","issn-type":"print"},{"key":"26","series-title":"Studies in Advanced Mathematics","isbn-type":"print","volume-title":"A guide to distribution theory and Fourier transforms","author":"Strichartz, Robert S.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0849382734"},{"issue":"5","key":"27","doi-asserted-by":"publisher","first-page":"2347","DOI":"10.1023\/A:1011313229137","article-title":"Sharp estimates for errors of numerical differentiation type formulas on trigonometric polynomials","volume":"105","author":"Vinogradov, O. L.","year":"2001","journal-title":"J. Math. Sci. (New York)","ISSN":"https:\/\/id.crossref.org\/issn\/1072-3374","issn-type":"print"},{"issue":"6","key":"28","doi-asserted-by":"publisher","first-page":"1461","DOI":"10.1088\/0266-5611\/18\/6\/301","article-title":"A numerical differentiation method and its application to reconstruction of discontinuity","volume":"18","author":"Wang, Y. B.","year":"2002","journal-title":"Inverse Problems","ISSN":"https:\/\/id.crossref.org\/issn\/0266-5611","issn-type":"print"},{"issue":"2","key":"29","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1093\/biomet\/82.2.385","article-title":"Jump and sharp cusp detection by wavelets","volume":"82","author":"Wang, Yazhen","year":"1995","journal-title":"Biometrika","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3444","issn-type":"print"},{"issue":"1","key":"30","doi-asserted-by":"publisher","first-page":"21","DOI":"10.2307\/2695765","article-title":"Numerical integration of periodic functions: a few examples","volume":"109","author":"Weideman, J. A. C.","year":"2002","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-253\/S0025-5718-05-01767-9\/S0025-5718-05-01767-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-253\/S0025-5718-05-01767-9\/S0025-5718-05-01767-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:33:27Z","timestamp":1776782007000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-253\/S0025-5718-05-01767-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,6,23]]},"references-count":30,"journal-issue":{"issue":"253","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["S0025-5718-05-01767-9"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-05-01767-9","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":[[2005,6,23]]}}}