{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T05:18:07Z","timestamp":1778044687369,"version":"3.51.4"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2011,2,22]],"date-time":"2011-02-22T00:00:00Z","timestamp":1298332800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2011,9]]},"DOI":"10.1007\/s11075-011-9447-8","type":"journal-article","created":{"date-parts":[[2011,2,21]],"date-time":"2011-02-21T07:36:13Z","timestamp":1298273773000},"page":"53-83","source":"Crossref","is-referenced-by-count":50,"title":["Error analysis of Jacobi derivative estimators for noisy signals"],"prefix":"10.1007","volume":"58","author":[{"given":"Da-Yan","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Olivier","family":"Gibaru","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wilfrid","family":"Perruquetti","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2011,2,22]]},"reference":[{"key":"9447_CR1","volume-title":"Handbook of Mathematical Functions","year":"1965","unstructured":"Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Dover, New York (1965)"},{"issue":"4","key":"9447_CR2","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1109\/81.382477","volume":"42","author":"MA Al-Alaoui","year":"1995","unstructured":"Al-Alaoui, M.A.: A class of second-order integrators and low-pass differentiators. IEEE Trans. Circuits Syst. I 42(4), 220\u2013223 (1995)","journal-title":"IEEE Trans. Circuits Syst. I"},{"issue":"4","key":"9447_CR3","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1109\/82.378044","volume":"42","author":"CK Chen","year":"1995","unstructured":"Chen, C.K., Lee, J.H.: Design of high-order digital differentiators using L 1 error criteria. IEEE Trans. Circuits Syst. II 42(4), 287\u2013291 (1995)","journal-title":"IEEE Trans. Circuits Syst. II"},{"key":"9447_CR4","doi-asserted-by":"crossref","first-page":"1565","DOI":"10.1109\/TAC.2002.802740","volume":"47","author":"Y Chitour","year":"2002","unstructured":"Chitour, Y.: Time-varying high-gain observers for numerical differentiation. IEEE Trans. Automat. Contr. 47, 1565\u20131569 (2002)","journal-title":"IEEE Trans. Automat. Contr."},{"key":"9447_CR5","doi-asserted-by":"crossref","first-page":"1523","DOI":"10.1080\/002071799220029","volume":"72","author":"AM Dabroom","year":"1999","unstructured":"Dabroom, A.M., Khalil, H.K.: Discrete-time implementation of high-gain observers for numerical differentiation. Int. J. Control 72, 1523\u20131537 (1999)","journal-title":"Int. J. Control"},{"key":"9447_CR6","volume-title":"Proceedings of the IEEE Conference on Decision and Control","author":"S Diop","year":"2000","unstructured":"Diop, S., Grizzle, J.W., Chaplais, F.: On numerical differentiation algorithms for nonlinear estimation. In: Proceedings of the IEEE Conference on Decision and Control. IEEE Press, New York (2000). Paper CD001876"},{"key":"9447_CR7","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1016\/j.crma.2006.02.037","volume":"342","author":"M Fliess","year":"2006","unstructured":"Fliess, M.: Analyse non standard du bruit. C.R. Acad. Sci. Paris Ser. I 342, 797\u2013802 (2006)","journal-title":"C.R. Acad. Sci. Paris Ser. I"},{"key":"9447_CR8","doi-asserted-by":"crossref","unstructured":"Fliess, M.: Critique du rapport signal \u00e0 bruit en communications num\u00e9riques \u2013 Questioning the signal to noise ratio in digital communications. In: International Conference in Honor of Claude Lobry, ARIMA (Revue africaine d\u2019informatique et de Math\u00e9matiques appliqu\u00e9es), vol. 9, pp. 419\u2013429. Available at http:\/\/hal.inria.fr\/inria-00311719\/en\/ (2008)","DOI":"10.46298\/arima.1908"},{"issue":"339","key":"9447_CR9","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1016\/j.crma.2004.10.003","volume":"I","author":"M Fliess","year":"2004","unstructured":"Fliess, M., Join, C., Mboup, M., Sira-Ram\u00edrez, H.: Compression diff\u00e9rentielle de transitoires bruit\u00e9s. C. R. Acad. Sci. (Paris) I(339), 821\u2013826 (2004)","journal-title":"C. R. Acad. Sci. (Paris)"},{"key":"9447_CR10","unstructured":"Fliess, M., Sira-Ram\u00edrez, H.: Closed-loop parametric identification for continuous-time linear systems via new algebraic techniques. In: Garnier, H., Wang, L. (eds.) Identification of Continuous-Time Models from Sampled Data, pp. 363\u2013391. Springer (2008). Available at http:\/\/hal.inria.fr\/inria-00114958\/en\/"},{"key":"9447_CR11","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1051\/cocv:2003008","volume":"9","author":"M Fliess","year":"2003","unstructured":"Fliess, M., Sira-Ram\u00edrez, H.: An algebraic framework for linear identification. ESAIM Control Optim. Calc. Variat. 9, 151\u2013168 (2003)","journal-title":"ESAIM Control Optim. Calc. Variat."},{"key":"9447_CR12","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.1088\/0266-5611\/11\/6\/009","volume":"11","author":"DN H\u00e0o","year":"1995","unstructured":"H\u00e0o, D.N., Schneider, A., Reinhardt, H.J.: Regularization of a non-characteristic Cauchy problem for a parabolic equation. Inverse Problems 11, 1247\u20131264 (1995)","journal-title":"Inverse Problems"},{"key":"9447_CR13","volume-title":"Signals and Systems","author":"S Haykin","year":"2002","unstructured":"Haykin, S., Van Veen, B.: Signals and Systems, 2nd edn. Wiley, New York (2002)","edition":"2"},{"key":"9447_CR14","doi-asserted-by":"crossref","first-page":"2055","DOI":"10.1109\/TAC.2003.819303","volume":"48","author":"S Ibrir","year":"2003","unstructured":"Ibrir, S.: Online exact differentiation and notion of asymptotic algebraic observers. IEEE Trans. Automat. Contr. 48, 2055\u20132060 (2003)","journal-title":"IEEE Trans. Automat. Contr."},{"key":"9447_CR15","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/j.automatica.2003.09.020","volume":"40","author":"S Ibrir","year":"2004","unstructured":"Ibrir, S.: Linear time-derivatives trackers. Automatica 40, 397\u2013405 (2004)","journal-title":"Automatica"},{"key":"9447_CR16","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/S0377-0427(99)00358-1","volume":"126","author":"IR Khan","year":"2000","unstructured":"Khan, I.R., Ohba, R.: New finite difference formulas for numerical differentiation. J. Comput. Appl. Math. 126, 269\u2013276 (2000)","journal-title":"J. Comput. Appl. Math."},{"key":"9447_CR17","doi-asserted-by":"crossref","first-page":"924","DOI":"10.1080\/0020717031000099029","volume":"76","author":"A Levant","year":"2003","unstructured":"Levant, A.: Higher-order sliding modes, differentiation and output-feedback control. Int. J. Control 76, 924\u2013941 (2003)","journal-title":"Int. J. Control"},{"key":"9447_CR18","volume-title":"Applied Analysis","author":"C Lanczos","year":"1956","unstructured":"Lanczos, C.: Applied Analysis. Prentice-Hall, Englewood Cliffs (1956)"},{"key":"9447_CR19","unstructured":"Liu, D.Y., Gibaru, O., Perruquetti, W.: Error analysis for a class of numerical differentiator: application to state observation. In: 48th IEEE Conference on Decision and Control, China (2009). Available at http:\/\/hal.inria.fr\/inria-00437129\/en\/"},{"key":"9447_CR20","unstructured":"Liu, D.Y., Gibaru, O., Perruquetti, W., Fliess, M., Mboup, M.: An error analysis in the algebraic estimation of a noisy sinusoidal signal. In: 16th Mediterranean Conference on Control and Automation (MED\u2019 2008), Ajaccio (2008). Available at http:\/\/hal.inria.fr\/inria-00300234\/en\/"},{"key":"9447_CR21","doi-asserted-by":"crossref","unstructured":"Liu, D.Y., Gibaru, O., Perruquetti, W.: Differentiation by integration with Jacobi polynomials. J. Comput. Appl. Math. (2010). doi: 10.1016\/j.cam.2010.12.023 . Available at http:\/\/hal.inria.fr\/inria-00550160","DOI":"10.1016\/j.cam.2010.12.023"},{"key":"9447_CR22","volume-title":"Probability Theory","author":"M Lo\u00e8ve","year":"1963","unstructured":"Lo\u00e8ve, M.: Probability Theory, 3rd edn. van Nostrand, New York (1963)","edition":"3"},{"key":"9447_CR23","first-page":"397","volume-title":"Approximation and Computation, Int. Ser. Numer. Math., vol. 119","author":"JN Lyness","year":"1994","unstructured":"Lyness, J.N.: Finite-part integrals and the Euler\u2013Maclaurin expansion. In: Zahar, R.V.M. (ed.) Approximation and Computation, Int. Ser. Numer. Math., vol. 119, pp. 397\u2013407. Birkh\u00e4user, Basel (1994)"},{"key":"9447_CR24","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1080\/00036810802555441","volume":"88","author":"M Mboup","year":"2009","unstructured":"Mboup, M.: Parameter estimation for signals described by differential equations. Appl. Anal. 88, 29\u201352 (2009)","journal-title":"Appl. Anal."},{"key":"9447_CR25","doi-asserted-by":"crossref","unstructured":"Mboup, M., Join, C., Fliess, M.: A revised look at numerical differentiation with an application to nonlinear feedback control. In: 15th Mediterranean Conference on Control and Automation (MED\u201907). Athenes, Greece (2007)","DOI":"10.1109\/MED.2007.4433728"},{"issue":"4","key":"9447_CR26","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1007\/s11075-008-9236-1","volume":"50","author":"M Mboup","year":"2009","unstructured":"Mboup, M., Join, C., Fliess, M.: Numerical differentiation with annihilators in noisy environment. Numer. Algorithms 50(4), 439\u2013467 (2009)","journal-title":"Numer. Algorithms"},{"key":"9447_CR27","doi-asserted-by":"crossref","DOI":"10.1002\/9781118033210","volume-title":"The Mollification Method and the Numerical Solution of Ill-Posed Problems","author":"DA Murio","year":"1993","unstructured":"Murio, D.A.: The Mollification Method and the Numerical Solution of Ill-Posed Problems. Wiley, New York (1993)"},{"key":"9447_CR28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0898-1221(98)00001-7","volume":"35","author":"DA Murio","year":"1998","unstructured":"Murio, D.A., Mej\u00eda, C.E., Zhan, S.: Discrete mollification and automatic numerical differentiation. Comput. Math. Appl. 35, 1\u201316 (1998)","journal-title":"Comput. Math. Appl."},{"key":"9447_CR29","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/j.cam.2006.11.035","volume":"212","author":"G Nakamura","year":"2008","unstructured":"Nakamura, G., Wang, S., Wang, Y.: Numerical differentiation for the second order derivatives of functions of two variables. J. Comput. Appl. Math. 212, 341\u2013358 (2008)","journal-title":"J. Comput. Appl. Math."},{"issue":"10","key":"9447_CR30","first-page":"55","volume":"24","author":"R Qu","year":"1996","unstructured":"Qu, R.: A new approach to numerical differentiation and integration. Math. Comput. 24(10), 55\u201368 (1996)","journal-title":"Math. Comput."},{"issue":"12","key":"9447_CR31","doi-asserted-by":"crossref","first-page":"2779","DOI":"10.1109\/TCSI.2006.883877","volume":"53","author":"CM Rader","year":"2006","unstructured":"Rader, C.M., Jackson, L.B.: Approximating noncausal IIR digital filters having arbitrary poles, including new Hilbert transformer designs, via forward\/backward block recursion. IEEE Trans. Circuits Syst. I 53(12), 2779\u20132787 (2006)","journal-title":"IEEE Trans. Circuits Syst. I"},{"key":"9447_CR32","doi-asserted-by":"crossref","first-page":"1131","DOI":"10.1090\/S0025-5718-01-01307-2","volume":"70","author":"AG Ramm","year":"2001","unstructured":"Ramm, A.G., Smirnova, A.B.: On stable numerical differentiation. Math. Comput. 70, 1131\u20131153 (2001)","journal-title":"Math. Comput."},{"key":"9447_CR33","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1016\/j.cam.2004.10.016","volume":"177","author":"SK Rangarajana","year":"2005","unstructured":"Rangarajana, S.K., Purushothaman, S.P.: Lanczos\u2019 generalized derivative for higher orders. J. Comput. Appl. Math. 177, 461\u2013465 (2005)","journal-title":"J. Comput. Appl. Math."},{"key":"9447_CR34","volume-title":"Digital Signal Processing","author":"RA Roberts","year":"1987","unstructured":"Roberts, R.A., Mullis, C.T.: Digital Signal Processing. Addison-Wesley, Reading (1987)"},{"key":"9447_CR35","volume-title":"Stochastic Processes","author":"E Parzen","year":"1962","unstructured":"Parzen, E.: Stochastic Processes. Holden-Day, San Francisco (1962)"},{"key":"9447_CR36","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1109\/TCST.2006.876917","volume":"14","author":"YX Su","year":"2006","unstructured":"Su, Y.X., Zheng, C.H., Mueller, P.C., Duan, B.Y.: A simple improved velocity estimation for low-speed regions based on position measurements only. IEEE Trans. Control Syst. Technol. 14, 937\u2013942 (2006)","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"9447_CR37","volume-title":"Orthogonal Polynomials","author":"G Szeg\u00f6","year":"1967","unstructured":"Szeg\u00f6, G.: Orthogonal Polynomials, 3rd edn. AMS, Providence (1967)","edition":"3"},{"key":"9447_CR38","doi-asserted-by":"crossref","first-page":"1461","DOI":"10.1088\/0266-5611\/18\/6\/301","volume":"18","author":"Y Wang","year":"2002","unstructured":"Wang, Y., Jia, X., Cheng, J.: A numerical differentiation method and its application to reconstruction of discontinuity. Inverse Problems 18, 1461\u20131476 (2002)","journal-title":"Inverse Problems"},{"key":"9447_CR39","doi-asserted-by":"crossref","first-page":"941","DOI":"10.1016\/j.cam.2010.01.056","volume":"234","author":"Z Wang","year":"2010","unstructured":"Wang, Z., Wen, R.: Numerical differentiation for high orders by an integration method. J. Comput. Appl. Math. 234, 941\u2013948 (2010)","journal-title":"J. Comput. Appl. Math."},{"key":"9447_CR40","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1088\/0266-5611\/21\/2\/013","volume":"21","author":"T Wei","year":"2005","unstructured":"Wei, T., Hon, Y.C., Wang, Y.: Reconstruction of numerical derivatives from scattered noisy data. Inverse Problems 21, 657\u2013672 (2005)","journal-title":"Inverse Problems"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-011-9447-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-011-9447-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-011-9447-8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,19]],"date-time":"2021-11-19T05:41:04Z","timestamp":1637300464000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-011-9447-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,2,22]]},"references-count":40,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2011,9]]}},"alternative-id":["9447"],"URL":"https:\/\/doi.org\/10.1007\/s11075-011-9447-8","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,2,22]]}}}