{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T13:07:09Z","timestamp":1773234429396,"version":"3.50.1"},"reference-count":61,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2006,10,18]],"date-time":"2006-10-18T00:00:00Z","timestamp":1161129600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2008,5]]},"DOI":"10.1007\/s10444-006-9021-4","type":"journal-article","created":{"date-parts":[[2006,10,17]],"date-time":"2006-10-17T18:35:33Z","timestamp":1161110133000},"page":"301-329","source":"Crossref","is-referenced-by-count":40,"title":["Frames in spaces with finite rate of innovation"],"prefix":"10.1007","volume":"28","author":[{"given":"Qiyu","family":"Sun","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2006,10,18]]},"reference":[{"key":"9021_CR1","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1137\/S0036144501386986","volume":"43","author":"A. Aldroubi","year":"2001","unstructured":"Aldroubi, A., Gr\u00f6chenig, K.: Nonuniform sampling and reconstruction in shift-invariant space. SIAM Rev. 43, 585\u2013620 (2001)","journal-title":"SIAM Rev."},{"key":"9021_CR2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00041-001-0001-2","volume":"7","author":"A. Aldroubi","year":"2001","unstructured":"Aldroubi, A., Sun, Q., Tang, W.-S.: $p$ -frames and shift invariant subspaces of $L^p$ . J. Fourier Anal. Appl. 7, 1\u201321 (2001)","journal-title":"J. Fourier Anal. Appl."},{"key":"9021_CR3","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/s00041-005-4003-3","volume":"11","author":"A. Aldroubi","year":"2005","unstructured":"Aldroubi, A., Sun, Q., Tang, W.-S.: Convolution, average sampling and a Calderon resolution of the identity for shift-invariant spaces. J. Fourier Anal. Appl. 11, 215\u2013244 (2005)","journal-title":"J. Fourier Anal. Appl."},{"key":"9021_CR4","unstructured":"Balan, R., Casazza, P.G., Heil, C., Landau, Z.: Density, overcompleteness and localization of frames I: theorey; II: Gabor system. Preprint (2004)"},{"key":"9021_CR5","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1007\/BF01077924","volume":"24","author":"A.G. Baskakov","year":"1990","unstructured":"Baskakov, A.G.: Wiener's theorem and asymptotic estimates for elements in inverse matrices. Funktsional Anal. o. Prilozhen. 24, 64\u201365 (1990)","journal-title":"Funktsional Anal. o. Prilozhen."},{"key":"9021_CR6","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1006\/acha.1997.0237","volume":"5","author":"J.J. Benedetto","year":"1998","unstructured":"Benedetto, J.J., Li, S.: The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmon. Anal. 5, 389\u2013427 (1998)","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"9021_CR7","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1006\/jfan.2000.3635","volume":"177","author":"M. Bownik","year":"2000","unstructured":"Bownik, M.: The structure of shift-invariant subspaces of $L\\sp 2(R\\sp n)$ . J. Funct. Anal. 177, 282\u2013309 (2000)","journal-title":"J. Funct. Anal."},{"key":"9021_CR8","doi-asserted-by":"crossref","unstructured":"Bownik, M.: Anisotropic Hardy spaces and wavelets. Mem. Am. Math. Soc. 781, 22 pp (2003)","DOI":"10.1090\/memo\/0781"},{"key":"9021_CR9","first-page":"185","volume":"3","author":"P.L. Butzer","year":"1983","unstructured":"Butzer, P.L.: A survey of the Whittaker\u2013Shannon sampling theorem and some of its extensions. J. Math. Res. Expo. 3, 185\u2013212 (1983)","journal-title":"J. Math. Res. Expo."},{"key":"9021_CR10","doi-asserted-by":"crossref","first-page":"129","DOI":"10.11650\/twjm\/1500407227","volume":"4","author":"P. Casazza","year":"2000","unstructured":"Casazza, P.: The art of frame theory. Taiwan. J. Math. 4, 129\u2013201 (2000)","journal-title":"Taiwan. J. Math."},{"key":"9021_CR11","doi-asserted-by":"crossref","unstructured":"Casazza, P., Han, D., Larson, D.R.: Frames for Banach spaces. In: The Functional and Harmonic Analysis of Wavelets and Frames (San Antonio, Texas, 1999), Contemporary Mathematics, vol. 247, pp. 149\u2013182. (1999)","DOI":"10.1090\/conm\/247\/03801"},{"key":"9021_CR12","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1023\/A:1022882423107","volume":"19","author":"H.-L. Chen","year":"2003","unstructured":"Chen, H.-L., Peng, S.-L.: Localization of dual periodic scaling and wavelet functions. Adv. Comput. Math. 19, 195\u2013210 (2003)","journal-title":"Adv. Comput. Math."},{"key":"9021_CR13","doi-asserted-by":"crossref","unstructured":"Christensen, O.: An Introduction to Frames and Riesz Bases. Birkh\u00e4user, Boston (2003)","DOI":"10.1007\/978-0-8176-8224-8"},{"key":"9021_CR14","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1023\/A:1021364413257","volume":"18","author":"O. Christensen","year":"2003","unstructured":"Christensen, O., Stoeva, D.T.: $p$ -frames in separable Banach spaces. Adv. Comput. Math. 18, 117\u2013126 (2003)","journal-title":"Adv. Comput. Math."},{"key":"9021_CR15","doi-asserted-by":"crossref","unstructured":"Chui, C.K.: An Introduction to Wavelets. Academic, Boston, Massachusetts (1992)","DOI":"10.1063\/1.4823126"},{"key":"9021_CR16","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.acha.2004.02.004","volume":"17","author":"C.K. Chui","year":"2004","unstructured":"Chui, C.K., He, W., St\u00f6ckler, J.: Nonstationary tight wavelet frames: I. Bounded intervals; II. Unbounded intervals. Appl. Comput. Harmon. Anal. 17, 141\u2013197 (2004); 18, 25\u201366 (2005)","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"9021_CR17","doi-asserted-by":"crossref","unstructured":"Chui, C.K., Sun, Q.: Affine frame decompositions and shift-invariant spaces. Appl. Comput. Harmon. Anal. 20, 74\u2013107 (2006)","DOI":"10.1016\/j.acha.2005.09.003"},{"key":"9021_CR18","doi-asserted-by":"crossref","first-page":"1745","DOI":"10.1137\/S003614109427429X","volume":"27","author":"A. Cohen","year":"1996","unstructured":"Cohen, A., Dyn, N.: Nonstationary subdivision schemes and multiresolution analysis. SIAM J. Math. Anal. 27, 1745\u20131769 (1996)","journal-title":"SIAM J. Math. Anal."},{"key":"9021_CR19","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1006\/acha.1993.1005","volume":"1","author":"A. Cohen","year":"1993","unstructured":"Cohen, A., Daubechies, I., Vial, P.: Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1, 54\u201381 (1993)","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"9021_CR20","doi-asserted-by":"crossref","unstructured":"Coifman, R., Maggioni, M.: Diffusion wavelets. Appl. Comput. Harmon. Anal. 21, 53\u201394 (2006)","DOI":"10.1016\/j.acha.2006.04.004"},{"key":"9021_CR21","doi-asserted-by":"crossref","unstructured":"Coifman, R., Weiss, G.: Analyses Harmoniques Noncommutative sur Certains Espaces Homogenes. Springer, Berlin Heidelberg New York (1971)","DOI":"10.1007\/BFb0058946"},{"key":"9021_CR22","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1109\/TAP.2002.1011221","volume":"50","author":"R.J.-M. Cramer","year":"2002","unstructured":"Cramer, R.J.-M., Scholtz, R.A., Win, M.Z.: Evaluation of an ultra wide-band propagation channel. IEEE Trans. Antennas Propag. 50, 561\u2013569 (2002)","journal-title":"IEEE Trans. Antennas Propag."},{"key":"9021_CR23","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970104","volume-title":"Ten Lectures on Wavelets. CBMF Conference Series in Applied Mathematics, 61","author":"I. Daubechies","year":"1992","unstructured":"Daubechies, I.: Ten Lectures on Wavelets. CBMF Conference Series in Applied Mathematics, 61. SIAM, Philadelphia (1992)"},{"key":"9021_CR24","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s00365-004-0592-3","volume":"22","author":"M. Fornasier","year":"2005","unstructured":"Fornasier, M., Gr\u00f6chenig, K.: Intrinsic localization of frames. Constr. Approx. 22, 395\u2013415 (2005)","journal-title":"Constr. Approx."},{"key":"9021_CR25","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/s00041-005-4053-6","volume":"11","author":"M. Fornasier","year":"2005","unstructured":"Fornasier, M., Rauhut, H.: Continuous frames, function spaces, and the discretization problem. J. Fourier Anal. Appl. 11, 245\u2013287 (2005)","journal-title":"J. Fourier Anal. Appl."},{"key":"9021_CR26","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1007\/BF02510419","volume":"7","author":"L. Grafakos","year":"2001","unstructured":"Grafakos, L., Lennerd, C.: Characterization of $L_p(R_n)$ using Gabor frames. J. Fourier Anal. Appl. 7, 101\u2013126 (2001)","journal-title":"J. Fourier Anal. Appl."},{"key":"9021_CR27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01321715","volume":"112","author":"K. Gr\u00f6chenig","year":"1991","unstructured":"Gr\u00f6chenig, K.: Describing functions: atomic decompositions versus frames. Monatsh. Math. 112, 1\u201342 (1991)","journal-title":"Monatsh. Math."},{"key":"9021_CR28","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0003-1","volume-title":"Foundation of Time-frequency Analysis","author":"K. Gr\u00f6chenig","year":"2001","unstructured":"Gr\u00f6chenig, K.: Foundation of time-frequency analysis. Birkh\u00e4user, Boston (2001)"},{"key":"9021_CR29","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/s00041-004-8007-1","volume":"10","author":"K. Gr\u00f6chenig","year":"2004","unstructured":"Gr\u00f6chenig, K.: Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl. 10, 105\u2013132 (2004)","journal-title":"J. Fourier Anal. Appl."},{"key":"9021_CR30","unstructured":"Gr\u00f6chenig, K.: Time-frequency analysis of Sj\u00f6strand class. Rev. Mat. Iberoam. (To appear)"},{"key":"9021_CR31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0894-0347-03-00444-2","volume":"17","author":"K. Gr\u00f6chenig","year":"2003","unstructured":"Gr\u00f6chenig, K., Leinert, M.: Wiener's lemma for twisted convolution and Gabor frames. J. Am. Math. Soc. 17, 1\u201318 (2003)","journal-title":"J. Am. Math. Soc."},{"key":"9021_CR32","doi-asserted-by":"crossref","unstructured":"Gr\u00f6chenig, K., Leinert, M.: Symmetry of matrix algebras and symbolic calculus for infinite matrices. Trans. Am. Math. Soc. 358, 2695\u20132711 (2006)","DOI":"10.1090\/S0002-9947-06-03841-4"},{"key":"9021_CR33","unstructured":"Hao, Y., Marziliano, P., Vetterli, M., Blu, T.: Sampling and reconstruction of ECG as a signal with a finite rate of innovation. (Submitted to IEEE Trans. Biomed. Eng.)"},{"key":"9021_CR34","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1090\/S0273-0979-1985-15293-0","volume":"12","author":"J.R. Higgins","year":"1985","unstructured":"Higgins, J.R.: Five short stories about the cardinal series. Bull. Am. Math. Soc. 12, 45\u201389 (1985)","journal-title":"Bull. Am. Math. Soc."},{"key":"9021_CR35","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1016\/S0294-1449(16)30287-6","volume":"7","author":"S. Jaffard","year":"1990","unstructured":"Jaffard, S.: Properi\u00e9t\u00e9s des matrices bien localis\u00e9es pr\u00e9s de leur diagonale et quelques applications. Ann. Inst. Henri Poincar\u00e9 7, 461\u2013476 (1990)","journal-title":"Ann. Inst. Henri Poincar\u00e9"},{"key":"9021_CR36","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1007\/s00041-001-4017-4","volume":"1","author":"A.J.E.M. Janssen","year":"1995","unstructured":"Janssen, A.J.E.M.: Duality and biorthogonality for Weyl\u2013Heisenberg frames. J. Fourier Anal. Appl. 1, 403\u2013436 (1995)","journal-title":"J. Fourier Anal. Appl."},{"key":"9021_CR37","doi-asserted-by":"crossref","first-page":"1565","DOI":"10.1109\/PROC.1977.10771","volume":"65","author":"A. Jerri","year":"1977","unstructured":"Jerri, A.: The Shannon sampling theorem \u2013 Its various extensions and applications: A tutorial review. Proc. IEEE 65, 1565\u20131596 (1977)","journal-title":"Proc. IEEE"},{"key":"9021_CR38","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/BF02762276","volume":"103","author":"R.-Q. Jia","year":"1998","unstructured":"Jia, R.-Q.: Shift-invariant spaces and linear operator equations. Isr. J. Math. 103, 259\u2013288 (1998)","journal-title":"Isr. J. Math."},{"key":"9021_CR39","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/B978-0-12-438660-0.50036-4","volume-title":"Curves and Surfaces (Chamonix-Mont-Blanc, 1990)","author":"R.-Q. Jia","year":"1991","unstructured":"Jia, R.-Q., Micchelli, C.A.: Using the refinement equations for the construction of pre-wavelets. II. Powers of two. In: Curves and Surfaces (Chamonix-Mont-Blanc, 1990), pp. 209\u2013246. Academic, Boston, Massachusetts (1991)"},{"key":"9021_CR40","unstructured":"Kusuma, J., Maravic, I., Vetterli, M.: Sampling with finite rate of innovation: channel and timing estimatation for UWB and GPS. In: IEEE Conference on Communication 2003. Anchorage, AK"},{"key":"9021_CR41","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/0001-8708(79)90012-4","volume":"33","author":"R. Macias","year":"1979","unstructured":"Macias, R., Segovia, C.: Lipschitz functions on spaces of homogenous type. Adv. Math. 33, 257\u2013270 (1979)","journal-title":"Adv. Math."},{"key":"9021_CR42","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/0001-8708(79)90013-6","volume":"33","author":"R. Macias","year":"1979","unstructured":"Macias, R., Segovia, C.: A decomposition into atoms of distribution on spaces of homogenous type. Adv. Math. 33, 271\u2013309 (1979)","journal-title":"Adv. Math."},{"key":"9021_CR43","doi-asserted-by":"crossref","unstructured":"Mallat, S.G.: A Wavelet Tour of Signal Processing. Academic (1999)","DOI":"10.1016\/B978-012466606-1\/50008-8"},{"key":"9021_CR44","doi-asserted-by":"crossref","first-page":"2788","DOI":"10.1109\/TSP.2005.850321","volume":"53","author":"I. Maravic","year":"2005","unstructured":"Maravic, I., Vetterli, M.: Sampling and reconstruction of signals with finite rate of innovation in the presence of noise. IEEE Trans. Signal Process. 53, 2788\u20132805 (2005)","journal-title":"IEEE Trans. Signal Process."},{"key":"9021_CR45","doi-asserted-by":"crossref","first-page":"3462","DOI":"10.1109\/78.887038","volume":"48","author":"P. Marziliano","year":"2000","unstructured":"Marziliano, P., Vetterli, M.: Reconstruction of irregular sampled discrete-time band-limited signals with unknown sampling locations. IEEE Trans. Signal Process. 48, 3462\u20133471 (2000)","journal-title":"IEEE Trans. Signal Process."},{"key":"9021_CR46","doi-asserted-by":"crossref","unstructured":"Marziliano, P., Vetterli, M., Blu, T.: Sampling and exact reconstruction of band-limited signals with shot noise. IEEE Trans. Information Theory. 5, 2230\u20132233 (2006)","DOI":"10.1109\/TIT.2006.872844"},{"key":"9021_CR47","unstructured":"Marziliano, P.: Sampling innovations. PhD thesis, Swiss Federal Institute of Technology (April 2001)"},{"key":"9021_CR48","volume-title":"Ondelettes et Operateurs","author":"Y. Meyer","year":"1990","unstructured":"Meyer, Y.: Ondelettes et Operateurs. Herman, Paris (1990)"},{"key":"9021_CR49","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/BF02392539","volume":"155","author":"A. Nagel","year":"1985","unstructured":"Nagel, A., Stein, E., Waigner, S.: Ball and metrics defined by vector fields I: Basic properties. Acta Math. 155, 103\u2013147 (1985)","journal-title":"Acta Math."},{"key":"9021_CR50","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jath.1994.1001","volume":"76","author":"G. Plonka","year":"1994","unstructured":"Plonka, G.: Periodic spline interpolation with shifted nodes. J. Approx. Theory 76, 1\u201320 (1994)","journal-title":"J. Approx. Theory"},{"key":"9021_CR51","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1215\/S0012-7094-97-08913-4","volume":"89","author":"A. Ron","year":"1997","unstructured":"Ron, A., Shen, Z.: Weyl\u2013Heisenberg frames and Riesz bases in $L_2(R^d)$ . Duke Math. J. 89, 237\u2013282 (1997)","journal-title":"Duke Math. J."},{"key":"9021_CR52","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00365-004-0563-8","volume":"22","author":"A. Ron","year":"2005","unstructured":"Ron, A., Shen, Z.: Generalized shift-invariant systems. Constr. Approx. 22, 1\u201345 (2005)","journal-title":"Constr. Approx."},{"key":"9021_CR53","volume-title":"Spline Functions: Basic Theory","author":"L.L. Schumaker","year":"1981","unstructured":"Schumaker, L.L.: Spline Functions: Basic Theory. Wiley & Sons, New York (1981)"},{"key":"9021_CR54","doi-asserted-by":"crossref","first-page":"185","DOI":"10.4310\/MRL.1994.v1.n2.a6","volume":"1","author":"J. Sj\u00f6strand","year":"1994","unstructured":"Sj\u00f6strand, J.: An algebra of pseudodifferential operators. Math. Res. Lett. 1, 185\u2013192 (1994)","journal-title":"Math. Res. Lett."},{"key":"9021_CR55","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1006\/jmaa.2001.7479","volume":"261","author":"Q. Sun","year":"2001","unstructured":"Sun, Q.: Stability of the shifts of global supported distributions. J. Math. Anal. Appl. 261, 113\u2013125 (2001)","journal-title":"J. Math. Anal. Appl."},{"key":"9021_CR56","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1016\/j.crma.2005.03.002","volume":"340","author":"Q. Sun","year":"2005","unstructured":"Sun, Q.: Wiener's lemma for infinite matrices with polynomial off-diagonal decay. C. R. Acad. Sci. Paris, Ser I 340, 567\u2013570 (2005)","journal-title":"C. R. Acad. Sci. Paris, Ser I"},{"key":"9021_CR57","doi-asserted-by":"crossref","unstructured":"Sun, Q.: Wiener's lemma for infinite matrices. Trans. Am. Math. Soc. (accepted)","DOI":"10.1090\/S0002-9947-07-04303-6"},{"key":"9021_CR58","unstructured":"Sun, Q.: Non-uniform average sampling and reconstruction of signals with finite rate of innovation. SIAM J. Math. Anal. (submitted)"},{"key":"9021_CR59","volume-title":"An Introduction to Multiband Wavelets","author":"Q. Sun","year":"2000","unstructured":"Sun, Q., Bi, N., Huang, D.: An Introduction to Multiband Wavelets. Zhejiang University Press, Hangzhou (2000)"},{"key":"9021_CR60","doi-asserted-by":"crossref","first-page":"1417","DOI":"10.1109\/TSP.2002.1003065","volume":"50","author":"M. Vetterli","year":"2002","unstructured":"Vetterli, M., Marziliano, P., Blu, T.: Sampling signals with finite rate of innovation. IEEE Trans. Signal Process. 50, 1417\u20131428 (2002)","journal-title":"IEEE Trans. Signal Process."},{"key":"9021_CR61","volume-title":"An Introduction to Nonharmonic Fourier Series","author":"R.M. Young","year":"1980","unstructured":"Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic, New York (1980)"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-006-9021-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-006-9021-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-006-9021-4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T17:08:09Z","timestamp":1559149689000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-006-9021-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,10,18]]},"references-count":61,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2008,5]]}},"alternative-id":["9021"],"URL":"https:\/\/doi.org\/10.1007\/s10444-006-9021-4","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,10,18]]}}}