{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T14:16:41Z","timestamp":1773152201705,"version":"3.50.1"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2015,6,24]],"date-time":"2015-06-24T00:00:00Z","timestamp":1435104000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"Portuguese Government through the FCT","award":["SFRH\/BPD\/101139\/2014"],"award-info":[{"award-number":["SFRH\/BPD\/101139\/2014"]}]},{"name":"Brazilian Government through the CNPq","award":["470019\/2013-1"],"award-info":[{"award-number":["470019\/2013-1"]}]},{"name":"Direcci\u00f3n General de Investigaci\u00f3n Cient\u00edfica y T\u00e9cnica, Ministerio de Econom\u00eda y Competitividad of Spain","award":["MTM2012\u201336732\u2013C03\u201301"],"award-info":[{"award-number":["MTM2012\u201336732\u2013C03\u201301"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2016,3]]},"DOI":"10.1007\/s11075-015-0017-3","type":"journal-article","created":{"date-parts":[[2015,6,22]],"date-time":"2015-06-22T21:56:12Z","timestamp":1435010172000},"page":"699-714","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Zeros of para\u2013orthogonal polynomials and linear spectral transformations on the unit circle"],"prefix":"10.1007","volume":"71","author":[{"given":"K.","family":"Castillo","sequence":"first","affiliation":[]},{"given":"F.","family":"Marcell\u00e1n","sequence":"additional","affiliation":[]},{"given":"M. N.","family":"Rebocho","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2015,6,24]]},"reference":[{"key":"17_CR1","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.jat.2005.09.001","volume":"137","author":"AF Beardon","year":"2005","unstructured":"Beardon, A.F., Driver, K.A.: The zeros of linear combinations of orthogonal polynomials. J. Approx. Theory 137, 179\u2013186 (2005)","journal-title":"J. Approx. Theory"},{"key":"17_CR2","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.apnum.2003.10.001","volume":"48","author":"C Brezinski","year":"2004","unstructured":"Brezinski, C., Driver, K.A., Redivo-Zaglia, M.: Quasi-orthogonality with applications to some families of classical orthogonal polynomials. Appl. Numer. Math. 48, 157\u2013168 (2004)","journal-title":"Appl. Numer. Math."},{"key":"17_CR3","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1090\/S0002-9939-1952-0047828-8","volume":"3","author":"FF Bonsall","year":"1952","unstructured":"Bonsall, F.F., Marden, M.: Zeros of self-inverse polynomials. Proc. Amer. Math. Soc. 3, 471\u2013475 (1952)","journal-title":"Proc. Amer. Math. Soc."},{"key":"17_CR4","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1137\/S089547989223050X","volume":"16","author":"A Bunse-Gerstner","year":"1995","unstructured":"Bunse-Gerstner, A., He, C.: On a Sturm sequence of polynomials for unitary Hessenberg matrices. SIAM J. Matrix Anal. Appl. 16, 1043\u20131055 (1995)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"17_CR5","doi-asserted-by":"crossref","unstructured":"Cachafeiro, A., Marcell\u00e1n, F.: Orthogonal polynomials and jump modifications. In: 1986 Lecture Notes in Math. 1329, Orthogonal Polynomials and their Applications, Segovia, pp. 236\u2013240. Springer, Berlin (1988)","DOI":"10.1007\/BFb0083362"},{"key":"17_CR6","doi-asserted-by":"crossref","unstructured":"Cachafeiro, A., Marcell\u00e1n, F.: Asymptotics for the ratio of the leading coefficients of orthogonal polynomials associated with a jump modification. In: Lecture Notes in Math. 1354, Approximation and Optimization, Havana, 1987, pp. 111\u2013117. Springer, Berlin (1988)","DOI":"10.1007\/BFb0089587"},{"key":"17_CR7","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1216\/rmjm\/1181072574","volume":"23","author":"A Cachafeiro","year":"1993","unstructured":"Cachafeiro, A., Marcell\u00e1n, F.: Modifications of Toeplitz matrices: Jump functions. Rocky Mountain J. Math. 23, 521\u2013531 (1993)","journal-title":"Rocky Mountain J. Math."},{"key":"17_CR8","first-page":"447","volume":"8","author":"MJ Cantero","year":"2002","unstructured":"Cantero, M.J., Moral, L., Vel\u00e1zquez, L.: Measures and para\u2013orthogonal polynomials on the unit circle. East J. Approx. 8, 447\u2013464 (2002)","journal-title":"East J. Approx."},{"key":"17_CR9","unstructured":"Castillo, K., Cruz\u2013Barroso, R., Perdomo\u2013P\u00edo, F.: On a spectral theorem in the para-orthogonality theory. Submitted"},{"key":"17_CR10","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1090\/S0025-5718-2012-02655-X","volume":"82","author":"K Castillo","year":"2013","unstructured":"Castillo, K., Marcell\u00e1n, F.: Generators of rational Spectral Transformations for non-trivial \ud835\udc36 $\\mathcal {C}$ -functions. Math. Comp. 82, 1057\u20131068 (2013)","journal-title":"Math. Comp."},{"key":"17_CR11","doi-asserted-by":"crossref","first-page":"1663","DOI":"10.1016\/j.apnum.2012.05.006","volume":"62","author":"K Castillo","year":"2012","unstructured":"Castillo, K., Mello, M.V., Rafaeli, F.R.: Monotonicity and asymptotics of zeros of Sobolev type orthogonal polynomials: A general case. Appl. Numer. Math. 62, 1663\u20131671 (2012)","journal-title":"Appl. Numer. Math."},{"key":"17_CR12","volume-title":"An Introduction to Orthogonal Polynomials","author":"TS Chihara","year":"1978","unstructured":"Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978)"},{"key":"17_CR13","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1515\/crll.1858.55.61","volume":"55","author":"EB Christoffel","year":"1858","unstructured":"Christoffel, E.B.: \u00dcber die Gaussische Quadratur und eine Verallgemeinerung derselben. J. Reine Angew. Math. 55, 61\u201382 (1858)","journal-title":"J. Reine Angew. Math."},{"issue":"3","key":"17_CR14","first-page":"5","volume":"4","author":"G Darboux","year":"1878","unstructured":"Darboux, G.: M\u00e9moire sur l\u2019approximation des fonctions de tr\u00e8s-grands nombres, et sur une classe \u00e9tendue de d\u00e9veloppements en s\u00e9rie. Liouville J. 4(3), 5\u201356; 377\u2013416 (1878)","journal-title":"Liouville J."},{"key":"17_CR15","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/j.cam.2006.02.041","volume":"202","author":"L Daruis","year":"2007","unstructured":"Daruis, L., Hern\u00e1ndez, J., Marcell\u00e1n, F.: Spectral transformations for Hermitian Toeplitz matrices. J. Comput. Appl. Math. 202, 155\u2013176 (2007)","journal-title":"J. Comput. Appl. Math."},{"key":"17_CR16","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1216\/rmjm\/1181069970","volume":"33","author":"L Daruis","year":"2003","unstructured":"Daruis, L., Nj\u00e5stad, O., Van Assche, W.: Para-orthogonal polynomials in frequency analysis. Rocky Mountain. J. Math. 33, 629\u2013645 (2003)","journal-title":"Rocky Mountain. J. Math."},{"key":"17_CR17","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1109\/TASSP.1986.1164830","volume":"34","author":"P Delsarte","year":"1986","unstructured":"Delsarte, P., Genin, Y.: The split Levinson algorithm. IEEE Trans. Acoust. Speech Signal Process 34, 470\u2013478 (1986)","journal-title":"IEEE Trans. Acoust. Speech Signal Process"},{"key":"17_CR18","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1137\/0519050","volume":"19","author":"P Delsarte","year":"1988","unstructured":"Delsarte, P., Genin, Y.: The tridiagonal approach to Szego\u2019s orthogonal polynomials, Toeplitz linear systems, and related interpolation problems. SIAM J. Math. Anal. 19, 718\u2013735 (1988)","journal-title":"SIAM J. Math. Anal."},{"key":"17_CR19","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1137\/0612018","volume":"12","author":"P Delsarte","year":"1991","unstructured":"Delsarte, P., Genin, Y.: Tridiagonal approach to the algebraic environment of Toeplitz matrices, Part I: Basic results. SIAM J. Matrix Anal. Appl. 12, 220\u2013238 (1991)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"17_CR20","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1137\/0612031","volume":"12","author":"P Delsarte","year":"1991","unstructured":"Delsarte, P., Genin, Y.: Tridiagonal approach to the algebraic environment of Toeplitz matrices, Part II: Zeros and eigenvalues problems. SIAM J. Matrix Anal. Appl. 12, 432\u2013448 (1991)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"17_CR21","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1007\/s00211-007-0100-3","volume":"107","author":"K Driver","year":"2007","unstructured":"Driver, K., Jordaan, K.: Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials. Numer. Math. 107, 615\u2013624 (2007)","journal-title":"Numer. Math."},{"key":"17_CR22","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/0024-3795(93)90469-5","volume":"188\/189","author":"L Elsner","year":"1993","unstructured":"Elsner, L., He, C.: Perturbation and interlace theorems for the unitary eigenvalue problem. Lineal Algebra Appl. 188\/189, 207\u2013230 (1993)","journal-title":"Lineal Algebra Appl."},{"key":"17_CR23","first-page":"1","volume":"104","author":"YaL Geronimus","year":"1954","unstructured":"Geronimus, Ya. L.: Polynomials orthogonal on a circle and their applications. Amer. Math. Soc. Translation 104, 1\u201379 (1954). Translation of the Russian original 1948","journal-title":"Amer. Math. Soc. Translation"},{"key":"17_CR24","doi-asserted-by":"crossref","first-page":"742","DOI":"10.2307\/1969232","volume":"47","author":"J(akaYaL) Geronimus","year":"1946","unstructured":"Geronimus, J. (aka Ya. L.): On the trigonometric moment problem. Ann. Math. 47, 742\u2013761 (1946)","journal-title":"Ann. Math."},{"key":"17_CR25","unstructured":"Geronimus, Ya. L.: Orthogonal polynomials: Estimates, asymptotic formulas, and series of polynomials orthogonal on the unit circle and on an interval. Authorized translation from the Russian Consultants Bureau, New York (1961)"},{"key":"17_CR26","first-page":"1","volume":"5","author":"E Godoy","year":"1991","unstructured":"Godoy, E., Marcell\u00e1n, F.: An analogue of the Christoffel formula for polynomial modification of a measure on the unit circle. Boll. Un. Mat. Ital. A 5, 1\u201312 (1991)","journal-title":"Boll. Un. Mat. Ital. A"},{"key":"17_CR27","doi-asserted-by":"crossref","first-page":"930","DOI":"10.4153\/CJM-1993-052-1","volume":"45","author":"E Godoy","year":"1993","unstructured":"Godoy, E., Marcell\u00e1n, F.: Orthogonal polynomials and rational modifications of measures. Canad. J. Math. 45, 930\u2013943 (1993)","journal-title":"Canad. J. Math."},{"key":"17_CR28","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1023\/A:1019765002077","volume":"96","author":"L Golinskii","year":"2002","unstructured":"Golinskii, L.: Quadrature formula and zeros of para-orthogonal polynomials on the unit circle. Acta Math. Hungar. 96, 169\u2013186 (2002)","journal-title":"Acta Math. Hungar."},{"key":"17_CR29","doi-asserted-by":"crossref","first-page":"782","DOI":"10.1137\/S0895479803438201","volume":"26","author":"S Helsen","year":"2005","unstructured":"Helsen, S., Kuijlaars, A.B.J., Van Barel, M.: Convergence of the isometric Arnoldi process. SIAM J. Matrix Anal. Appl. 26, 782\u2013809 (2005)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"17_CR30","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/j.cam.2014.02.017","volume":"267","author":"M Humet","year":"2014","unstructured":"Humet, M., Van Barel, M.: Algorithms for the Geronimus transformation for orthogonal polynomials on the unit circle. J. Comput. Appl. Math. 267, 195\u2013217 (2014)","journal-title":"J. Comput. Appl. Math."},{"key":"17_CR31","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1112\/blms\/21.2.113","volume":"21","author":"WB Jones","year":"1989","unstructured":"Jones, W.B., Nj\u00e5stad, O., Thron, W.J.: Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle. Bull. London Math. Soc. 21, 113\u2013152 (1989)","journal-title":"Bull. London Math. Soc."},{"key":"17_CR32","first-page":"9","volume":"7","author":"X Li","year":"1999","unstructured":"Li, X., Marcell\u00e1n, F.: Representation of orthogonal polynomials for modified measures. Comm. Anal. Theory Cont. Fract. 7, 9\u201322 (1999)","journal-title":"Comm. Anal. Theory Cont. Fract."},{"issue":"VII","key":"17_CR33","first-page":"455","volume":"7","author":"F Marcell\u00e1n","year":"1992","unstructured":"Marcell\u00e1n, F., P\u00e9rez, T.E., Pi\u00f1ar, M.A.: On zeros of Sobolev-type orthogonal polynomials. Rend. Mat. 7(VII), 455\u2013473 (1992)","journal-title":"Rend. Mat."},{"key":"17_CR34","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/S0096-3003(01)00079-0","volume":"128","author":"F Marcell\u00e1n","year":"2002","unstructured":"Marcell\u00e1n, F., Moral, L.: Sobolev-type orthogonal polynomials on the unit circle. Appl. Math. Comput. 128, 329\u2013363 (2002)","journal-title":"Appl. Math. Comput."},{"key":"17_CR35","doi-asserted-by":"crossref","unstructured":"Nevai, P.: Orthogonal polynomials. Mem. Amer. Math. Soc. 18(213) (1979)","DOI":"10.1090\/memo\/0213"},{"key":"17_CR36","volume-title":"Analytic theory of polynomials, London Mathematical Society Monographs. New Series, vol. 26","author":"QI Rahman","year":"2002","unstructured":"Rahman, Q.I., Schmeisser, G.: Analytic theory of polynomials, London Mathematical Society Monographs. New Series, vol. 26. The Clarendon Press, Oxford University Press, Oxford (2002)"},{"key":"17_CR37","unstructured":"Obrechkoff, N.: Zeros of polynomials, Bulgarian Academic Monographs (7), Marin Drinov Academic Publishing House, Sofia, 2003. (English translation of the original text, published in Bulgarian in 1963)"},{"key":"17_CR38","doi-asserted-by":"crossref","first-page":"1685","DOI":"10.1090\/S0025-5718-2011-02414-2","volume":"80","author":"F Peherstorfer","year":"2011","unstructured":"Peherstorfer, F.: Positive trigonometric quadrature formulas and quadrature on the unit circle. Math. Comp. 80, 1685\u20131701 (2011)","journal-title":"Math. Comp."},{"key":"17_CR39","first-page":"465","volume":"30","author":"J Von Neumann","year":"1929","unstructured":"Von Neumann, J., Wigner, E.: \u00dcber merkw\u00fcrdige diskrete Eigenwerte. Phys. Z. 30, 465\u2013467 (1929)","journal-title":"Phys. Z."},{"key":"17_CR40","unstructured":"Simon, B.: Orthogonal polynomials on the unit circle, 2 vols., Amer. Math. Soc. Coll. Publ. Series, vol. 54, Amer. Math. Soc., Providence, R. I. (2005)"},{"key":"17_CR41","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1016\/j.jmaa.2006.06.076","volume":"329","author":"B Simon","year":"2007","unstructured":"Simon, B.: Rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle. J. Math. Anal. Appl. 329, 376\u2013382 (2007)","journal-title":"J. Math. Anal. Appl."},{"key":"17_CR42","volume-title":"Szego\u2019s, theorem and its descendants: Spectral theory for L2 perturbations of orthogonal polynomials","author":"B Simon","year":"2011","unstructured":"Simon, B.: Szego\u2019s, theorem and its descendants: Spectral theory for L 2 perturbations of orthogonal polynomials. Princeton University Press, Princeton and Oxford (2011)"},{"key":"17_CR43","unstructured":"Szego, G.: Orthogonal polynomials, Amer. Math. Soc. Coll. Publ., Vol. 23, Amer. Math. Soc., Providence, R. I., 1939; 4t h edition (1975)"},{"key":"17_CR44","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1016\/j.jat.2006.12.007","volume":"146","author":"ML Wong","year":"2007","unstructured":"Wong, M.L.: First and second kind paraorthogonal polynomials and their zeros. J. Approx. Theory 146, 282\u2013293 (2007)","journal-title":"J. Approx. Theory"},{"key":"17_CR45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00365-008-9024-0","volume":"30","author":"ML Wong","year":"2009","unstructured":"Wong, M.L.: Generalized bounded variation and inserting point masses. Constr. Approx. 30, 1\u201315 (2009)","journal-title":"Constr. Approx."},{"key":"17_CR46","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/S0377-0427(97)00130-1","volume":"85","author":"A Zhedanov","year":"1997","unstructured":"Zhedanov, A.: Rational spectral transformations and orthogonal polynomials. J. Comput. Appl. Math. 85, 67\u201386 (1997)","journal-title":"J. Comput. Appl. Math."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-015-0017-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-015-0017-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-015-0017-3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,27]],"date-time":"2019-08-27T11:21:00Z","timestamp":1566904860000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-015-0017-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,24]]},"references-count":46,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2016,3]]}},"alternative-id":["17"],"URL":"https:\/\/doi.org\/10.1007\/s11075-015-0017-3","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,6,24]]}}}