Elements of Pólya-Schur theory in the finite difference setting
HTML articles powered by AMS MathViewer
- by Petter Brändén, Ilia Krasikov and Boris Shapiro
- Proc. Amer. Math. Soc. 144 (2016), 4831-4843
- DOI: https://doi.org/10.1090/proc/13115
- Published electronically: April 19, 2016
- PDF | Request permission
Abstract:
The Pólya-Schur theory describes the class of hyperbolicity preservers, i.e., the class of linear operators acting on univariate polynomials and preserving real-rootedness. We attempt to develop an analog of Pólya-Schur theory in the setting of linear finite difference operators. We study the class of linear finite difference operators preserving the set of real-rooted polynomials whose mesh (i.e., the minimal distance between the roots) is at least one. In particular, we prove a finite difference version of the classical Hermite-Poulain theorem and several results about discrete multiplier sequences.References
- Julius Borcea and Petter Brändén, Pólya-Schur master theorems for circular domains and their boundaries, Ann. of Math. (2) 170 (2009), no. 1, 465–492. MR 2521123, DOI 10.4007/annals.2009.170.465
- J. Borcea and P. Brändén, Multivariate Pólya-Schur classification problems in the Weyl algebra, Proc. Lond. Math. Soc. (3) 101 (2010), no. 1, 73–104. MR 2661242, DOI 10.1112/plms/pdp049
- Francesco Brenti, Unimodal, log-concave and Pólya frequency sequences in combinatorics, Mem. Amer. Math. Soc. 81 (1989), no. 413, viii+106. MR 963833, DOI 10.1090/memo/0413
- David A. Cardon, Extended Laguerre inequalities and a criterion for real zeros, Progress in analysis and its applications, World Sci. Publ., Hackensack, NJ, 2010, pp. 143–149. MR 2758000, DOI 10.1142/9789814313179_{0}019
- Thomas Craven and George Csordas, Problems and theorems in the theory of multiplier sequences, Serdica Math. J. 22 (1996), no. 4, 515–524. MR 1483603
- Thomas Craven and George Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977), no. 4, 801–817. MR 568321
- Thomas Craven and George Csordas, Composition theorems, multiplier sequences and complex zero decreasing sequences, Value distribution theory and related topics, Adv. Complex Anal. Appl., vol. 3, Kluwer Acad. Publ., Boston, MA, 2004, pp. 131–166. MR 2173299, DOI 10.1007/1-4020-7951-6_{6}
- Matthew Chasse and George Csordas, Discrete analogues of the Laguerre inequalities and a conjecture of I. Krasikov, Ann. Sci. Math. Québec 36 (2012), no. 1, 77–91 (2013) (English, with English and French summaries). MR 3113292
- George Csordas and Richard S. Varga, Necessary and sufficient conditions and the Riemann hypothesis, Adv. in Appl. Math. 11 (1990), no. 3, 328–357. MR 1061423, DOI 10.1016/0196-8858(90)90013-O
- David W. Farmer and Robert C. Rhoades, Differentiation evens out zero spacings, Trans. Amer. Math. Soc. 357 (2005), no. 9, 3789–3811. MR 2146650, DOI 10.1090/S0002-9947-05-03721-9
- S. Fisk, Polynomials, roots, and interlacing, arXiv:math/0612833v2.
- William H. Foster and Ilia Krasikov, Inequalities for real-root polynomials and entire functions, Adv. in Appl. Math. 29 (2002), no. 1, 102–114. MR 1921546, DOI 10.1016/S0196-8858(02)00005-2
- A. O. Gel′fond, Calculus of finite differences, International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, 1971. Translated from the Russian. MR 0342890
- Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR 2191786, DOI 10.1017/CBO9781107325982
- Olga Katkova, Boris Shapiro, and Anna Vishnyakova, Multiplier sequences and logarithmic mesh, C. R. Math. Acad. Sci. Paris 349 (2011), no. 1-2, 35–38 (English, with English and French summaries). MR 2755692, DOI 10.1016/j.crma.2010.11.031
- Ilia Krasikov, Discrete analogues of the Laguerre inequality, Anal. Appl. (Singap.) 1 (2003), no. 2, 189–197. MR 1976615, DOI 10.1142/S0219530503000120
- Nikola Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963 (German). MR 0164003
- Merrell L. Patrick, Some inequalities concerning Jacobi polynomials, SIAM J. Math. Anal. 2 (1971), 213–220. MR 288330, DOI 10.1137/0502018
- Merrell L. Patrick, Extensions of inequalities of the Laguerre and Turán type, Pacific J. Math. 44 (1973), 675–682. MR 315176, DOI 10.2140/pjm.1973.44.675
- J. Schur and G. Pólya, Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math. 144 (1914), 89–113 (German). MR 1580897, DOI 10.1515/crll.1914.144.89
- Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, London Mathematical Society Monographs. New Series, vol. 26, The Clarendon Press, Oxford University Press, Oxford, 2002. MR 1954841
- A. Stoyanoff, Sur un theoreme de M Marcel Riesz, Nouvelles Annales de Mathematique 1 (1926), 97–99.
- G. Szegö, Bemerkungen zu einem Satz von J. H. Grace über die Wurzeln algebraischer Gleichungen, Math. Z. 13 (1922), no. 1, 28–55 (German). MR 1544526, DOI 10.1007/BF01485280
- Gyula Sz.-Nagy, Über Polynome mit lauter reelen Nullstellen, Acta Math. Acad. Sci. Hungar. 1 (1950), 225–228 (German, with Russian summary). MR 46479, DOI 10.1007/BF02021314
- David G. Wagner, Total positivity of Hadamard products, J. Math. Anal. Appl. 163 (1992), no. 2, 459–483. MR 1145841, DOI 10.1016/0022-247X(92)90261-B
- David G. Wagner, Zeros of reliability polynomials and $f$-vectors of matroids, Combin. Probab. Comput. 9 (2000), no. 2, 167–190. MR 1762787, DOI 10.1017/S0963548399004162
- Peter Walker, Separation of the zeros of polynomials, Amer. Math. Monthly 100 (1993), no. 3, 272–273. MR 1212834, DOI 10.2307/2324460
- Peter Walker, Bounds for the separation of real zeros of polynomials, J. Austral. Math. Soc. Ser. A 59 (1995), no. 3, 330–342. MR 1355224, DOI 10.1017/S144678870003723X
Bibliographic Information
- Petter Brändén
- Affiliation: Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
- MR Author ID: 721471
- Email: pbranden@kth.se
- Ilia Krasikov
- Affiliation: Department of Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, United Kingdom
- MR Author ID: 218954
- Email: mastiik@brunel.ac.uk
- Boris Shapiro
- Affiliation: Department of Mathematics, Stockholm University, S-10691, Stockholm, Sweden
- MR Author ID: 212628
- Email: shapiro@math.su.se
- Received by editor(s): October 21, 2014
- Received by editor(s) in revised form: September 30, 2015, and January 15, 2016
- Published electronically: April 19, 2016
- Communicated by: Walter Van Assche
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4831-4843
- MSC (2010): Primary 26C10; Secondary 30C15
- DOI: https://doi.org/10.1090/proc/13115
- MathSciNet review: 3544533