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February  2020, 25(2): 529-544. doi: 10.3934/dcdsb.2019252

A Favard type theorem for Hurwitz polynomials

Abdon E. Choque-Rivero 1,, and  Iván Area 2,

1. 

Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Ciudad Universitaria, C.P. 58048, Morelia, Michoacán, México
2. 

Departamento de Matemática Aplicada Ⅱ, E.E. Aeronáutica e do Espazo, Universidade de Vigo, 32004-Ourense, Spain

* Corresponding author: Abdon E. Choque-Rivero

Dedicated to Prof. Juan J. Nieto on the occasion of his 60th birthday

Received  February 2019 Revised  April 2019 Published  November 2019

A Favard type theorem for Hurwitz polynomials is proposed. This result is a sufficient condition for a sequence of polynomials of increasing degree to be a sequence of Hurwitz polynomials. As in the Favard celebrated theorem, the three-term recurrence relation is used. Some examples of Hurwitz sequences are also presented. Additionally, a characterization of constructing a family of orthogonal polynomials on $ [0, \infty) $ by two couples of numerical sequences $ ({A_{1, j}, B_{1, j}}) $ and $ ({A_{2, j}, B_{2, j}}) $ is stated.

Keywords: Hurwitz polynomials, orthogonal polynomials, Markov moments, three-term recurrence relation coefficients.
Mathematics Subject Classification: Primary: 12D10, 26C10, 33C45, 34K20, 39A30.
Citation: Abdon E. Choque-Rivero, Iván Area. A Favard type theorem for Hurwitz polynomials. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 529-544. doi: 10.3934/dcdsb.2019252
References:
[1]

M. Abramowitz and I. A. Stegun, Stein, Josef Table errata: Handbook of mathematical functions with formulas, graphs, and mathematical tables, Math. Comp., 24 (1970), 503.  Google Scholar

[2]

P. Batra, Componentwise products of totally non-negative matrices generated by functions in the Laguerre-Pólya class, Applied and computational matrix analysis, Springer Proc. Math. Stat., Springer, Cham, 192 (2017), 151–163.  Google Scholar

[3] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York-London, 1963.   Google Scholar
[4]

T. S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and its Applications, Vol. 13. Gordon and Breach Science Publishers, New York-London-Paris, 1978.  Google Scholar

[5]

A. E. Choque-Rivero, On Dyukarev's resolvent matrix for a truncated Stieltjes matrix moment problem under the view of orthogonal matrix polynomials, Linear Algebra Appl., 474 (2015), 44-109.  doi: 10.1016/j.laa.2015.01.027.  Google Scholar

[6]

A. E. Choque-Rivero, On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials, Linear Algebra Appl., 476 (2015), 56-84.  doi: 10.1016/j.laa.2015.03.001.  Google Scholar

[7]

A. E. Choque-Rivero, The Kharitonov theorem and robust stabilization via orthogonal polynomials, Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh., 86 (2017), 49-78.   Google Scholar

[8]

A. E. Choque-Rivero, Hurwitz polynomials and orthogonal polynomials generated by Routh-Markov parameters, Mediterr. J. Math., 15 (2018), Art. 40, 15 pp. doi: 10.1007/s00009-018-1083-2.  Google Scholar

[9]

B. N. Datta, Application of Hankel matrices of Markov parameters to the solutions of the Routh-Hurwitz and the Schur-Cohn problems, J. Math. Anal. Appl., 68 (1979), 276-290.  doi: 10.1016/0022-247X(79)90115-X.  Google Scholar

[10]

H. Dette and W. J. Studden, Matrix measures, moment spaces and Favard's theorem for the interval [0, 1] and [0, ∞), Linear Algebra Appl., 345 (2002), 169-193.  doi: 10.1016/S0024-3795(01)00493-1.  Google Scholar

[11]

Y. M. Dyukarev, Indeterminacy criteria for the Stieltjes matrix moment problem, Math. Notes, 75 (2004), 66-82.  doi: 10.1023/B:MATN.0000015022.02925.bd.  Google Scholar

[12]

J. Favard, Sur les polynômes de Tchebicheff, C. R. Acad. Sci. Paris, 200 (1935), 2052-2053.   Google Scholar

[13]

F. R. Gantmacher, The Theory of Matrices. Vols. 1, 2, Chelsea Publishing Co., New York, 1959.  Google Scholar

[14]

Y. Genin, Hurwitz sequences of polynomials, Philips Research Reports, 30 (1975), 89-102.   Google Scholar

[15]

D. Gómez-Ullate, N. Kamran and R. Milson, An extended class of orthogonal polynomials defined by a Sturm-Liouville problem, J. Math. Anal. Appl., 359 (2009), 352-367.  doi: 10.1016/j.jmaa.2009.05.052.  Google Scholar

[16]

D. Gómez-Ullate, N. Kamran and R. Milson, An extension of Bochner's problem: Exceptional invariant subspaces, J. Approx. Theory, 162 (2010), 987-1006.  doi: 10.1016/j.jat.2009.11.002.  Google Scholar

[17]

N. Guglielmi and E. Hairer, Order stars and stability for delay differential equations, Numerische Mathematik, 83 (1999), 371-383.  doi: 10.1007/s002110050454.  Google Scholar

[18]

J. Hale, Theory of Functional Differential Equations, Second edition, Applied Mathematical Sciences, Vol. 3. Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[19]

C. V. Hollot, Kharitonov-like results in the space of Markov parameters, IEEE Transactions on Automatic Control, 34 (1989), 536-538.  doi: 10.1109/9.24206.  Google Scholar

[20]

O. Holtz and M. Tyaglov, Structured matrices, continued fractions, and root localization of polynomials, SIAM Rev., 54 (2012), 421-509.  doi: 10.1137/090781127.  Google Scholar

[21]

A. Hurwitz, Uber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Mathematische Annalen, 46 (1895), 273-284.  doi: 10.1007/BF01446812.  Google Scholar

[22]

V. Katsnelson, Stieltjes functions and Hurwitz stable entire functions, Complex Anal. Oper. Theory, 5 (2011), 611-630.  doi: 10.1007/s11785-011-0146-1.  Google Scholar

[23]

M. G. Kreǐn and M. A. Naimark, The method of symmetric and Hermitian forms int the theory of the separation of the roots of algebraic equations, Linear and Multilinear Algebra, 10 (1981), 265-308.  doi: 10.1080/03081088108817420.  Google Scholar

[24]

M. G. Kreǐn and A. A. Nudel'man, The Markov Moment Problem and Extremal Problems, Mathematical Monographs, Vol. 50. American Mathematical Society, Providence, R.I., 1977.  Google Scholar

[25]

M. M. Postnikov, Stable Polynomials, Nauka, Moscow, 1981,176 pp.  Google Scholar

[26]

M. Prevost and T. Rivoal, Remainder Padé approximants for the exponential function, Constructive Approximation, 25 (2007), 109-123.  doi: 10.1007/s00365-006-0635-6.  Google Scholar

[27]

G. Szegö, Orthogonal Polynomials, Fourth edition, American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I., 1975.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Stein, Josef Table errata: Handbook of mathematical functions with formulas, graphs, and mathematical tables, Math. Comp., 24 (1970), 503.  Google Scholar

[2]

P. Batra, Componentwise products of totally non-negative matrices generated by functions in the Laguerre-Pólya class, Applied and computational matrix analysis, Springer Proc. Math. Stat., Springer, Cham, 192 (2017), 151–163.  Google Scholar

[3] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York-London, 1963.   Google Scholar
[4]

T. S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and its Applications, Vol. 13. Gordon and Breach Science Publishers, New York-London-Paris, 1978.  Google Scholar

[5]

A. E. Choque-Rivero, On Dyukarev's resolvent matrix for a truncated Stieltjes matrix moment problem under the view of orthogonal matrix polynomials, Linear Algebra Appl., 474 (2015), 44-109.  doi: 10.1016/j.laa.2015.01.027.  Google Scholar

[6]

A. E. Choque-Rivero, On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials, Linear Algebra Appl., 476 (2015), 56-84.  doi: 10.1016/j.laa.2015.03.001.  Google Scholar

[7]

A. E. Choque-Rivero, The Kharitonov theorem and robust stabilization via orthogonal polynomials, Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh., 86 (2017), 49-78.   Google Scholar

[8]

A. E. Choque-Rivero, Hurwitz polynomials and orthogonal polynomials generated by Routh-Markov parameters, Mediterr. J. Math., 15 (2018), Art. 40, 15 pp. doi: 10.1007/s00009-018-1083-2.  Google Scholar

[9]

B. N. Datta, Application of Hankel matrices of Markov parameters to the solutions of the Routh-Hurwitz and the Schur-Cohn problems, J. Math. Anal. Appl., 68 (1979), 276-290.  doi: 10.1016/0022-247X(79)90115-X.  Google Scholar

[10]

H. Dette and W. J. Studden, Matrix measures, moment spaces and Favard's theorem for the interval [0, 1] and [0, ∞), Linear Algebra Appl., 345 (2002), 169-193.  doi: 10.1016/S0024-3795(01)00493-1.  Google Scholar

[11]

Y. M. Dyukarev, Indeterminacy criteria for the Stieltjes matrix moment problem, Math. Notes, 75 (2004), 66-82.  doi: 10.1023/B:MATN.0000015022.02925.bd.  Google Scholar

[12]

J. Favard, Sur les polynômes de Tchebicheff, C. R. Acad. Sci. Paris, 200 (1935), 2052-2053.   Google Scholar

[13]

F. R. Gantmacher, The Theory of Matrices. Vols. 1, 2, Chelsea Publishing Co., New York, 1959.  Google Scholar

[14]

Y. Genin, Hurwitz sequences of polynomials, Philips Research Reports, 30 (1975), 89-102.   Google Scholar

[15]

D. Gómez-Ullate, N. Kamran and R. Milson, An extended class of orthogonal polynomials defined by a Sturm-Liouville problem, J. Math. Anal. Appl., 359 (2009), 352-367.  doi: 10.1016/j.jmaa.2009.05.052.  Google Scholar

[16]

D. Gómez-Ullate, N. Kamran and R. Milson, An extension of Bochner's problem: Exceptional invariant subspaces, J. Approx. Theory, 162 (2010), 987-1006.  doi: 10.1016/j.jat.2009.11.002.  Google Scholar

[17]

N. Guglielmi and E. Hairer, Order stars and stability for delay differential equations, Numerische Mathematik, 83 (1999), 371-383.  doi: 10.1007/s002110050454.  Google Scholar

[18]

J. Hale, Theory of Functional Differential Equations, Second edition, Applied Mathematical Sciences, Vol. 3. Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[19]

C. V. Hollot, Kharitonov-like results in the space of Markov parameters, IEEE Transactions on Automatic Control, 34 (1989), 536-538.  doi: 10.1109/9.24206.  Google Scholar

[20]

O. Holtz and M. Tyaglov, Structured matrices, continued fractions, and root localization of polynomials, SIAM Rev., 54 (2012), 421-509.  doi: 10.1137/090781127.  Google Scholar

[21]

A. Hurwitz, Uber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Mathematische Annalen, 46 (1895), 273-284.  doi: 10.1007/BF01446812.  Google Scholar

[22]

V. Katsnelson, Stieltjes functions and Hurwitz stable entire functions, Complex Anal. Oper. Theory, 5 (2011), 611-630.  doi: 10.1007/s11785-011-0146-1.  Google Scholar

[23]

M. G. Kreǐn and M. A. Naimark, The method of symmetric and Hermitian forms int the theory of the separation of the roots of algebraic equations, Linear and Multilinear Algebra, 10 (1981), 265-308.  doi: 10.1080/03081088108817420.  Google Scholar

[24]

M. G. Kreǐn and A. A. Nudel'man, The Markov Moment Problem and Extremal Problems, Mathematical Monographs, Vol. 50. American Mathematical Society, Providence, R.I., 1977.  Google Scholar

[25]

M. M. Postnikov, Stable Polynomials, Nauka, Moscow, 1981,176 pp.  Google Scholar

[26]

M. Prevost and T. Rivoal, Remainder Padé approximants for the exponential function, Constructive Approximation, 25 (2007), 109-123.  doi: 10.1007/s00365-006-0635-6.  Google Scholar

[27]

G. Szegö, Orthogonal Polynomials, Fourth edition, American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I., 1975.  Google Scholar

Figure 1.  Zeros of the polynomials $ f_{n}(z) $ for $ \alpha = \sqrt{2} $ and $ n = 1, 2, 3, 4, 5 $: $ f_{1}(z) $ in black, $ f_{2}(z) $ in blue, $ f_{3}(z) $ in magenta, $ f_{4}(z) $ in orange, and $ f_{5}(z) $ in red
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Figure 2.  Zeros of the polynomials $ f_{n}(z) $ for $ n = 1, 2, 3, 4, 5 $: $ f_{1}(z) $ in black color, $ f_{2}(z) $ in blue, $ f_{3}(z) $ in magenta, $ f_{4}(z) $ in orange, and $ f_{5}(z) $ in red
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Figure 3.  Zeros of the polynomials $ f_{n}(z) $ for $ \alpha = 3 $ and $ n = 1, 2, 3, 4, 5 $: $ f_{1}(z) $ in black, $ f_{2}(z) $ in blue, $ f_{3}(z) $ in magenta, $ f_{4}(z) $ in orange, and $ f_{5}(z) $ in red
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