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

A Favard type theorem for Hurwitz polynomials

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  February 2020 Early access  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.

Citation: Abdon E. Choque-Rivero, Iván Area. A Favard type theorem for Hurwitz polynomials. Discrete and 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.

[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.

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

D. Gómez-UllateN. 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.

[16]

D. Gómez-UllateN. 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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[27]

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

show all references

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

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.

[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.

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

D. Gómez-UllateN. 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.

[16]

D. Gómez-UllateN. 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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[27]

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

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
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
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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