Geodesic
Generalized Markov--Bernstein Inequalities and Stability of Dynamical Systems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 251-267

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We analyze the Markov–Bernstein type inequalities between the norms of functions and of their derivatives for complex exponential polynomials. We establish a relation between the sharp constants in these inequalities and the stability problem for linear switching systems. In particular, the maximal discretization step is estimated. We prove the monotonicity of the sharp constants with respect to the exponents, provided those exponents are real. This gives asymptotically tight uniform bounds and the general form of the extremal polynomial. The case of complex exponent is left as an open problem.
Keywords: exponential polynomial, Bernstein inequality, inequality between derivative, Chebyshev system, stability, Lyapunov exponent, Lyapunov functions, dynamical switching system.
Mots-clés : quasipolynomial 
@article{TM_2022_319_a15,
     author = {Vladimir Yu. Protasov},
     title = {Generalized {Markov--Bernstein} {Inequalities} and {Stability} of {Dynamical} {Systems}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {251--267},
     publisher = {mathdoc},
     volume = {319},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2022_319_a15/}
}
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Vladimir Yu. Protasov. Generalized Markov--Bernstein Inequalities and Stability of Dynamical Systems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 251-267. http://geodesic.mathdoc.fr/item/TM_2022_319_a15/