Muñoz-Fernández, Gustavo Adolfo and Sánchez, V.M. and Seoane Sepúlveda, Juan Benigno (2010) Estimates on the Derivative of a Polynomial with a Curved Majorant Using Convex Techniques. Journal of Convex Analysis, 17 (1). pp. 241-252. ISSN 0944-6532
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A mapping phi : [-1, 1] -> [0, infinity) is a curved majorant for a polynomial p in one real variable if vertical bar p(x)vertical bar <= phi(x) for all x is an element of [-1, 1]. If P(n)(phi)(R) is the set of all one real variable polynomials of degree at most n having the curved majorant phi, then we study the problem of determining, explicitly, the best possible constant M(n)(phi)(x) in the inequality vertical bar p'(x)vertical bar <= M(n)(phi)(x)parallel to p parallel to, for each fixed x is an element of [-1, 1], where p is an element of p(n)(phi)(R) and parallel to p parallel to is the sup norm of p over the interval [-1, 1]. These types of estimates are known as Bernstein type inequalities for polynomials with a curved majorant. The cases treated in this manuscript, namely phi(x) = root 1 - x(2) or phi(x) = vertical bar x vertical bar for an x is an element of [-1, 1] (circular and linear majorant respectively), were first studied by Rahman in . In that reference the author provided, for each n is an element of N, the maximum of M(n)(phi)(x) over [-1, 1] as well as an upper bound for M(n)(phi)(x) for each x is an element of [-1, 1], where phi is either a circular or a linear majorant. Here we provide sharp Bernstein inequalities for some specific families of polynomials having a linear or circular majorant by means of classical convex analysis techniques (in particular we use the Krein-Milman approach).
|Uncontrolled Keywords:||Bernstein type inequality; circular and linear majorants; extreme points|
|Subjects:||Sciences > Mathematics > Functional analysis and Operator theory|
R. M. Aron, M. Klimek: Supremum norms for quadratic polynomials, Arch. Math. 76 (2001) 73–80.
A. A. Markov: On a problem of D. I. Mendeleev, Zap. Im. Akad. Nauk. 62 (1889) 1–24 (in Russian).
A. A. Markov: On a question by D. I. Mendeleev, available at: http://www.math. technion.ac.il/hat/papers.html.
S. Bernstein: Sur l’ordre de la meilleure approximation des fonctions continues par des polynomes de degr´e donn´e, Acad. Roy. Belg. Cl. Sci. M´em. 4 (1912) 1–103.
S. Bernstein: Collected Works: Vol. I. The Constructive Theory of Functions (1905–1939), Atomic Energy Commission, Springfield (1958).
R. P. Boas: Inequalities for the derivatives of polynomials, Math. Mag. 42 (1969) 165–174.
G. A. Muñoz-Fernández, Y. Sarantopoulos: Bernstein and Markov-type inequalities for polynomials on real Banach spaces, Math. Proc. Camb. Philos. Soc. 133 (2002) 515–530.
G. A. Muñoz-Fernández, Y. Sarantopoulos, J. B. Seoane-Sepúlveda: An application of the Krein-Milman theorem to Bernstein and Markov inequalities, J. Convex Analysis 15
G. A. Muñoz-Fernández, J. B. Seoane-Sepúlveda: Geometry of Banach spaces of trinomials, J. Math. Anal. Appl. 340 (2008) 1069–1087.
Q. I. Rahman: On a problem of Tur´an about polynomials with curved majorants, Trans. Amer. Math. Soc. 163 (1972) 447–455.
E. V. Voronovskaja: The functional of the first derivative and improvement of a theorem of A. A. Markov, Izv. Akad. Nauk SSSR, Ser. Mat. 23 (1959) 951–962 (in Russian).
E. V. Voronovskaja: The Functional Method and its Applications, AMS, Providence(1970).
|Deposited On:||30 Oct 2012 10:00|
|Last Modified:||21 Feb 2013 17:41|
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