Atlas: Bernstein-Markov type inequalities in real Banach spaces. by Gustavo A. Munoz

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Functional Analysis Valencia 2000
July 3-7, 2000
Technical University of Valencia (UPV) and University of Valencia (UV)
Valencia, Spain

Organizers
R.M. Aron (Kent State U., USA), K.D. Bierstedt (U. Paderborn, Germany), J. Bonet (UPV), J. Cerdà (U. Barcelona, Spain), H. Jarchow (U. Zürich, Switzerland), M. Maestre (UV), J. Schmets (U. Liège, Belgium)

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Bernstein-Markov type inequalities in real Banach spaces.
by
Gustavo A. Muñoz
Universidad Complutense, Madrid.
Coauthors: Yannis Sarantopoulos (National Technical University, Athens)

It was proved in 1892 by V. A. Markov that if p is a real polynomial in one variable of degree not greater than n and p^(k) is its kth derivative, then for every x in [-1, 1] we have

|p^(k)(x)| <= T_n^(k)(1)
max
-1 <= t <= 1
|p(t)|,

where T_n is the nth Chebyshev polynomial of the first kind defined by T_n(x)=cos(n arccosx), for every x in [-1, 1]. In this work we generalize Markov's inequality for any derivative of a polynomial on a real Hilbert space. This answers a question raised by L. A. Harris in his commentary to Problem 74 in The Scottish Book. We also give some estimates in the general case of real Banach spaces and we generalize some other related inequalities.

(T)

Date received: April 14, 2000