Bernstein's Polynomial Inequalities and Functional Analysis
نویسنده
چکیده
This expository article shows how classical inequalities for the derivative of polynomials can be proved in real and complex Hilbert spaces using only elementary arguments from functional analysis. As we shall see, there is a surprising interconnection between an equality of norms for symmetric multilinear mappings due to Banach and an inequality for the derivative of trigonometric polynomials due to van der Corput and Schaake. We encounter little extra difficulty in establishing our inequalities in several or infinite dimensions. After giving the definitions of polynomials and derivatives in normed linear spaces, we establish a lemma of Hörmander, which is an extension of a theorem of Laguerre to complex vector spaces. This powerful lemma is the key to the proofs of the polynomial inequalities we discuss; however, its proof is a simple argument relying only on the fundamental theorem of algebra. Following de Bruijn (who considered only the case of the complex plane), we deduce a theorem which obtains discs inside the range of a complex-valued polynomial on the closed unit ball of a complex Hilbert space. Here the size of the disc is determined by the value of the derivative. An easy consequence is an extension to complex Hilbert spaces of an estimate of Malik on the derivative of polynomials whose roots lie outside a given disc. (Malik’s estimate generalized a conjecture of Erdös that was proved by Lax.) Another consequence is an extension to complex Hilbert spaces of the classical complex form of Bernstein’s inequality. Still another consequence is an inequality for the derivative of a polynomial on a complex Hilbert space whose real part has a known bound on the closed unit ball. When the Hilbert space is the complex plane, this inequality contains an inequality of Szegö and leads to an inequality of van der Corput and Schaake for trigonometric polynomials, which is a strengthened form of the Bernstein inequality. Using methods of van der Corput and Schaake, we deduce an inequality for
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