Estimates for the Lorentz degree of polynomials
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In 1916 S. N. Bernstein observed that every polynomial p having no zeros in (-1, 1) can be written in the form ∑j = 0d aj(1 - x)j(1 + x)d - j with all aj ≥ 0 or all aj ≤ 0. The smallest natural number d for which such a representation holds is called the Lorentz degree of p and it is denoted by d(p). The Lorentz degree d(p) can be much larger than the ordinary degree deg(p). In this paper lower bounds are given for the Lorentz degree of certain special polynomials which show the sharpness of some earlier results of T. Erdélyi and J. Szabados. As a by-product, we give a short proof of Markov and Bernstein type inequalities for the derivatives of polynomials of the above form, established by G. G. Lorentz in 1963. The Lorentz degree of trigonometric polynomials is also introduced and some analogues of our algebraic results are established. © 1991.
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