On some problems of approximations

Szegő, Gábor (1964) On some problems of approximations. A MAGYAR TUDOMÁNYOS AKADÉMIA MATEMATIKAI KUTATÓ INTÉZETÉNEK KÖZLEMÉNYEI, 9 (1-2). pp. 3-9.

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Abstract

The problems dealt with in the present Note are connected with the classical inequalities of A. Markov and S. Bernstein (cf. [ 3 ] ) . We formulate them as follows. Let f(x) be a polynomial of degree n satisfying the condition | f(x) | ≤ 1 in the finite real interval a ≤ x ≤ b. We have then in the same interval (1.1) | f'(x) | ≤ 2 / b - a * n², | f'(x) | ≤ [(x - a) (b - x)]⁻¹/² * n. Both bounds are sharp as it can be shown by the example f(x) = Tₙ 2 x - a / b - a - 1 where Tₙ is Ohebyshev's polynomial. We shall deal with the following four problems. Problem 1: Let us consider all polynomials f(x) of a fixed degree n not vanishing identically. Introducing the norm (1.2) ||f|| = max e⁻x |f(x)|, x ≥ 0, we seek the maximum Mₙ of the ratio ||f'|| : ||f||. Problem 2: Let X₀ be a fixed constant, X₀ ≥ 0. Considering the same class of polynomials and the same norm as in Problem 1, we seek the maximum Mₙ(X₀) of the ratio |f'(X₀)| : ||f|| Problem 3: Again we consider the set of all polynomials f(x) = ⁿΣᵥ=₀ aᵥ xᵛ and the same norm ||f|| as in the previous Problems. We seek the maximum Gₙ of the ratio |aₙ| : ||f||. Problem 4: Let us consider all polynomials f(x) = ⁿΣᵥ=₀ aᵥ xᵛ of the fixed degree n satisfying the condition |f(x)| ≤ 1 in the interval — 1 ≤ x ≤ 1. We seek the maximum Hₙ of (1.3) max |ⁿΣᵥ=₀ (v + 1) aᵥ xᵛ|, -1 ≤ x ≤ 1. In this Note we derive upper and lower bounds for the maxima defined in these Problems; especially in the cases 1, 2, 4 we determine the correct order of magnitude as n ⭢ ∞. The explicit evaluation of these maxima seems to be rather difficult. Our results can be formulated as follows: (1.4) Problem 1 : Mₙ ~ n. Problem 2: Mₙ(0) ~ n , Mₙ(X₀) ~ n¹/² if X₀ > 0. Problem 3: An¹/³ < 2-ⁿ n!, Gₙ < Bn¹/². Problem 4 : Hₙ ~ log n. Here A and В are positive constants independent of n. The symbol aₙ ~ bₙ means always that the ratio |aₙ/bₙ| is bounded away from 0 and Occasionally, we use the symbol aₙ ∼/= bₙ if the ratio aₙ/bₙ tends to 1 as n ⭢ ∞. These Problems (except 3) arose in conversations with Professor Paul Turán during his stay at Stanford University in the first half of the year 1963. I owe him also some simplifications and other valuable comments to the proofs. Problem 4 has originated in the joint research of S. Knapowski — P. Turán on primes in certain arithmetic progressions [2]. Problem 3 is slightly different in character from the inequalities (1.1); it is the analog of the famous problem of Chebyshev characterizing the Chebyshev polynomials Tₙ(x).

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