How Far Is an Ultraflat Sequence of Unimodular Polynomials from Being Conjugate-reciprocal?
نویسندگان
چکیده
In this paper we study ultraflat sequences (Pn) of unimodular polynomials Pn ∈ Kn in general, not necessarily those produced by Kahane in his paper [Ka]. We examine how far is a sequence (Pn) of unimodular polynomials Pn ∈ Kn from being conjugate reciprocal. Our main results include the following. Theorem. Given a sequence (εn) of positive numbers tending to 0, assume that (Pn) is a (εn)-ultraflat sequence of unimodular polynomials Pn ∈ Kn. The coefficients of Pn are denoted by ak,n, that is, Pn(z) = n X k=0 ak,nz k , , k = 0, 1, . . . , n, n = 1, 2, . . . .
منابع مشابه
On the Real Part of Ultraflat Sequences of Unimodular Polynomials
Let Pn(z) = ∑n k=0 ak,nz k ∈ C[z] be a sequence of unimodular polynomials (|ak,n| = 1 for all k, n) which is ultraflat in the sense of Kahane, i.e., lim n→∞ max |z|=1 ∣
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Let Pn(z) = ∑n k=0 ak,nz k ∈ C [z] be a sequence of unimodular polynomials (|ak,n| = 1 for all k, n) which is ultraflat in the sense of Kahane, i.e., lim n→∞ max |z|=1 ∣∣∣(n + 1)−1/2|Pn(z)| − 1∣∣∣ = 0 . We prove the following conjecture of Saffari (1991): ∑n k=0 ak,nan−k,n = o(n) as n → ∞, that is, the polynomial Pn(z) and its “conjugate reciprocal” P ∗ n(z) = ∑n k=0 an−k,nz k become “nearly or...
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