On the oscillations of the modulus of Rudin-Shapiro polynomials around the middle of their ranges (original) (raw)

On the Oscillation of the Modulus of the Rudin-Shapiro Polynomials Around the Middle of Their Ranges

2021

Let either Rk(t) := |Pk(e it)|2 or Rk(t) := |Qk(e it)|2, where Pk and Qk are the usual Rudin-Shapiro polynomials of degree n − 1 with n = 2k. The graphs of Rk on the period suggest many zeros of the equation Rk(t) = n in a dense fashion. Let N(I, Rk − n) denote the number of zeros of Rk − n in an interval I := [α, β] ⊂ [0, 2π]. Improving earlier results stated only for I := [0, 2π], in this paper we show that n|I| 8π − 4 π (n log n) − 1 ≤ N(I, Rk − n) ≤ n|I| π + 16 π (n log n) , k ≥ 2 , for every I := [α, β] ⊂ [0, 2π], where |I| = β − α denotes the length of the interval I.

On the zeros of orthogonal polynomials on the unit circle

2011

Let zn{z_n}zn be a sequence in the unit disk zinmathbbC:∣z∣<1{z\in\mathbb{C}:|z|<1}zinmathbbC:∣z∣<1. It is known that there exists a unique positive Borel measure in the unit circle zinmathbbC:∣z∣=1{z\in\mathbb{C}:|z|=1}zinmathbbC:∣z∣=1 such that the orthogonal polynomials Phin{\Phi_n}Phin satisfy [\Phi_n(z_n)=0] for each n=1,2,...n=1,2,...n=1,2,.... Characteristics of the orthogonality measure and asymptotic properties of the orthogonal polynomial are given in terms of asymptotic behavior of the sequence zn{z_n}zn. Particular attention is paid to periodic sequence of zeros zn{z_n}zn of period two and three.

On The Zeros of a Polynomial in a Given Circle

2013

In this paper we discuss the problem of finding the number of zeros of a polynomial in a given circle when the coefficients of the polynomial or their real or imaginary parts are restricted to certain conditions. Our results in this direction generalize some well-known results in the theory of the distribution of zeros of polynomials.