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Register a new userImproved results on the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle
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Authors
Tamas Erdelyi
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Let either $R_k(t) := |P_k(e^{it})|^2$ or $R_k(t) := |Q_k(e^{it})|^2$, where $P_k$ and $Q_k$ are the usual Rudin-Shapiro polynomials of degree $n-1$ with $n=2^k$. In a recent paper we combined close to sharp upper bounds for the modulus of the autocorrelation coefficients of the Rudin-Shapiro polynomials with a deep theorem of Littlewood to prove that there is an absolute constant $A>0$ such that the equation $R_k(t) = (1+eta )n$ has at least $An^{0.5394282}$ distinct zeros in $[0,2pi)$ whenever $eta$ is real and $|eta| < 2^{-11}$. In this paper we show that the equation $R_k(t)=(1+eta)n$ has at least $(1/2-|eta|-varepsilon)n/2$ distinct zeros in $[0,2pi)$ for every $eta in (-1/2,1/2)$, $varepsilon > 0$, and sufficiently large $k geq k_{eta,varepsilon}$.
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In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. Binary sequences with low autocorrelation coefficients are of interest in radar, sonar, and communication systems. In this paper we show that the Mahler measure of the Rudin-Shapiro polynomials of degree $n=2^k-1$ is asymptotically $(2n/e)^{1/2}$, as it was conjectured by B. Saffari in 1985. Our approach is based heavily on the Saffari and Montgomery conjectures proved recently by B. Rodgers.
We consider the class of Rudin-Shapiro-like polynomials, whose $L^4$ norms on the complex unit circle were studied by Borwein and Mossinghoff. The polynomial $f(z)=f_0+f_1 z + cdots + f_d z^d$ is identified with the sequence $(f_0,f_1,ldots,f_d)$ of its coefficients. From the $L^4$ norm of a polynomial, one can easily calculate the autocorrelation merit factor of its associated sequence, and conversely. In this paper, we study the crosscorrelation properties of pairs of sequences associated to Rudin-Shapiro-like polynomials. We find an explicit formula for the crosscorrelation merit factor. A computer search is then used to find pairs of Rudin-Shapiro-like polynomials whose autocorrelation and crosscorrelation merit factors are simultaneously high. Pursley and Sarwate proved a bound that limits how good this combined autocorrelation and crosscorrelation performance can be. We find infinite families of polynomials whose performance approaches quite close to this fundamental limit.
For the weight function $W_mu(x) = (1-|x|^2)^mu$, $mu > -1$, $lambda > 0$ and $b_mu$ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product $$ la f,g ra = {b_mu [int_{BB^d} f(x) g(x) W_mu(x) dx + lambda int_{BB^d} abla f(x) cdot abla g(x) W_mu(x) dx]} $$ are constructed in terms of spherical harmonics and a sequence of Sobolev orthog onal polynomials of one variable. The latter ones, hence, the orthogonal polynomials with respect to $la cdot,cdotra$, can be generated through a recursive formula.
A joint algebraic interpretation of the biorthogonal Askey polynomials on the unit circle and of the orthogonal Jacobi polynomials is offered. It ties their bispectral properties to an algebra called the meta-Jacobi algebra $mmathfrak{J}$.
We shall establish two-side explicit inequalities, which are asymptotically sharp up to a constant factor, on the maximum value of $|H_k(x)| e^{-x^2/2},$ on the real axis, where $H_k$ are the Hermite polynomials.
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