SFU NTAG Seminar: Daniel Tarnu

The Rudin-Shapiro polynomials $p_{m}$ were first studied by Rudin, Shapiro, and Golay independently nearly 80 years ago and are defined recursively by $p_{0}(x) = q_{0}(x) = 1$ and 
$$ p_{m}(x) = p_{m-1}(x) + x^{2^{m-1}} q_{m-1}(x), $$
$$ q_{m}(x) = p_{m-1}(x) - x^{2^{m-1}} q_{m-1}(x). $$
This class of polynomials benefits from rich structure and is of special interest as a subset of Littlewood polynomials (i.e., polynomials with coefficients in $\{ -1, 1 \}$), in part due to their having small $L^{4}$ norm, which is desirable and almost always unsatisfied by Littlewood polynomials in general. In application, uses are found for the Rudin-Shapiro polynomials in varied contexts such as radio and spectrometry.

If we let $p_{m}(x) = \sum_{j=0}^{2^{m}-1} a_{j}x^{j}$, the sequence $(a_{0}, a_{1}, \dots, a_{2^{m}-1})$ is called the $m$-th Rudin-Shapiro sequence. We denote by $C_{m}(k)$ the aperiodic autocorrelation at shift $k$ of the $m$-th Rudin-Shapiro sequence:
$$ C_{m}(k) = \sum_{j=0}^{2^{m}-1} a_{j}a_{j+k}, $$
where it is understood that $a_{j} = 0$ for $j \notin [0, 2^{m}-1]$. These autocorrelations have been studied extensively. It is often difficult to determine or approximate $C_{m}(k)$ for any given $m$ and $k$, but using the structure of $p_{m}$, bounds on partial moments of the $C_{m}(k)$ can be deduced. We give the precise orders of $\sum_{0 < k \leq x} (C_{m}(k))^{2} $ and $\max_{0 < k \leq x} |C_{m}(k)|$, and asymptotic bounds for $\sum_{0 < k \leq x} |C_{m}(k)|$. Furthermore, we construct an analogue of $|C_{m}(k)|$ on $[0,1]$ and show that its maximum value occurs uniquely at $x = \frac{2}{3}$, supporting our conjecture that the maximum value of $|C_{m}(k)|$ occurs uniquely at some $k_{m}^{\ast}$ with $\lim_{m \to \infty} \frac{k_{m}^{\ast}}{2^{m}} = \frac{2}{3}$. This is joint work with Stephen Choi.