arXiv subjects

Tamás Erdélyi

Publications and source records attributed to Tamás Erdélyi.

24 records · Page 2Linked to original sources

arXiv2017

The asymptotic value of the Mahler measure of the Rudin-Shapiro polynomials

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.

math.CA↗

arXiv2017

Improved lower bounds for the Mahler measure of the Fekete polynomials

We show that there is an absolute constant $c > 1/2$ such that the Mahler measure of the Fekete polynomials $f_p$ of the form $$f_p(z) := \sum_{k=1}^{p-1}{\left( \frac kp \right)z^k}\,,$$ (where the coefficients are the usual Legendre symbols) is at least $c\sqrt{p}$ for all sufficiently large primes $p$. This improves the lower bound $\left(\frac 12 - \varepsilon\right)\sqrt{p}$ known before for the Mahler measure of the Fekete polynomials $f_p$ for all sufficiently large primes $p \geq c_{\varepsilon}$. Our approach is based on the study of the zeros of the Fekete polynomials on the unit circle.

math.CV↗

arXiv2013

Inverse Bernstein inequalities and min-max-min problems on the unit circle

We give a short and elementary proof of an inverse Bernstein-type inequality found by S. Khrushchev for the derivative of a polynomial having all its zeros on the unit circle. The inequality is used to show that equally-spaced points solve a min-max-min problem for the logarithmic potential of such polynomials. Using techniques recently developed for polarization (Chebyshev-type) problems, we show that this optimality also holds for a large class of potentials, including the Riesz potentials $1/r^s$ with $s>0.$

math.MG↗

arXiv2011

Limits on Preserving Quantum Coherence using Multi-Pulse Control

We explore the physical limits of pulsed dynamical decoupling methods for decoherence control as determined by finite timing resources. By focusing on a decohering qubit controlled by arbitrary sequences of $π$-pulses, we establish a non-perturbative quantitative upper bound to the achievable coherence for specified maximum pulsing rate and noise spectral bandwidth. We introduce numerically optimized control "bandwidth-adapted" sequences that saturate the performance bound, and show how they outperform existing sequences in a realistic excitonic-qubit system where timing constraints are significant. As a byproduct, our analysis reinforces the impossibility of fault-tolerance accuracy thresholds for generic open quantum systems under purely reversible error control.

quant-ph↗

arXiv2010

The size of exponential sums on intervals of the real line

We prove that there is a constant $c > 0$ depending only on $M \geq 1$ and $μ\geq 0$ such that $$\int_y^{y+a}{|g(t)| \, dt} \geq \exp (-c/(aδ))\,, a \in (0,π]\,,$$ for every $g$ of the form $$g(t) = \sum_{j=0}^n{a_j e^{iλ_jt}}, a_j \in {\Bbb C}, \enskip |a_j| \leq Mj^μ\,, \enskip |a_0|=1\,, \enskip n \in {\Bbb N} \,,$$ where the exponents $λ_j \in {\Bbb C}$ satisfy $$\text{\rm Re}(λ_0) = 0\,, \qquad \text{\rm Re}(λ_j) \geq jδ> 0\,, j=1,2,\ldots\,,$$ and for every subinterval $[y,y+a]$ of the real line. Establishing inequalities of this variety is motivated by problems in physics.

math.CA↗

arXiv1995

Müntz spaces and Remez inequalities

Two relatively long-standing conjectures concerning Müntz polynomials are resolved. The central tool is a bounded Remez type inequality for non-dense Müntz spaces.

math.CA↗