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We derive bounds and asymptotics for the maximum Riesz polarization quantity M p n (A) := max x 1 ,x 2 ,... ,x n ∈A min x∈A n j=1

for every f P S and a P 0; 1, where S denotes the collection of all analytic Research supported in part by NSERC of Canada.

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 nonperturbative 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 by-product, our analysis reinforces the impossibility of fault-tolerance accuracy thresholds for generic open quantum systems under purely reversible error control.Building on the discovery of spin-echo and multiple-pulse techniques in nuclear magnetic resonance [1], dynamical decoupling (DD) methods for open quantum systems [2] have become a versatile tool for decoherence control in quantum engineering and fault-tolerant quantum computation. DD involves "open-loop" (feedback-free) quantum control based on the application of a time-dependent Hamiltonian which, in the simplest setting, effects a predetermined sequence of unitary operations (pulses) drawn from a basic repertoire. Physically, DD relies on the ability to access control time scales that are short relative to the correlation time scale of the interaction to be removed. The reduction in decoherence is achieved perturbatively, by ensuring that sufficiently high orders of the error-inducing Hamiltonian are removed. Recently, a number of increasingly powerful pulsed DD schemes have been proposed and validated in the laboratory. Uhrig DD (UDD) sequences [3], for instance, perturbatively cancel pure dephasing in a single qubit up to an arbitrarily high order n while using a minimal number (n) of pulses, paving the way to further optimization for given sequence duration [4,5] and/or specific noise environments [6], to nearly optimal protocols for generic single-qubit decoherence [7]. Experimentally, UDD has been employed to prolong coherence time in systems ranging from trapped ions [4,5,8] and atomic ensembles [9] to spin-based devices [10], and to enhance contrast in magnetic resonance imaging of tissue [11].In a realistic DD setting, the achievable performance is inevitably influenced by errors due to limited control as well as by deviations from the intended decoherence model. Since it is conceivable that both model uncertainty and pulse nonidealities can be largely removed by more accurate system identification and control design, some of these limitations may be regarded as nonfundamental in nature. Compositepulse [12] and pulse-shaping [13] techniques can be used, for instance, to cancel to high accuracy the effects of both systematic control errors and finite-width corrections. We argue, however, that, even in a situation where pulses may be assumed perfect and instantaneous, an ultimate constraint is implied by the fact that the rate at which control ope...

Abstract. The results of this paper show that many types of polynomials cannot be small on subarcs of the unit circle in the complex plane. A typical result of the paper is the following. Let Fn denote the set of polynomials of degree at most n with coefficients from {−1, 0, 1}. There are absolute constants c 1 > 0, c 2 > 0, and c 3 > 0 such thatfor every subarc A of the unit circle ∂D := {z ∈ C : |z| = 1} with length 0 < a < c 3 .The lower bound results extend to the class of f of the formwith varying nonnegative integers m ≤ n. It is shown that functions f of the above form cannot be arbitrarily small uniformly on subarcs of the circle. However, this does not extend to sets of positive measure. It shown that it is possible to find a polynomial of the above form that is arbitrarily small on as much of the boundary (in the sense of linear Lebesgue measure) as one likes.An easy to formulate corollary of the results of this paper is the following.Corollary. Let A be a subarc of the unit circle with length ℓ(A) = a. If (p k ) is a sequence of monic polynomials that tends to 0 in L 1 (A), then the sequence H(p k ) of heights tends to ∞.The results of this paper are dealing with (extensions of) classes much studied by Littlewood and many others in regards to the various conjectures of Littlewood concerning growth and flatness of unimodular polynomials on the unit circle ∂D. Hence the title of the paper.1991 Mathematics Subject Classification. 11J54, 11B83.

It is proven that for any system of n points z1, . . . , zn on the (complex) unit circle, there exists another point z of norm 1, such thatTwo proofs are presented: one uses a characterisation of equioscillating rational functions, while the other is based on Bernstein's inequality.

Abstract. The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system {xxo, xx¡, ...} with respect to Lebesgue measure on [0,1]. In this paper, differential and integral recurrence formulae for the Müntz-Legendre polynomials are obtained. Interlacing and lexicographical properties of their zeros are studied, and the smallest and largest zeros are universally estimated via the zeros of Laguerre polynomials. The uniform convergence of the Christoffel functions is proved equivalent to the nondenseness of the Müntz space on [0, 1 ], which implies that in this case the orthogonal Müntz-Legendre polynomials tend to 0 uniformly on closed subintervals of [0, 1 ). Some inequalities for Müntz polynomials are also investigated, most notably, a sharp L2 Markov inequality is proved.

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