On the Rate of Convergence for the Pseudospectral Optimal Control of Feedback Linearizable Systems

In this paper, we prove a theorem on the rate of convergence for the optimal cost computed using PS methods. It is a first proved convergence rate in the literature of PS optimal control. In addition to the high-order convergence rate, two theorems are proved for the existence and convergence of the approximate solutions. This paper contains several essential differences from existing papers on PS optimal control as well as some other direct computational methods. The proofs do not use necessary conditions of optimal control. Furthermore, we do not make coercivity type of assumptions. As a result, the theory does not require the local uniqueness of optimal solutions. In addition, a restrictive assumption on the cluster points of discrete solutions made in existing convergence theorems are removed.Comment: 28 pages, 3 figures, 1 tabl

An asymptotically Gaussian bound on the Rademacher tails

An explicit upper bound on the tail probabilities for the normalized Rademacher sums is given. This bound, which is best possible in a certain sense, is asymptotically equivalent to the corresponding tail probability of the standard normal distribution, thus affirming a longstanding conjecture by Efron. Applications to sums of general centered uniformly bounded independent random variables and to the Student test are presented.Comment: The discussion and references are expanded; the proofs of Lemmas 2.2 and 2.3 are simplifie

Elementary proofs of Paley-Wiener theorems for the Dunkl transform on the real line

We give an elementary proof of the Paley-Wiener theorem for smooth functions for the Dunkl transforms on the real line, establish a similar theorem for L^2-functions and prove identities in the spirit of Bang for L^p-functions. The proofs seem to be new also in the special case of the Fourier transform.Comment: 9 pp., LaTeX, no figures; final version, to appear in Int. Math. Res. No

Sparse non-negative super-resolution -- simplified and stabilised

The convolution of a discrete measure, $x=\sum_{i=1}^ka_i\delta_{t_i}$, with a local window function, $\phi(s-t)$, is a common model for a measurement device whose resolution is substantially lower than that of the objects being observed. Super-resolution concerns localising the point sources $\{a_i,t_i\}_{i=1}^k$ with an accuracy beyond the essential support of $\phi(s-t)$, typically from $m$ samples $y(s_j)=\sum_{i=1}^k a_i\phi(s_j-t_i)+\eta_j$, where $\eta_j$ indicates an inexactness in the sample value. We consider the setting of $x$ being non-negative and seek to characterise all non-negative measures approximately consistent with the samples. We first show that $x$ is the unique non-negative measure consistent with the samples provided the samples are exact, i.e. $\eta_j=0$, $m\ge 2k+1$ samples are available, and $\phi(s-t)$ generates a Chebyshev system. This is independent of how close the sample locations are and {\em does not rely on any regulariser beyond non-negativity}; as such, it extends and clarifies the work by Schiebinger et al. and De Castro et al., who achieve the same results but require a total variation regulariser, which we show is unnecessary. Moreover, we characterise non-negative solutions $\hat{x}$ consistent with the samples within the bound $\sum_{j=1}^m\eta_j^2\le \delta^2$. Any such non-negative measure is within ${\mathcal O}(\delta^{1/7})$ of the discrete measure $x$ generating the samples in the generalised Wasserstein distance, converging to one another as $\delta$ approaches zero. We also show how to make these general results, for windows that form a Chebyshev system, precise for the case of $\phi(s-t)$ being a Gaussian window. The main innovation of these results is that non-negativity alone is sufficient to localise point sources beyond the essential sensor resolution.Comment: 59 pages, 7 figure