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Regularized sampling of multiband signals  [PDF]
J. Selva
Mathematics , 2010, DOI: 10.1109/TSP.2010.2057248
Abstract: This paper presents a regularized sampling method for multiband signals, that makes it possible to approach the Landau limit, while keeping the sensitivity to noise at a low level. The method is based on band-limited windowing, followed by trigonometric approximation in consecutive time intervals. The key point is that the trigonometric approximation "inherits" the multiband property, that is, its coefficients are formed by bursts of non-zero elements corresponding to the multiband components. It is shown that this method can be well combined with the recently proposed synchronous multi-rate sampling (SMRS) scheme, given that the resulting linear system is sparse and formed by ones and zeroes. The proposed method allows one to trade sampling efficiency for noise sensitivity, and is specially well suited for bounded signals with unbounded energy like those in communications, navigation, audio systems, etc. Besides, it is also applicable to finite energy signals and periodic band-limited signals (trigonometric polynomials). The paper includes a subspace method for blindly estimating the support of the multiband signal as well as its components, and the results are validated through several numerical examples.
A note on circular trace formulae  [PDF]
Irina Nenciu
Mathematics , 2006,
Abstract: We find a finite CMV matrix whose eigenvalues coincide with the Dirichlet data of a circular periodic problem. As a consequence, we obtain circular analogues of the classical trace formulae for periodic Jacobi matrices.
Note on Explicit Formulae for the Bures Metric  [PDF]
J. Dittmann
Physics , 1998, DOI: 10.1088/0305-4470/32/14/007
Abstract: The aim of this paper is to derive explicit formulae for the Riemannian Bures metric on the manifold of (finite dimensional) nondegenerate density matrices. The computation of the Bures metric using the presented equations does not require any diagonalization procedure and uses matrix products, determinants and traces, only.
Asymptotic sampling formulae for Lambda-coalescents  [PDF]
Julien Berestycki,Nathanael Berestycki,Vlada Limic
Mathematics , 2012,
Abstract: We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a Lambda-coalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to infinity. Some of our results hold in the case of a general Lambda-coalescent that comes down from infinity, but we obtain more precise information under a regular variation assumption. In this case, we obtain results of independent interest for the time at which a mutation uniformly chosen at random was generated. This exhibits a phase transition at \alpha=3/2, where \alpha \in(1,2) is the exponent of regular variation.
Random Sampling Using Shannon Interpolation and Poisson Summation Formulae  [PDF]
Xiao Z. Wang,Wei E. I. Sha
Mathematics , 2009,
Abstract: This report mainly focused on the basic concepts and the recovery methods for the random sampling. The recovery methods involve the orthogonal matching pursuit algorithm and the gradient-based total variation strategy. In particular, a fast and efficient observation matrix filling technique was implemented by the classic Shannon interpolation and Poisson summation formulae. The numerical results for the trigonometric signal, the Gaussian-modulated sinusoidal pulse, and the square wave were demonstrated and discussed. The work may give some help for future work in theoretical study and practical implementation of the random sampling.
A Note on Information-Directed Sampling and Thompson Sampling  [PDF]
Li Zhou
Computer Science , 2015,
Abstract: This note introduce three Bayesian style Multi-armed bandit algorithms: Information-directed sampling, Thompson Sampling and Generalized Thompson Sampling. The goal is to give an intuitive explanation for these three algorithms and their regret bounds, and provide some derivations that are omitted in the original papers.

Wang Lei-heng,

计算数学 , 1998,
Abstract: In this note, using precise estimates for the regularized functionals, the upper and lower monotone approximations, in the sense of energy, of the regularized methods for the simplified contact problem with Coulumb friction are presented,and a posteriori error estimate is obtained.
A Note on Operator Sampling and Fractional Fourier Transform  [PDF]
Qingyue Zhang
Mathematical Problems in Engineering , 2011, DOI: 10.1155/2011/303460
Abstract: This paper presents that the kernel of the fractional Fourier transform (FRFT) satisfies the operator version of Kramer's Lemma (Hong and Pfander, 2010), which gives a new applicability of Kramer's Lemma. Moreover, we give a new sampling formulae for reconstructing the operators which are bandlimited in the FRFT sense. 1. Introduction and Notations Sampling theory for operators motivated by the operator identification problem in communications engineering has been developed during the last few years [1–4]. In [4], Hong and Pfander gave an operator version of Kramer's Lemma (see [4, Theorem 25]). But they did not give any explicit kernel satisfying the hypotheses in [4, Theorem 25] other than the Fourier kernel. In this paper, we present that the kernel of the fractional Fourier transform satisfies the hypotheses in [4, Theorem 25]. Therefore, we give a new applicability of Kramer's method. The FRFT—a generalization of the Fourier transform (FT)—has received much attention in recent years due to its numerous applications, including signal processing, quantum physics, communications, and optics [5–7]. Hong and Pfander studied the sampling theorem on the operators which are bandlimited in the FT sense (see [4]). In this paper, we generalize their results to bandlimited operators in the FRFT sense. For , its FRFT is defined by where , and the transform kernel is given by where is Dirac distribution function over , , and . The inverse FRFT is the FRFT at angle , given by where the bar denotes the complex conjugation. Whenever , (1.2) reduces to the FT. Through this paper, we assume that . In FRFT domain, the function space with bandwidth is defined by For the sake of simplicity, when , is written as . In the following, we use the notation if there exist positive constants and such that for all objects in the set . Let be a Hilbert space and be a sequence in . The set is said to be a frame [8, 9] for if Let with . is a set sampling for if 2. The Properties of the Kernel of FRFT In this section, we consider under what conditions is a frame for for every . The following theorem gives a necessary and sufficient condition for to be a frame for for every . Theorem 2.1. For any and . is a frame for if and only if is a frame for . Remark 2.2. By Theorem 2.1, when taking appropriate , is a frame for each . Therefore, we give a kernel satisfying the hypotheses in [4, Theorem 25], which gives a new applicability of Kramer's Lemma. To prove Theorem 2.1, we need to introduce the following results. Lemma 2.3. is a set of sampling for if and only if is a frame for for
Approximate sampling formulae for general finite-alleles models of mutation  [PDF]
Anand Bhaskar,John A. Kamm,Yun S. Song
Quantitative Biology , 2011, DOI: 10.1239/aap/1339878718
Abstract: Many applications in genetic analyses utilize sampling distributions, which describe the probability of observing a sample of DNA sequences randomly drawn from a population. In the one-locus case with special models of mutation such as the infinite-alleles model or the finite-alleles parent-independent mutation model, closed-form sampling distributions under the coalescent have been known for many decades. However, no exact formula is currently known for more general models of mutation that are of biological interest. In this paper, models with finitely-many alleles are considered, and an urn construction related to the coalescent is used to derive approximate closed-form sampling formulas for an arbitrary irreducible recurrent mutation model or for a reversible recurrent mutation model, depending on whether the number of distinct observed allele types is at most three or four, respectively. It is demonstrated empirically that the formulas derived here are highly accurate when the per-base mutation rate is low, which holds for many biological organisms.
A note on transformations on auxiliary variable in survey sampling  [PDF]
Rajesh Singh,Mukesh Kumar
Statistics , 2012, DOI: 10.3233/MAS-2011-0154
Abstract: In this note, we address the doubts of Singh (2001) and Gupta and Shabbir (2008) on the transformations of auxiliary variables by adding unit free constants. The original contribution by Sisodia and Dwivedi (1981) is correct.
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