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We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of √n lie in NP intersect coNP. The result (almost) subsumes the three mutually-incomparable previous results regarding these lattice problems: Banaszczyk [1993], Goldreich and Goldwasser [2000], and Aharonov and Regev [2003]. Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier series over the lattice. This technique might be useful elsewhere -- we demonstrate this by giving a simple and efficient algorithm for one other lattice problem (CVPP) improving on a previous result of Regev [2003]. An interesting fact is that our result emerged from a "dequantization" of our previous quantum result in Aharonov and Regev [2003]. This route to proving purely classical results might be beneficial elsewhere. [ABSTRACT FROM AUTHOR]

Here we develop an option pricing method based on Legendre series expansion of the density function. The key insight, relying on the close relation of the characteristic function with the series coefficients, allows to recover the density function rapidly and accurately. Based on this representation for the density function, approximations formulas for pricing European type options are derived. To obtain highly accurate result for European call option, the implementation involves integrating high degree Legendre polynomials against exponential function. Some numerical instabilities arise because of serious subtractive cancellations in its formulation (96) in Proposition A.1. To overcome this difficulty, we rewrite this quantity as solution of a second-order linear difference equation and solve it using a robust and stable algorithm from Olver. Derivation of the pricing method has been accompanied by an error analysis. Errors bounds have been derived and the study relies more on smoothness properties which are not provided by the payoff functions, but rather by the density function of the underlying stochastic models. This is particularly relevant for options pricing where the payoffs of the contract are generally not smooth functions. The numerical experiments on a class of models widely used in quantitative finance show exponential convergence. [ABSTRACT FROM AUTHOR]

Some approximation properties of the Cesàro means of trigonometric Fourier series of the metric of space L([ a, b]) , 1 ≤ p < + ∞, are studied. [ABSTRACT FROM AUTHOR]

Some approximate properties of the Cesàro means of conjugate trigonometric Fourier series are studied. [ABSTRACT FROM AUTHOR]

In the circular restricted three-body problem (CR3BP), transit orbit is a class of orbit which can pass through the bottleneck region of the zero velocity curve and escapes from the vicinity of the primary or the secondary. This kind of orbit plays a very important role in the design of space exploration missions. A kind of low-energy interplanetary transfer, which is called Interplanetary Superhighway (IPS), can be realized by utilizing transit orbits. To use the transit orbit in actual mission design, a key issue is to find an algorithm which can separate the states corresponding to transit orbits from the states corresponding to other types of orbits rapidly. In fact, the distribution of transit orbit in the phase space has been investigated by numerical method, and a Fourier series approximation method has been introduced to describe the boundary of transit orbits. However, the Fourier series approximation method needs several hundred sets of Fourier series. The coefficients of these Fourier series are neither easy to be computed nor convenient to be stored, which makes the method can hardly be used in actual mission design. In this paper, the support vector machine (SVM) is used to classify the trajectories in the CR3BP. Using the Gaussian kernel, the 6-dimensional states in the CR3BP are mapped into an infinite-dimensional space, and the bound of the transit orbits is described by a hyperplane. A training data generation method is introduced, which reduces the size of training data by generating the states near the hyperplane. The numerical results show that the proposed algorithm gives the good correct rate of classification, and its computing speed is much faster than that of the Fourier series approximation method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper modified variants of the sparse Fourier transform algorithms from Iwen (2010) [32] are presented which improve on the approximation error bounds of the original algorithms. In addition, simple methods for extending the improved sparse Fourier transforms to higher dimensional settings are developed. As a consequence, approximate Fourier transforms are obtained which will identify a near-optimal k-term Fourier series for any given input function, , in time (neglecting logarithmic factors). Faster randomized Fourier algorithm variants with runtime complexities that scale linearly in the sparsity parameter k are also presented. [Copyright &y& Elsevier]

The most challenging problem in the current block-based statistical static timing analysis (SSTA) is how to handle the max operation efficiently and accurately. Existing SSTA techniques suffer from limited modeling capability by using a linear delay model with Gaussian distribution, or have scalability problems due to expensive operations involved to handle non-Gaussian variation sources or nonlinear delays. To overcome these limitations, we propose efficient algorithms to handle the max operation in SSTA with both quadratic delay dependency and non-Gaussian variation sources simultaneously. Based on such algorithms, we develop an SSTA flow with quadratic delay model and non-Gaussian variation sources. All the atomic operations, max and add, are calculated efficiently via either closed-form formulas or low dimension (at most 2-D) lookup tables. We prove that the complexity of our algorithm is linear in both variation sources and circuit sizes, hence our algorithm scales well for large designs. Compared to Monte Carlo simulation for non-Gaussian variation sources and nonlinear delay models, our approach predicts the mean, standard deviation and 95% percentile point with less than 2% error, and the skewness with less than 10% error. [ABSTRACT FROM PUBLISHER]

Abstract: A rapidly convergent series, based on Fourier expansion of the exponential multiplier, is presented for highly accurate approximation of the Voigt function (VF). The corresponding algorithm enables the rapid calculation, required for its implementation as a subprogram in an interpolation approach. The numerical analysis of this VF approximation suggests that it may be more accurate than 10−9 in the Humlíček regions 3 and 4. [Copyright &y& Elsevier]

Cet article porte sur l'estimation de fonctions à partir d'observations (t [ABSTRACT FROM AUTHOR]

The filtered backprojection (FBP) method has recently been used for almost all the practical CT scanners because of its simple principle and high accuracy in reconstruction of images. However, it is difficult to use this method for some applications such as a real time display in which images are adjusted during displaying images, since a backprojection of the FBP method requires a very large amount of computation. This paper introduces a method of backprojection in which the projection data compensated by a filter are expanded using Fourier series, and the sum of the series is evaluated by a high-speed synthesis algorithm of complex exponential using Gaussian functions and the fast Fourier transform (FFT) method. The theory is explained based on a conventional FBP method; then the usefulness of the proposed method is demonstrated. The results show that the proposed method can realize interpolations with various frequency characteristics requiring about one-sixth the amount of computation compared with a conventional backprojection using a linear interpolation. [ABSTRACT FROM AUTHOR]