100 research outputs found

    Constructing Cubature Formulas of Degree 5 with Few Points

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    This paper will devote to construct a family of fifth degree cubature formulae for nn-cube with symmetric measure and nn-dimensional spherically symmetrical region. The formula for nn-cube contains at most n2+5n+3n^2+5n+3 points and for nn-dimensional spherically symmetrical region contains only n2+3n+3n^2+3n+3 points. Moreover, the numbers can be reduced to n2+3n+1n^2+3n+1 and n2+n+1n^2+n+1 if n=7n=7 respectively, the later of which is minimal.Comment: 13 page

    Approximate Approximations from scattered data

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    The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function on a uniform grid to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe the application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.Comment: 29 pages, 17 figure

    On square positive extensions and cubature formulas

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    AbstractWe consider linear functionals Ld defined over Pd⊂P≔R[x1,…,xn] and study how to extend Ld to a square positive L:P→R, i.e. to a linear functional L with L(p2)⩾0 for all p∈P. We use the connection of square positive functionals to real ideals, to orthogonal polynomials, and to moment matrices. Finally, we report how such extension techniques are used to construct and analyse cubature formulas

    Radon needlet thresholding

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    We provide a new algorithm for the treatment of the noisy inversion of the Radon transform using an appropriate thresholding technique adapted to a well-chosen new localized basis. We establish minimax results and prove their optimality. In particular, we prove that the procedures provided here are able to attain minimax bounds for any Lp\mathbb {L}_p loss. It s important to notice that most of the minimax bounds obtained here are new to our knowledge. It is also important to emphasize the adaptation properties of our procedures with respect to the regularity (sparsity) of the object to recover and to inhomogeneous smoothness. We perform a numerical study that is of importance since we especially have to discuss the cubature problems and propose an averaging procedure that is mostly in the spirit of the cycle spinning performed for periodic signals

    Inversion of noisy Radon transform by SVD based needlet

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    A linear method for inverting noisy observations of the Radon transform is developed based on decomposition systems (needlets) with rapidly decaying elements induced by the Radon transform SVD basis. Upper bounds of the risk of the estimator are established in LpL^p (1≤p≤∞1\le p\le \infty) norms for functions with Besov space smoothness. A practical implementation of the method is given and several examples are discussed
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