100 research outputs found
Constructing Cubature Formulas of Degree 5 with Few Points
This paper will devote to construct a family of fifth degree cubature
formulae for -cube with symmetric measure and -dimensional spherically
symmetrical region. The formula for -cube contains at most points
and for -dimensional spherically symmetrical region contains only
points. Moreover, the numbers can be reduced to and if
respectively, the later of which is minimal.Comment: 13 page
Approximate Approximations from scattered data
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
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
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 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
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 () norms for functions
with Besov space smoothness. A practical implementation of the method is given
and several examples are discussed
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