1,132 research outputs found
Non-uniform spline recovery from small degree polynomial approximation
We investigate the sparse spikes deconvolution problem onto spaces of
algebraic polynomials. Our framework encompasses the measure reconstruction
problem from a combination of noiseless and noisy moment measurements. We study
a TV-norm regularization procedure to localize the support and estimate the
weights of a target discrete measure in this frame. Furthermore, we derive
quantitative bounds on the support recovery and the amplitudes errors under a
Chebyshev-type minimal separation condition on its support. Incidentally, we
study the localization of the knots of non-uniform splines when a Gaussian
perturbation of their inner-products with a known polynomial basis is observed
(i.e. a small degree polynomial approximation is known) and the boundary
conditions are known. We prove that the knots can be recovered in a grid-free
manner using semidefinite programming
A new class of trigonometric B-Spline Curves
We construct one-frequency trigonometric spline curves with a de Boor-like algorithm for evaluation and analyze their shape-preserving properties. The convergence to quadratic B-spline curves is also analyzed. A fundamental tool is the concept of the normalized B-basis, which has optimal shape-preserving properties and good symmetric properties
Fast multi-dimensional scattered data approximation with Neumann boundary conditions
An important problem in applications is the approximation of a function
from a finite set of randomly scattered data . A common and powerful
approach is to construct a trigonometric least squares approximation based on
the set of exponentials . This leads to fast numerical
algorithms, but suffers from disturbing boundary effects due to the underlying
periodicity assumption on the data, an assumption that is rarely satisfied in
practice. To overcome this drawback we impose Neumann boundary conditions on
the data. This implies the use of cosine polynomials as basis
functions. We show that scattered data approximation using cosine polynomials
leads to a least squares problem involving certain Toeplitz+Hankel matrices. We
derive estimates on the condition number of these matrices. Unlike other
Toeplitz+Hankel matrices, the Toeplitz+Hankel matrices arising in our context
cannot be diagonalized by the discrete cosine transform, but they still allow a
fast matrix-vector multiplication via DCT which gives rise to fast conjugate
gradient type algorithms. We show how the results can be generalized to higher
dimensions. Finally we demonstrate the performance of the proposed method by
applying it to a two-dimensional geophysical scattered data problem
Optimal Uniform Convergence Rates and Asymptotic Normality for Series Estimators Under Weak Dependence and Weak Conditions
We show that spline and wavelet series regression estimators for weakly
dependent regressors attain the optimal uniform (i.e. sup-norm) convergence
rate of Stone (1982), where is the number of
regressors and is the smoothness of the regression function. The optimal
rate is achieved even for heavy-tailed martingale difference errors with finite
th absolute moment for . We also establish the asymptotic
normality of t statistics for possibly nonlinear, irregular functionals of the
conditional mean function under weak conditions. The results are proved by
deriving a new exponential inequality for sums of weakly dependent random
matrices, which is of independent interest.Comment: forthcoming in Journal of Econometric
Tchebycheffian B-splines in isogeometric Galerkin methods
Tchebycheffian splines are smooth piecewise functions whose pieces are drawn
from (possibly different) Tchebycheff spaces, a natural generalization of
algebraic polynomial spaces. They enjoy most of the properties known in the
polynomial spline case. In particular, under suitable assumptions,
Tchebycheffian splines admit a representation in terms of basis functions,
called Tchebycheffian B-splines (TB-splines), completely analogous to
polynomial B-splines. A particularly interesting subclass consists of
Tchebycheffian splines with pieces belonging to null-spaces of
constant-coefficient linear differential operators. They grant the freedom of
combining polynomials with exponential and trigonometric functions with any
number of individual shape parameters. Moreover, they have been recently
equipped with efficient evaluation and manipulation procedures. In this paper,
we consider the use of TB-splines with pieces belonging to null-spaces of
constant-coefficient linear differential operators as an attractive substitute
for standard polynomial B-splines and rational NURBS in isogeometric Galerkin
methods. We discuss how to exploit the large flexibility of the geometrical and
analytical features of the underlying Tchebycheff spaces according to
problem-driven selection strategies. TB-splines offer a wide and robust
environment for the isogeometric paradigm beyond the limits of the rational
NURBS model.Comment: 35 pages, 18 figure
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