https://www.ricam.oeaw.ac.at/files/reports/12/rep12-08.pdf

Multigrid methods for isogeometric discretization.

We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a kmonotone density g0 at a fixed point x0 when k> 2. We find that the jth derivative of the estima- tors at x0 converges at the rate n −(k−j)/(2k+1) for j =0 ,...,k � 1. The limiting distribution depends on an almost surely uniquely de- fined stochastic process Hk that stays above (below) the k-fold in- tegral of Brownian motion plus a deterministic drift, when k is even (odd). Both the MLE and LSE are known to be splines of degree k � 1 with simple knots. Establishing the order of the random gap τ + n � τ − n where τ ± n denote two successive knots, is a key ingredient of the proof of the main results. We show that this "gap problem" can be solved if a conjecture about the upper bound on the error in a particular Hermite interpolation via odd-degree splines holds.

/pdf/estimation-of-a-k-monotone-density-limit-distribution-theory-3zi0tpmbjp.pdf

Estimation of a k-monotone density: limit distribution theory and the Spline connection

This book describes in detail the key algorithms needed for computing with spline functions and illustrates their use in solving several basic problems in numerical analysis, including function approximation, numerical quadrature, data fitting, and the numerical solution of PDEs. The focus is on computational methods for bivariate splines on triangulations in the plane and on the sphere, although both univariate and tensor-product splines are also discussed. The book contains numerous examples and figures to illustrate the methods and their performance. All of the algorithms in the book have been coded in a separate MATLAB package available for license. The package can be used to run all of the examples in the book and also provides readers with the essential tools needed to create software for their own applications. In addition to the included bibliography, a list of over 100 pages of additional references can be found on the book's website. Audience: This book is designed for mathematicians, engineers, scientists, and anyone else wanting to make use of spline functions for numerical computation.

/pdf/spline-functions-computational-methods-1rbw5zbh86.pdf

Spline Functions: Computational Methods

Nonstationary tight wavelet frames, I: Bounded intervals

/pdf/the-l-norm-of-the-l2-spline-projector-is-bounded-3masdxi6sx.pdf

The L∞-norm of the L2-spline projector is bounded independently of the knot sequence: A proof of de Boor's conjecture

New Upper Bound for the B-Spline Basis Condition Number

Monotone approximation of functions by trigonometric polynomials

Inequalities of Kolmogorov type and estimates of spline interpolation on periodic classes W 2 m

New results are obtained on the approximation of elements of Sobolev classes Wpl in the Lq metric by interpolating splines of order 2m - 1 and deficiency 1, defined on nonuniform nets Δn The results are stated in terms of global and local properties of Δn, and depend mainly on an integral representation of the error

A. Yu. Shadrin

Papers

The L∞-norm of the L2-spline projector is bounded independently of the knot sequence: A proof of de Boor's conjecture

New Upper Bound for the B-Spline Basis Condition Number

Monotone approximation of functions by trigonometric polynomials

Inequalities of Kolmogorov type and estimates of spline interpolation on periodic classes W 2 m

On the approximation of functions by interpolating splines defined on nonuniform nets