On the exact estimates of the best spline approximations of functions.
On the Fejér means of bounded Ciesielski systems
We investigate the bounded Ciesielski systems, which can be obtained from the spline systems of order (m,k) in the same way as the Walsh system arises from the Haar system. It is shown that the maximal operator of the Fejér means of the Ciesielski-Fourier series is bounded from the Hardy space to if 1/2 < p < ∞ and m ≥ 0, |k| ≤ m + 1. Moreover, it is of weak type (1,1). As a consequence, the Fejér means of the Ciesielski-Fourier series of a function f converges to f a.e. if f ∈ L₁ as n...
On the local linear independence of translates of a box spline
On the total positivity of the truncated power kernel
On the two dimensional spline interpolation of Hermite-type.
Operator norm bounds and error bounds for quadratic spline interpolation
Optimal discrete approximation of continuous linear operators applicable to control problems
Optimal interpolatory splines using -spline representation
Optimal polygonal interpolation
Optimal quadratic interpolatory splines on general knotset
Optimal Rules with Polynomial Precision for Hilbert Spaces Possessing Reproducing Kernel Functions.
Oscillation Matrices with Spline Smoothing.
Penalized Least Squares Fitting
* Supported by the Army Research Office under grant DAAD-19-02-10059.Bounds on the error of certain penalized least squares data fitting methods are derived. In addition to general results in a fairly abstract setting, more detailed results are included for several particularly interesting special cases, including splines in both one and several variables.
Piecewise L-Splines.
Piecewise Polynomial Spaces and Geometric Continuity of Curves.
Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization
We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough (L∞) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution H) minimizing the L2 norm of the source...
Polynomial and spline estimators of the distribution function with prescribed accuracy
Dvoretzky-Kiefer-Wolfowitz type inequalities for some polynomial and spline estimators of distribution functions are constructed. Moreover, hints on the corresponding algorithms are given as well.
Polynomials as Linear Combinations of Multivariate B-Splines.
Probabilistic and analytic formulas for the periodic splines interpolating with multiple nodes
Properties of bounded orthonormal spline bases