* 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.
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...
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.
We give some new properties of refinable measures and survey results on their asymptotic normality. We also give a survey on the asymptotically optimal time-frequency localisation of refinable measures and associated wavelets.