Related papers: Some Sharp Error Bounds for Multivariate Linear In…
New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the…
In this paper a spline based integral approximation is utilized to propose a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The approximations can be improved…
We develop a new approximation theory for linear and quadratic interpolation models, suitable for use in convex-constrained derivative-free optimization (DFO). Most existing model-based DFO methods for constrained problems assume the…
In this paper we provide a priori error estimates with explicit constants for both the $L^2$-projection and the Ritz projection onto spline spaces of arbitrary smoothness defined on arbitrary grids. This extends the results recently…
In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider…
We investigate error orders for integral limit approximations to traces of products of Toeplitz matrices generated by integrable functions on $[-\pi,\pi]$ having some singularities at the origin. Even though a sharp error order of the above…
Graph-based approximation methods are of growing interest in many areas, including transportation, biological and chemical networks, financial models, image processing, network flows, and more. In these applications, often a basis for the…
Richardson extrapolation is a classical technique from numerical analysis that can improve the approximation error of an estimation method by combining linearly several estimates obtained from different values of one of its hyperparameters,…
Extrapolation is a well-known technique for solving convex optimization and variational inequalities and recently attracts some attention for non-convex optimization. Several recent works have empirically shown its success in some machine…
The inverse tangent function can be bounded by different inequalities, for example by Shafer's inequality. In this publication, we propose a new sharp double inequality, consisting of a lower and an upper bound, for the inverse tangent…
The recovery of the intrinsic geometric structures of data collections is an important problem in data analysis. Supervised extensions of several manifold learning approaches have been proposed in the recent years. Meanwhile, existing…
We study the space of bandlimited Lipschitz functions in one variable. In particular we provide a geometrical description of the natural interpolating and sampling sequences for this space. We also find a description of the trace of such…
Classical Laguerre spectral approximations are highly effective on the half-line when the target function is smooth in the usual polynomial scale. However, their accuracy deteriorates for nonsmooth functions. Such behavior appears naturally…
The paper is concerned with sharp estimates of constants in Poincare type inequalities for functions having zero mean value on the boundary of a Lipschitz domain or on a measurable part of it. These estimates are useful for various…
In this paper we study local error bound moduli for a locally Lipschitz and regular function via its outer limiting subdifferential set. We show that the distance of 0 from the outer limiting subdifferential of the support function of the…
We introduce an interpolation--regression operator for polynomial approximation on the unit sphere $\mathbb{S}^2$ from discrete samples. The approximant is a spherical polynomial of degree $r$ which interpolates the data on a prescribed…
Effective verification and validation techniques for modern scientific machine learning workflows are challenging to devise. Statistical methods are abundant and easily deployed, but often rely on speculative assumptions about the data and…
To minimize or upper-bound the value of a function "robustly", we might instead minimize or upper-bound the "epsilon-robust regularization", defined as the map from a point to the maximum value of the function within an epsilon-radius. This…
We establish linear convergence rates for a certain class of extrapolated fixed point algorithms which are based on dynamic string-averaging methods in a real Hilbert space. This applies, in particular, to the extrapolated simultaneous and…
This paper studies the approximation capacity of neural networks with an arbitrary activation function and with norm constraint on the weights. Upper and lower bounds on the approximation error of these networks are computed for smooth…