Related papers: On the power of function values for the approximat…

We study the $L_2$-approximation of functions from a Hilbert space and compare the sampling numbers with the approximation numbers. The sampling number $e_n$ is the minimal worst case error that can be achieved with $n$ function values,…

We study the problem of estimating the average of a Lipschitz continuous function $f$ defined over a metric space, by querying $f$ at only a single point. More specifically, we explore the role of randomness in drawing this sample. Our goal…

We consider the problem of finding the best harmonic or analytic approximant to a given function. We discuss when the best approximant is unique, and what regularity properties the best approximant inherits from the original function. All…

This paper studies the probabilistic function approximation problem over reproducing kernel Hilbert spaces. We show the existence and uniqueness of the optimizer under mild assumptions. Furthermore, we generalize the celebrated representer…

Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and…

This article delves into the study of the theory of regularized learning in Banach spaces for linear-functional data. It encompasses discussions on representer theorems, pseudo-approximation theorems, and convergence theorems. Regularized…

Under investigation is the problem of finding the best approximation of a function in a Hilbert space subject to convex constraints and prescribed nonlinear transformations. We show that in many instances these prescriptions can be…

The approximation ratio has become one of the dominant measures in mechanism design problems. In light of analysis of algorithms, we define the \emph{smoothed approximation ratio} to compare the performance of the optimal mechanism and a…

We study the recovery of functions in various norms, including $L_p$ with $1\le p\le\infty$, based on function evaluations. We obtain worst case error bounds for general classes of functions in terms of the best $L_2$-approximation from a…

Determining the power distribution of the members of a shareholder meeting or a legislative committee is a well-known problem for many applications. In some cases it turns out that power is nearly proportional to relative voting weights,…

In this paper we analyze the approximation of stable linear time-invariant systems, like the Hilbert transform, by sampling series for bandlimited functions in the Paley-Wiener space $\mathcal{PW}_{\pi}^{1}$. It is known that there exist…

This works handles the inverse reinforcement learning problem in high-dimensional state spaces, which relies on an efficient solution of model-based high-dimensional reinforcement learning problems. To solve the computationally expensive…

In this work, we propose an extensive numerical study on approximating the absolute value function. The methods presented in this paper compute approximants in the form of rational functions and have been proposed relatively recently, e.g.,…

We prove that a Banach space has the uniform approximation property with proportional growth of the uniformity function iff it is a weak Hilbert space.

We develop a general, functional calculus approach to approximation of $C_0$-semigroups on Banach spaces by bounded completely monotone functions of their generators. The approach comprises most of well-known approximation formulas, yields…

Horn functions form a subclass of Boolean functions and appear in many different areas of computer science and mathematics as a general tool to describe implications and dependencies. Finding minimum sized representations for such functions…

This paper investigates the approximation of Gaussian random variables in Banach spaces, focusing on the high-probability bounds for the approximation of Gaussian random variables using finitely many observations. We derive non-asymptotic…

We consider the problem of approximating a function using Herglotz wave functions, which are a superposition of plane waves. When the discrepancy is measured in a ball, we show that the problem can essentially be solved by considering the…

We study multivariate integration and approximation for functions belonging to a weighted reproducing kernel Hilbert space based on half-period cosine functions in the worst-case setting. The weights in the norm of the function space depend…

Constructing or learning a function from a finite number of sampled data points (measurements) is a fundamental problem in science and engineering. This is often formulated as a minimum norm interpolation problem, regularized learning…