Related papers: Minimax principle and lower bounds in H$^{2}$-rati…

We derive lower bounds in best rational approximation of given degree to finite Blaschke products, in the Hardy space $H^2$ of the unit disk. We first consider approximation to $z^N$, and then move on to more general Blaschke products whose…

The maximum selection principle allows to give expansions, in an adaptive way, of functions in the Hardy space $\mathbf H_2$ of the disk in terms of Blaschke products. The expansion is specific to the given function. Blaschke factors and…

We generalize a classical result by A.Macintyre and W.Rogosinski on best $H^p$--approximation in $L^p$ of rational functions. For each inner function $\theta$ we give a description of $H^p$--badly approximable functions in $\bar \theta…

Using recently developed algorithms, we compute and compare best $L^2$ and $L^\infty$ rational approximations of analytic functions on the unit disk. Although there is some theory for these problems going back decades, this may be the first…

We investigate the approximation formulas that were proposed by Tanaka & Sugihara (2019), in weighted Hardy spaces, which are analytic function spaces with certain asymptotic decay. Under the criterion of minimum worst error of $n$-point…

We study two types of approximations of Lipschitz maps with derivatives of maximal slopes on Banach spaces. First, we characterize the Radon-Nikod\'ym property in terms of strongly norm attaining Lipschitz maps and maximal derivative…

In this paper we consider functions in the Hardy space $\mathbf{H}_2^{p\times q}$ defined in the unit disc of matrix-valued. We show that it is possible, as in the scalar case, to decompose those functions as linear combinations of suitably…

Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the "native" Hilbert space $\calh$ in which they are reproducing. Continuous kernels on compact domains have an expansion into…

This paper is concerned with minimizing a sum of rational functions over a compact set of high-dimension. Our approach relies on the second Lasserre's hierarchy (also known as the upper bounds hierarchy) formulated on the pushforward…

We study integration and $L^2$-approximation of functions of infinitely many variables in the following setting: The underlying function space is the countably infinite tensor product of univariate Hermite spaces and the probability measure…

Let $\rho_{n,m}(f;E)$ denote the error of best uniform rational approximation to a function $f$ analytic on a compact set $E\subset \mathbb{C}$ by rational functions whose numerator and denominator have degrees at most $n$ and $m$,…

We propose a simple and effective method for designing approximation formulas for weighted analytic functions. We consider spaces of such functions according to weight functions expressing the decay properties of the functions. Then, we…

We study the approximate minimization problem of weighted finite automata (WFAs): given a WFA, we want to compute its optimal approximation when restricted to a given size. We reformulate the problem as a rank-minimization task in the…

Let $E_{kk}^{(n)}$ denote the minimax (i.e., best supremum norm) error in approximation of $x^n$ on $[\kern .3pt 0,1]$ by rational functions of type $(k,k)$ with $k<n$. We show that in an appropriate limit $E_{kk}^{(n)} \sim 2\kern .3pt…

Estimation of convex functions finds broad applications in engineering and science, while convex shape constraint gives rise to numerous challenges in asymptotic performance analysis. This paper is devoted to minimax optimal estimation of…

In this Thesis we deal with problems regarding boundary behavior of analytic functions and approximation theory. We will begin by characterizing the set in which Blaschke products fail to have radial limits but have unrestricted limits on…

By making a seminal use of the maximum modulus principle of holomorphic functions we prove existence of $n$-best kernel approximation for a wide class of reproducing kernel Hilbert spaces of holomorphic functions in the unit disc, and for…

For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple…

Over the last decade, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the…

We consider functions $\omega$ on the unit circle $\mathbb T$ with a finite number of logarithmic singularities. We study the approximation of $\omega$ by rational functions and find an asymptotic formula for the distance in the BMO-norm…