Related papers: Optimal one-sided approximants of circular arc
Let G be a graph that may be drawn in the plane in such a way that all internal faces are centrally symmetric convex polygons. We show how to find a drawing of this type that maximizes the angular resolution of the drawing, the minimum…
In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the…
For interior-point algorithms in linear programming, it is well-known that the selection of the centering parameter is crucial for proving polynomility in theory and for efficiency in practice. However, the selection of the centering…
The purpose of the paper is to provide a characterization of the error of the best polynomial approximation of composite functions in weighted spaces. Such a characterization is essential for the convergence analysis of numerical methods…
Variational data assimilation technique applied to identification of optimal approximations of derivatives near boundary is discussed in frames of one-dimensional wave equation. Simplicity of the equation and of its numerical scheme allows…
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…
We estimate the number of zeros of a polynomial in $\mathbb{C}[z]$ within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erd{\'e}lyi, and Littmann~\cite{BE1}…
It is natural to expect the following loosely stated approximation principle to hold: a numerical approximation solution should be in some sense as smooth as its target exact solution in order to have optimal convergence. For piecewise…
Consider a rational family of planar rational curves in a certain region of interest. We are interested in finding an approximation to the implicit representation of the envelope. Since exact implicitization methods tend to be very costly,…
We study the approximation of maps into complex manifolds along with interpolation on certain compact subsets of the plane. Results are also obtained regarding approximation and interpolation of sections of holomorphic submersions.
We consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input--output pairs of an unknown…
The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…
In a previous paper with the same title, we gave an upper bound for the exponent of uniform rational approximation to a quadruple of $\mathbb{Q}$-linearly independent real numbers in geometric progression. Here, we explain why this upper…
Here we establish several results on the nonlocal curvature of planar curves. First we show how to express the nonlocal curvature of a curve relative to a point in terms of the nonlocal curvatures of simpler components of that curve…
We consider the numerical approximation of variational problems with orthotropic growth, that is those where the integrand depends strongly on the coordinate directions with possibly different growth in each direction. Under realistic…
In areas such as kernel smoothing and non-parametric regression there is emphasis on smooth interpolation and smooth statistical models. Splines are known to have optimal smoothness properties in one and higher dimensions. It is shown, with…
We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of simultaneous rational interpolants with a bounded number of poles. The conditions are expressed in terms of intrinsic…
This paper is our third step towards developing a theory of testing monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear monomials; and (2) how to find a maximum…
We present the first formulation of the optimal polynomial approximation of the solution of linear non-autonomous systems of ODEs in the framework of the so-called $\star$-product. This product is the basis of new approaches for the…
Considering simultaneous approximation to three numbers, we study the geometry of the sequence of best approximations. We provide a sharper lower bound for the ratio between ordinary and uniform exponent of Diophantine approximation,…