Related papers: Approximation by random complex polynomials and ra…
A landmark result from rational approximation theory states that $x^{1/p}$ on $[0,1]$ can be approximated by a type-$(n,n)$ rational function with root-exponential accuracy. Motivated by the recursive optimality property of Zolotarev…
A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a…
We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We then present a simple application, related to possible correlations between trace…
We study the close connection between rational functions that approximate a given Boolean function, and quantum algorithms that compute the same function using postselection. We show that the minimal degree of the former equals (up to a…
This paper presents a novel method for generating a single polynomial approximation that produces correctly rounded results for all inputs of an elementary function for multiple representations. The generated polynomial approximation has…
This paper describes a novel method to approximate the polynomial coefficients of regression functions, with particular interest on multi-dimensional classification. The derivation is simple, and offers a fast, robust classification…
In an isomorphic copy of the ring of symmetric polynomials we study some families of polynomials which are indexed by rational weight vectors. These families include well known symmetric polynomials, such as the elementary, homogeneous, and…
An extensive table of pairs of functions linked by the Legendre transformation is presented. Many special functions and formulas that are used in the sciences are included in the pairs. Formulations are provided for finding the Legendre…
In this chapter are given necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials. In addition, we also present the functional Rodrigues…
This expository article proves some results of Ferguson, on the approximation of continuous functions on a compact subset of R by polynomials with integral coefficients.
We introduce polystar bodies: compact starshaped sets whose gauge or radial functions are expressible by polynomials, enabling tractable computations, such as that of intersection bodies. We prove that polystar bodies are uniformly dense in…
For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd…
One of the main applications of free probability is to show that for appropriately chosen independent copies of $d$ random matrix models, any noncommutative polynomial in these $d$ variables has a spectral distribution that converges…
This is a survey article on selected topics in approximation theory. The topics either use techniques from the theory of several complex variables or arise in the study of the subject. The survey is aimed at readers having an acquaintance…
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…
We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using…
In this survey, we use (more or less) elementary means to establish the well-known result that for any given smooth multivariate function, the respective multivariate Bernstein polynomials converge to that function in all derivatives on…
Approximation theory is concerned with the ability to approximate functions by simpler and more easily calculated functions. The first question we ask in approximation theory concerns the {\it possibility of approximation}. Is the given…
A selection of the relevant theorems of Probability Theory that comes directly from Kolmogorov's axioms, Set Theory basic results, definitions and rules of inference are listed and proven in a systematic approach, aiming the student who…
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…