The Euclidean Algorithm is the often forgotten key to rational approximation techniques, including Taylor, Lagrange, Hermite, osculating, cubic spline, Chebyshev, Padé and other interpolation schemes. A unified view of these various interpolation techniques is eloquently expressed in terms of the concept of the spectral basis of a factor ring of polynomials. When these methods are applied to th...

We give a simple, geometric and explicit construction of bivariate interpolation at certain points in a square (called Padua points), giving compact formulas for their fundamental Lagrange polynomials. We show that the associated norms of the interpolation operator, i.e., the Lebesgue constants, have minimal order of growth of O((log n)2). To the best of our knowledge this is the first complete...

A t-out-of-n secret-sharing scheme allows an honest dealer D to distribute a secret s among n players, such that any subset of t players has no information about s, but every set of t + 1 players can collaboratively reconstruct the secret. The most famous secret-sharing scheme is Shamir’s Sharing Scheme [Sha79] (cf. Section 8.1.2). It uses polynomials to obtain the desired properties. Before pr...

f(xj) = p(xj), j = 1, . . . , n. This is sometimes called Lagrange interpolation to distinguish it from Hermite interpolation, see below. For the basics, see Chapter 7 in Heath. As in Heath, we let Pn be the (function) space of polynomials of degree at most n. We let C([a, b]) denote the set of continuous functions on the interval [a, b]. Similarly, we let C([a, b]) be the n times continuously ...

Several representations for the interpolating polynomial exist: Lagrange, Newton, orthogonal polynomials etc. Each representation is characterized by some basis functions. In this paper we investigate the transformations between the basis functions which map a specific representation to another. We show that for this purpose the LU and the QR decomposition of the Vandermonde matrix play a cruci...

• Asymptotics of the Lebesgue constant for Lagrange interpolation based on Lissajous-Chebyshev node points. partial sums Fourier series generated by anisotropically dilated rhombus. Formula Dirichlet kernel with frequencies in In this paper asymptotic formulas are given constants three special approximation processes related to ? 1 -partial series. particular, we consider polynomials points, rh...

Throughout this study, we continue the analysis of a recently found out Gibbs–Wilbraham phenomenon, being related to behavior Lagrange interpolation polynomials continuous absolute value function. Our study establishes error polynomial interpolants function |x| on [−1,1], using Chebyshev and Chebyshev–Lobatto nodal systems with an even number points. Moreover, respect odd cases, relevant change...

Space-time nite element methods for dynamic Signorini problems are discussed in this article. The discretization scheme is based on a mixed space-time formulation of the continuous problem, where the Lagrange multipliers represent the contact stress. To construct the trial space for the displacement and the velocity, we use piecewise polynomial and globally continuous basis functions in space a...