A new superfast O(n log n) complexity direct solver for real symmetric Toeplitz systems is presented. The algorithm is based on 1. the reduction to symmetric right-hand sides, 2. a polynomial interpretation in terms of Chebyshev polynomials, 3. an inversion formula involving real trigonometric transformations, and 4. an interpretation of the equations as a tangential interpolation problem. The ...

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 calculate the minimal surface bounded by four-sided figures whose projection on a plane is a rectangle, starting with the bilinear interpolation and using, for smoothness, the Chebyshev polynomial expansion in our discretized numerical algorithm to get closer to satisfying the zero mean curvature condition. We report values for both the bilinear and improved areas, suggesting a quantitative ...

Newton’s formula for constructing the interpolation polynomial is well–known. It makes use of divided differences. It was generalized around 1971–1973 by Mühlbach for interpolation by a linear family of functions forming a complete Chebyshev system. This generalization rests on a generalization of divided differences due to Popoviciu. In this paper, it is shown that Mühlbach’s formula is relate...

We study polynomial interpolation on a d-dimensional cube, where d is large. We suggest to use the least solution at sparse grids with the extrema of the Chebyshev polynomials. The polynomial exactness of this method is almost optimal. Our error bounds show that the method is universal, i.e., almost optimal for many different function spaces. We report on numerical experiments for d = 10 using ...

Interpolation was a topic in which Sharma was viewed as an almost uncontested world expert by his collaborators and many other colleagues. We survey recent results for exponential sums and linear combinations of shifted Gaussians which were obtained via interpolation. To illustrate the method exploiting the Pinkus-Smith Improvement Theorem for spans of Descartes systems, we present the proof of...

Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in [10]. The main results consist of a new derivation of the Gaussian type cubature for the product Chebyshev weight functions and associated interpolation polynomials on [−1, 1], as well as new results on [−1, 1]. In particular, compact formulas for the fu...

In this talk we present some results concerning the interpolation of discontinuous functions. First we consider an index of convergence associated to converging subsequences of a general sequence of real numbers (x n) n≥1. Namely, if L ∈ R, the index of convergence of (x n) n≥1 to L is defined by The definition can be extended to the cases L = ±∞ and also to the case where A ⊂ R. In the case wh...

Some new results on univariate interpolation by weak Cheby-shev spaces, using conditions of Schoenberg-Whitney type and the concept of almost interpolation sets, are given. x1. Introduction Let U denote a nite-dimensional subspace of real-valued functions deened on some set K. We are interested in describing those conngurations T = dimU jT = s: T is called an interpolation set (I-set) w.r.t. U....