• Corpus ID: 18178738

The Spectral Basis and Rational Interpolation

@article{Sobczyk2006TheSB,
  title={The Spectral Basis and Rational Interpolation},
  author={Garret Sobczyk},
  journal={arXiv: Numerical Analysis},
  year={2006}
}
  • G. Sobczyk
  • Published 18 February 2006
  • Mathematics
  • arXiv: Numerical Analysis
The Euclidean Algorithm is the often forgotten key to rational approximation techniques, including Taylor, Lagrange, Hermite, osculating, cubic spline, Chebyshev, Pade 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 the minimal polynomial of a matrix, they give a family of rational forms of functions of that matrix. 
View PDF on arXiv
18 Citations

Figures from this paper

18 Citations

Generalized Taylor's Theorem
The Euclidean algorithm makes possible a simple but powerful generalization of Taylor's theorem. Instead of expanding a function in a series around a single point, one spreads out the spectrum to…
Linear and Bilinear Forms
Geometric algebra is not used in this chapter. The material presented is closely related to the material in Sect. 7.1 but represents a change of viewpoint. Instead of talking about the reciprocal…
Differential Geometry of Curves
The study of the differential geometry of surfaces rightly begins with the study of curves, since a curve can be considered to be a 1-dimensional surface, and a more general surface can be considered…
Complex and Hyperbolic Numbers
The complex numbers were grudgingly accepted by Renaissance mathematicians because of their utility in solving the cubic equation.1 Whereas the complex numbers were discovered primarily for algebraic…
Vector Spaces and Matrices
We begin this chapter with the formal definition of a vector space. Multiplication of matrices is defined in terms of the product of a row vector and a column vector. Since the rules of matrix…
Mappings Between Surfaces
In this chapter we explore when a mapping will preserve geodesics, the shortest distances between two points on a surface, and when a mapping will preserve the angles between the intersection of two…
Linear Transformations on {\mathbb{R}}^{n}
The definition of a linear transformation on \({\mathbb{R}}^{n}\), and its natural extension to an outermorphism on all of the geometric algebra \({\mathbb{G}}_{n}\), is given. The tools of geometric…
Modular Number Systems
We begin by exploring the algebraic properties of the modular numbers, sometimes known as clock arithmetic, and the modular polynomials. The modular numbers and modular polynomials are based upon the…
Outer Product and Determinants
The outer product of vectors is closely related to the concept of a determinant. The outer products of vectors and determinants are very important tools in the study of the structure of a linear…
Systems of Linear Equations
We give a rather conventional treatment of the study of a system of linear equations by using the augmented matrix of the system. The so-called LU decomposition of a matrix is introduced, together…
...
...

References

SHOWING 1-10 OF 14 REFERENCES
The Missing Spectral Basis in Algebra and Number Theory
The justified fame of the Euclidean algorithm derives from the fact that it has a much larger realm of applicability than just the integers. In particular, let K be any field and let K[x] be the…
An Algebraic Approach to Convolutions and Transform Methods
We show that the Hopf algebra dual of the polynomials in one variable appears often in analysis, but under different disguises that include proper rational functions, exponential polynomials, shift…
A Hopf Algebra Structure on Rational Functions
Abstract We consider an indefinite inner product on the algebra of rational functions over the complex numbers, and we obtain a coproduct, which is dual of the usual multiplication, that gives a…
What Is the Role of Algebra in Applied
TLDR
Theorie der reducibeln ganzen Func-tionen von n Variabeln , Math.
Topics in Matrix Analysis
1. The field of values 2. Stable matrices and inertia 3. Singular value inequalities 4. Matrix equations and Kronecker products 5. Hadamard products 6. Matrices and functions.
Contemporary Abstract Algebra
Contemporary Abstract Algebra 7/e provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working…
Approximation and Optimization
Problem: Approximate f (x) by f a (x, w). Classical: f is known and f a is a restricted form. Learning: Only know a set of noisy points of f .
Generalized Vandermonde Determinants and Applications, Aportaciones Matematicas
  • Serie Comunicaciones
  • 2002
Introduction to numerical analysis: 2nd edition
Approximation and optimization : A. Gomez et al. (Eds), Lecture Notes in Mathematics, vol. 1354, Springer-Verlag, Berlin, 1988. XII + 277 pages, DM 42.50. ISBN 3-540-50443-5
...
...