# The Spectral Basis and Rational Interpolation

@article{Sobczyk2006TheSB, title={The Spectral Basis and Rational Interpolation}, author={Garret Sobczyk}, journal={arXiv: Numerical Analysis}, year={2006} }

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.

## Figures from this paper

## 18 Citations

Generalized Taylor's Theorem

- Mathematics
- 2007

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

- Mathematics
- 2013

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

- Mathematics
- 2013

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

- Mathematics
- 2013

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

- Mathematics
- 2013

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

- Mathematics
- 2013

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}

- Mathematics
- 2013

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

- Mathematics
- 2013

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

- Mathematics
- 2013

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

- Mathematics
- 2013

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

- MathematicsAm. Math. Mon.
- 2001

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

- Mathematics
- 1997

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

- Mathematics
- 1995

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

- Mathematics
- 2005

Theorie der reducibeln ganzen Func-tionen von n Variabeln , Math.

Topics in Matrix Analysis

- Mathematics
- 1991

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

- Physics
- 1986

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

- Mathematics
- 1988

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