# The Theory of Splines and Their Applications

## 1st Edition

### Mathematics in Science and Engineering: A Series of Monographs and Textbooks, Vol. 38

**Authors:**J. H. Ahlberg E. N. Nilson J. L. Walsh

**Editor:**Richard Bellman

**eBook ISBN:**9781483222950

**Imprint:**Academic Press

**Published Date:**1st January 1967

**Page Count:**296

## Description

The Theory of Splines and Their Applications discusses spline theory, the theory of cubic splines, polynomial splines of higher degree, generalized splines, doubly cubic splines, and two-dimensional generalized splines. The book explains the equations of the spline, procedures for applications of the spline, convergence properties, equal-interval splines, and special formulas for numerical differentiation or integration. The text explores the intrinsic properties of cubic splines including the Hilbert space interpretation, transformations defined by a mesh, and some connections with space technology concerning the payload of a rocket. The book also discusses the theory of polynomial splines of odd degree which can be approached through algebraically (which depends primarily on the examination in detail of the linear system of equations defining the spline). The theory can also be approached intrinsically (which exploits the consequences of basic integral relations existing between functions and approximating spline functions). The text also considers the second integral relation, raising the order of convergence, and the limits on the order of convergence. The book will prove useful for mathematicians, physicist, engineers, or academicians in the field of technology and applied mathematics.

## Table of Contents

Preface

Chapter I Introduction

1.1. What is a Spline?

1.2. Recent Developments in the Theory of Splines

Chapter II The Cubic Spline

2.1. Introduction

2.2. Existence, Uniqueness, and Best Approximation

2.3. Convergence

2.4. Equal Intervals

2.5. Approximate Differentiation and Integration

2.6. Curve Fitting

2.7. Approximate Solution of Differential Equations

2.8. Approximate Solution of Integral Equations

2.9. Additional Existence and Convergence Theorems

Chapter III Intrinsic Properties of Cubic Splines

3.1. The Minimum Norm Property

3.2. The Best Approximation Property

3.3. The Fundamental Identity

3.4. The First Integral Relation

3.5. Uniqueness

3.6. Existence

3.7. General Equations

3.8. Convergence of Lower-Order Derivatives

3.9. The Second Integral Relation

3.10. Raising the Order of Convergence

3.11. Convergence of Higher-Order Derivatives

3.12. Limits on the Order of Convergence

3.13. Hilbert Space Interpretation

3.14. Convergence in Norm

3.15. Canonical Mesh Bases and their Properties

3.16. Remainder Formulas

3.17. Transformations Defined by a Mesh

3.18. A Connection with Space Technology

Chapter IV The Polynomial Spline

4.1. Definition and Working Equations

4.2. Equal Intervals

4.3. Existence

4.4. Convergence

4.5. Quintic Splines of Deficiency 2, 3

4.6. Convergence of Periodic Splines on Uniform Meshes

Chapter V Intrinsic Properties of Polynomial Splines of Odd Degree

5.1. Introduction

5.2. The Fundamental Identity

5.3. The First Integral Relation

5.4. The Minimum Norm Property

5.5. The Best Approximation Property

5.6. Uniqueness

5.7. Defining Equations

5.8. Existence

5.9. Convergence of Lower-Order Derivatives

5.10. The Second Integral Relation

5.11. Raising the Order of Convergence

5.12. Convergence of Higher-Order Derivatives

5.13. Limits on the Order of Convergence

5.14. Hilbert Space Interpretation

5.15. Convergence in Norm

5.16. Canonical Mesh Bases and their Properties

5.17. Kernels and Integral Representations

5.18. Representation and Approximation of Linear Functionals

Chapter VI Generalized Splines

6.1. Introduction

6.2. The Fundamental Identity

6.3. The First Integral Relation

6.4. The Minimum Norm Property

6.5. Uniqueness

6.6. Defining Equations

6.7. Existence

6.8. Best Approximation

6.9. Convergence of Lower-Order Derivatives

6.10. The Second Integral Relation

6.11. Raising the Order of Convergence

6.12. Convergence of Higher-Order Derivatives

6.13. Limits on the Order of Convergence

6.14. Hilbert Space Interpretation

6.15. Convergence in Norm

6.16. Canonical Mesh Bases

6.17. Kernels and Integral Representations

6.18. Representation and Approximation of Linear Functionals

Chapter VII The Doubly Cubic Spline

7.1. Introduction

7.2. Partial Splines

7.3. Relation of Partial Splines to Doubly Cubic Splines

7.4. The Fundamental Identity

7.5. The First Integral Relation

7.6. The Minimum Norm Property

7.7. Uniqueness and Existence

7.8. Best Approximation

7.9. Cardinal Splines

7.10. Convergence Properties

7.11. The Second Integral Relation

7.12. The Direct Product of Hilbert Spaces

7.13. The Method of Cardinal Splines

7.14. Irregular Regions

7.15. Surface Representation

7.16. The Surfaces of Coons

Chapter VIII Generalized Splines in Two Dimensions

8.1. Introduction

8.2. Basic Definition

8.3. The Fundamental Identity

8.4. Types of Splines

8.5. The First Integral Relation

8.6. Uniqueness

8.7. Existence

8.8. Convergence

8.9. Hilbert Space Theory

Bibliography

Index

## Details

- No. of pages:
- 296

- Language:
- English

- Copyright:
- © Academic Press 1967

- Published:
- 1st January 1967

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483222950

## About the Authors

### J. H. Ahlberg

### E. N. Nilson

### J. L. Walsh

## About the Editor

### Richard Bellman

### Affiliations and Expertise

Departments of Mathematics, Electrical Engineering, and Medicine University of Southern California Los Angeles, California