Approximation Theory

From Taylor Polynomials to Wavelets

verfasst von: Ole Christensen, Khadija L. Christensen

Verlag: Birkhäuser Boston

Buchreihe : Applied and Numerical Harmonic Analysis

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book gives an elementary introduction to a classical area of mathemat ics - approximation theory - in a way that naturally leads to the modern field of wavelets. The main thread throughout the book is the idea of ap proximating "complicated expressions" with "simpler expressions," and how this plays a decisive role in many areas of modern mathematics and its applications. One of the main goals of the presentation is to make it clear to the reader that mathematics is a subject in a state of continuous evolution. This fact is usually difficult to explain to students at or near their second year of uni versity. Often, teachers do not have adequate elementary material to give to students as motivation and encouragement for their further studies. The present book will be of use in this context because the exposition demon strates the dynamic nature of mathematics and how classical disciplines influence many areas of modern mathematics and applications. The book may lead readers toward more advanced literature, such as the other pub lications in the Applied and Numerical Harmonic Analysis series (ANHA), by introducing ideas presented in several of those books in an elementary context. The focus here is on ideas rather than on technical details, and the book is not primarily meant to be a textbook.

Inhaltsverzeichnis

Frontmatter

1. Approximation with Polynomials

Abstract

In many applications of mathematics, we face functions which are far more complicated than the standard functions from classical analysis. Some of these functions can not be expressed in closed form via the standard functions, and some are only known implicitly or via their graph. Think, for example, of an electric circuit, where we are measuring the current at a certain point as a function of time: the outcome might be quite complicated, and best described via a graph.

Ole Christensen, Khadija L. Christensen

2. Infinite Series

Abstract

Motivated by (1.15) at the end of the previous chapter, we wish to define an infinite sum of power functions; however, before we can do so, we need to define an infinite sum of real (or complex) numbers. That is, our first task will be to consider an infinite sequence of numbers a₁, a₂,..., a_n,..., and examine when and how we can make sense of the sum

$${a_1} + {a_2} + {a_3} + \cdots + {a_n} + \cdots $$

Ole Christensen, Khadija L. Christensen

3. Fourier Analysis

Abstract

So far, we have mainly been dealing with power series representations, although we already saw the definition of an infinite series of more general functions. Unfortunately, only a relatively limited class of functions has a power series expansion, so often we need to seek other tools to represent functions.

Ole Christensen, Khadija L. Christensen

4. Wavelets and Applications

Abstract

In Section 2.5 we defined the sum function associated to a given family of functions f₀, f_l,... defined on the same set. However, in practice sum functions frequently appear in a different way: a certain class of functions is given, and we want to find “simple functions” f₀, f_l,... such that each function f in the class has an expansion

$$f\left( x \right) = \sum\limits_{n = 0}^\infty {{a_n}} {f_n}\left( x \right)$$

(4.1)

for some coefficients a_n. We note that this idea is similar to what we have seen in the context of power series and Fourier series: these cases correspond to the functions f_n being polynomials or trigonometric functions, respectively.

Ole Christensen, Khadija L. Christensen

5. Wavelets and their Mathematical Properties

Abstract

In this chapter we concentrate on the mathematical properties of wavelets, but still with more weight on an intuitive understanding than on technical details.

Ole Christensen, Khadija L. Christensen

Backmatter

Titel: Approximation Theory
verfasst von: Ole Christensen
Khadija L. Christensen
Verlag: Birkhäuser Boston
Electronic ISBN: 978-0-8176-4448-2
Print ISBN: 978-0-8176-3600-5
DOI: https://doi.org/10.1007/978-0-8176-4448-2