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2021 | Book

Continued Fractions and What Can Be Done with Them Read chapter Read first chapter

Continued Fractions and Signal Processing

Author: Prof. Tomas Sauer

Publisher: Springer International Publishing

Book Series : Springer Undergraduate Texts in Mathematics and Technology

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Besides their well-known value in number theory, continued fractions are also a useful tool in modern numerical applications and computer science. The goal of the book is to revisit the almost forgotten classical theory and to contextualize it for contemporary numerical applications and signal processing, thus enabling students and scientist to apply classical mathematics on recent problems. The books tries to be mostly self-contained and to make the material accessible for all interested readers. This provides a new view from an applied perspective, combining the classical recursive techniques of continued fractions with orthogonal problems, moment problems, Prony’s problem of sparse recovery and the design of stable rational filters, which are all connected by continued fractions.

Table of Contents

Frontmatter
Chapter 1. Continued Fractions and What Can Be Done with Them
Abstract
We begin with a less formal introduction to continued fractions and the underlying concepts to be used in this book. The goal of this chapter is to illustrate and highlight the main aspects of the topics that will be considered. This will hopefully also serve as a motivation to struggle with unavoidable details that will occur later, and aims to provide a guideline for the ideas, concepts and results ahead.
Tomas Sauer
Chapter 2. Continued Fractions of Real Numbers
Abstract
The most classical use of continued fractions is to describe rational numbers and then to approximate real numbers using continued fractions whose coefficients are nonnegative integers. After introducing the concept of convergents and their fundamental recurrence relations, we study real numbers as infinite continued fractions whose convergence issues will be resolved. The main advantage of continued fractions of an irrational number is that they provide the best approximations in a quantifiable sense, and that this approximation behavior even helps to distinguish the algebraic from the transcendental numbers. Moreover, continued fractions have a close relationship to musical harmony which will be pointed out as well.
Tomas Sauer
Chapter 3. Rational Functions as Continued Fractions of Polynomials
Abstract
In this chapter we switch from integers to the second most popular Euclidean Ring, namely univariate polynomials. Starting with the straightforward extension of the concept of continued fractions to arbitrary Euclidean rings, we specialize to polynomials and rational functions as well as the associated infinite objects which will be the formal Laurent series. Asking for the general existence of a representation of sequences and series using continued fractions, we will encounter fundamental and classical results by D. Bernoulli and Euler.
Tomas Sauer
Chapter 4. Continued Fractions and Gauss
Abstract
Based on continued fraction expansions for the formal Laurent series, we develop the connection between continued fractions and orthogonal polynomials in detail, and show the equivalence of orthogonal polynomials with a special class of continued fractions. In particular, this theory will reveal Gauss’ original proof for the construction of Gaussian quadrature rules and the role continued fractions played there. In addition, we will see the connection to moment problems, wherein families of polynomials of increasing degrees, with interlacing simple real zeros, can easily be generated by the recurrence relation of continued fractions.
Tomas Sauer
Chapter 5. Continued Fractions and Prony
Abstract
In modern terminology Prony’s problem consists of recovering a sparse exponential sum from integer samples. The interesting nonlinear part of this problem is in detecting the frequencies of the exponentials, which is done by considering Hankel matrices formed from the samples. Analyzing this method, we discover a striking equivalence between Hankel operators of finite rank, exponential (polynomial) sequences, and rational functions. This allows us to factorize the Hankel operators in a very natural way and to identify the Gauss quadrature formulas from the preceding chapter as flat extensions of the truncated moment sequence. This in turn is the same as obtaining a rank preserving extension of a Hankel matrix into an infinite Hankel operator. Furthermore the theory of this chapter allows us to also consider signed measures, which usually lead to indefinite inner products and functionals lacking square positivity.
Tomas Sauer
Chapter 6. Digital Signal Processing
Abstract
Digital signal processing deals with the study of how discrete signals, usually modeled as bi-infinite sequences, can be handled theoretically and practically. Such signals arise from the uniform sampling of continuous signals. The main concept of Signal Processing, the so-called LTI filter, can be implemented physically while mathematically it is represented as a convolution; therefore it is a well-known and classical mathematical object that fits well into the framework here. A more advanced concept are rational filters which contain an additional feedback loop that causes issues of stability and in turn relates to the location of the poles of the rational function that represents the filter. This chapter summarizes the necessary background in signal processing, including the main concepts of sampling, z-transforms and filters. Then we relate these fundamental concepts to difference equations. We also find relations to Prony’s problem in the context of sparse recovery.
Tomas Sauer
Chapter 7. Continued Fractions, Hurwitz and Stieltjes
Abstract
In the chapter on Signal Processing we found out that the stability of filters or difference equations is equivalent to the fact that certain polynomials have all their zeros inside the unit circle. In this chapter we consider polynomials with such restrictions. With a simple change of parameters, we see that polynomials with zeros inside the unit circle are equivalent to polynomials that have their zeros located in the left half-plane. This is what is known as Hurwitz polynomials, and they have once again close relations to continued fractions. Indeed a classical theorem by Stieltjes will characterize the Hurwitz polynomials using continued fractions which will also have a description using certain Hankel matrices and operators.
Tomas Sauer
Backmatter
Metadata
Title
Continued Fractions and Signal Processing
Author
Prof. Tomas Sauer
Copyright Year
2021
Electronic ISBN
978-3-030-84360-1
Print ISBN
978-3-030-84359-5
DOI
https://doi.org/10.1007/978-3-030-84360-1

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