Continued Fractions and Signal Processing

✍ Scribed by Tomas Sauer

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 275
• Series: Springer Undergraduate Texts in Mathematics and Technology
• Edition: 1
• Category: Library

✦ Synopsis

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

Preface
Acknowledgements
Contents
1 Continued Fractions and What Can Be Done with Them
1.1 The First and Fundamental Definition
1.2 Continued Fractions with Polynomials in Them
1.3 Continued Fractions and Moments
1.4 Continued Fractions and Prony
1.5 Digital Signal Processing
1.6 Zeros of Polynomials and Hurwitz
1.7 And What Else?
2 Continued Fractions of Real Numbers
2.1 Convergents and Continuants
2.1.1 Problems
2.2 Infinite Continued Fractions and Their Convergence
2.2.1 Problems
2.3 Continued Fractions with Integer Coefficients
2.3.1 Problems
2.4 Convergents as Best Approximants
2.5 Approximation Order, Quantitative Statements
2.5.1 Problems
2.6 Algebraic Numbers
2.6.1 Problems
2.7 Continued Fractions and Music
2.8 Problems
3 Rational Functions as Continued Fractions of Polynomials
3.1 A Starting Point with Some New Notation …
3.1.1 Problems
3.2 Euclidean Rings and Continued Fractions
3.2.1 Problems
3.3 Continued Fractions and the Extended Euclidean Algorithm
3.3.1 Problems
3.4 One Result of One Bernoulli
3.4.1 Problems
3.5 Power Series and Euler's Continued Fractions
3.5.1 Problems
3.6 Padé Approximation
3.6.1 Problems
4 Continued Fractions and Gauss
4.1 Orthogonal Polynomials, Continued Fractions and Moments
4.1.1 Problems
4.2 Gauss Quadrature
4.2.1 Problems
4.3 Sturm Chains
4.3.1 Problems
4.4 Computing the Zeros of Polynomials
4.4.1 Problems
5 Continued Fractions and Prony
5.1 Prony's Problem
5.1.1 Problems
5.2 Flat Extensions of Moment Sequences
5.2.1 Problems
5.3 Flat Extensions via Prony
5.3.1 Problems
6 Digital Signal Processing
6.1 Signals and Filters
6.1.1 Problems
6.2 Fourier and Sampling
6.2.1 Problems
6.3 Realization of Filters
6.4 Rational Filters and Stability
6.4.1 Problems
6.5 Stability of Difference Equations
6.5.1 Problems
6.6 Superresolution via Continued Fractions and a Determinant Identity
6.6.1 Problems
7 Continued Fractions, Hurwitz and Stieltjes
7.1 The Problems
7.2 Prelude: Zeros of Polynomials
7.2.1 Problems
7.3 Hurwitz Polynomials and Stieltjes' Theorem
7.4 Cauchy Index and the Argument of the Argument
7.4.1 Problems
7.5 The Routh–Hurwitz Theorem
7.5.1 Problems
7.6 The Routh Scheme or the Return of Sturm's Chains
7.6.1 Problems
7.7 Markov Numbers and Back to Moments
7.7.1 Problems
7.8 Hurwitz Polynomials and Total Positivity
7.8.1 Problems
Appendix References
Index