Tomas Sauer - Continued Fractions and Signal Processing
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- Book:Continued Fractions and Signal Processing
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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, Pronys problem of sparse recovery and the design of stable rational filters, which are all connected by continued fractions.
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Springer Undergraduate Texts in Mathematics and Technology (SUMAT) publishes textbooks aimed primarily at the undergraduate. Each text is designed principally for students who are considering careers either in the mathematical sciences or in technology-based areas such as engineering, finance, information technology and computer science, bioscience and medicine, optimization or industry. Texts aim to be accessible introductions to a wide range of core mathematical disciplines and their practical, real-world applications; and are fashioned both for course use and for independent study.
More information about this series at http://www.springer.com/series/7438
This Springer imprint is published by the registered company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Dedicated to the memory of my thesis advisor Hubert Berens who was a Doktorvater in the best meaning of the word.
and 
stand for, and that these two expressions actually mean the same rational number, although they do not represent the same meter in dance music. For the ancient Egyptians on the other hand, these fractions were not really what they wanted; they preferred to deal with pure reciprocals or unit fractions like 
or 
. The German word for that is Stammbruch; Googles automatic translation Stem break is a hint that artificial intelligence still has the potential for improvement. In Egyptian mathematics, fractions were actually written as sums of unit fractions and there even existed tables for such decompositions [53]. In this spirit, we (nowadays) consider 
with 
and write this fraction as 
; the numerator is larger than the denominator. This idea defines a recursive process and allows us to write each fraction formally as 
for some 
, at least if trivial ambiguities are excluded. Therefore, the embarrassment of multiple representations such as 
is avoided here. Moreover, any real number can be expressed, again uniquely, by an infinite continued fraction expansion 
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