Continued Fractions and Signal Processing

275 Pages·2021·3.759 MB·English

by  Tomas Sauer

Springer Undergraduate Texts
in Mathematics and Technology
Tomas Sauer
Continued
Fractions
and Signal
Processing
Springer Undergraduate Texts in Mathematics
and Technology
Series Editors
Helge Holden, Department of Mathematical Sciences, Norwegian University of
Science and Technology, Trondheim, Norway
KeriA.Kornelson,DepartmentofMathematics,UniversityofOklahoma,Norman,
OK, USA
Editorial Board
Lisa Goldberg, Department of Statistics, University of California, Berkeley,
Berkeley, CA, USA
Armin Iske, Department of Mathematics, University of Hamburg, Hamburg,
Germany
Palle E.T. Jorgensen, Department of Mathematics, University of Iowa, Iowa City,
IA, USA
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
Tomas Sauer
Continued Fractions
and Signal Processing
123
TomasSauer
UniversitätPassau
Passau, Germany
ISSN 1867-5506 ISSN 1867-5514 (electronic)
SpringerUndergraduate Textsin MathematicsandTechnology
ISBN978-3-030-84359-5 ISBN978-3-030-84360-1 (eBook)
https://doi.org/10.1007/978-3-030-84360-1
MathematicsSubjectClassification: 11A55,40A15,65D32
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature
SwitzerlandAG2021
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof
illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and
transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, expressed or implied, with respect to the material contained
hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard
tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Dedicated to the memory of my thesis advisor
Hubert Berens who was a “Doktorvater” in
the best meaning of the word.
Preface
Fractionsarestrangeanimals.Nowadays,everyoneisoratleastshouldbeawareof
what 3 and 6 stand for, and that these two expressions actually mean the same
4 8
rationalnumber,althoughtheydonotrepresentthesamemeterindancemusic.For
the ancient Egyptians on the other hand, these fractions were not really what they
wanted;theypreferredtodealwithpurereciprocalsorunitfractionslike1or1.The
5 9
GermanwordforthatisStammbruch;Google’sautomatictranslationStembreakis
ahintthatartificialintelligencestillhasthepotentialforimprovement.InEgyptian
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 a with a[b and write this fraction as
b
a c 1
¼a þ ¼a þ ;
b 0 b 0 b
c
where the fraction appearing in the denominator of the unit fraction on the
right-hand side is again similar to the type a; the numerator is larger than the
b
denominator. This idea defines a recursive process and allows us to write each
fraction formally as
a 1
¼a þ ¼:½a ;a ;a ;...(cid:2):
b 0 ½a ;a ;...(cid:2) 0 1 2
1 2
This is the definition of a continued fraction . It turns out that any rational
number, i.e., any fraction, has exactly one finite representation of the form
½a ;a ;...;a (cid:2)forsomen,atleastiftrivialambiguitiesareexcluded.Therefore,the
0 1 n
embarrassmentofmultiplerepresentationssuchas3¼6isavoidedhere.Moreover,
4 8
anyrealnumbercanbeexpressed,againuniquely,byaninfinitecontinuedfraction
expansion ½a ;a ;a ;...(cid:2); most prominently known is the famous golden ratio
0 1 2
vii
viii Preface
pffiffiffi
1þ 5
¼½1;1;1;...(cid:2);
2
whichwewillidentifyasthemostirrationalnumberpreciselyduetothefactthatit
has the simplest possible continued fraction expansion. According to [61] who in
turnrefersto[3],thiswasthewaytheancientGreeksintroducedrealnumbersonce
pffiffiffi pffiffiffi
theyrealizedthat 2¼½1;2;2;...(cid:2)isnotrational,thatis, 2cannotbewrittenas
a fraction p=q with p2Z and q2N.
Even after these very simple observations it should not come as a big surprise
any more that continued fractions play a fundamental role in Number Theory and
even in music. This will be the content of the first major chapter of this book. But
onceonedealswithrationalnumbersandstartstothinkbeyondnumbers,onemay
be tempted to also consider more general rational objects, in particular rational
functions in the sense of quotients of polynomials. Continued fractions with
polynomials in them were an important topic in the eighteenth and nineteenth
centuries with major contributions by Bernoulli, Euler, Gauss and many more.
Their contribution will occupy the major part of this book and will relate to con-
cepts like quadrature, moment problems and sparse recovery which all appear, not
quite accidentally in connection with Signal Processing, which will make these
classicalmathematicaltoolsrelevantformodernapplicationsaswell.Notthatthey
have ever been irrelevant, just to record that.
The last topic is to describe polynomials which have zeros that are restricted to
certainlocations,eitheroutsidetheunitcircleorinthelefthalf-plane.Thelatter,the
so-called Hurwitz polynomials have a really nice characterization in terms of
continued fractions.
Prerequisites and Recommended Reading
Thisbookemergedfromanintroductorylectureoncontinuedfractions,andstillis
supposed to act as elaborated lecture notes that view continued fractions from a
SignalProcessingperspectiveandfocusontheirroleinnumericalapplications.To
thatend,itstartswithanintroductorychapterthatsummarizesthemainconceptsto
beconsideredandoutlineswhatwillhappeninthechapterstofollowwhereresults
are worked out together with all the necessary details of sometimes a technical
nature.Thepresentationaimsatbeingmostlyself-containedandshouldnotrelyon
toomuchpreviousknowledge,exceptforthestandardbasicknowledgeinAnalysis
and Linear Algebra. Some more complex detailed aspects or material that is not
centraltothemainthemeandthereforedidnotmakeitintothepresentation,butare
providedintheexercisessothereadercanelaborateonthem.Theproblemsshould
be structured enough to provide sufficient information on the intermediate steps,
thereby helping to work out these additional facts and complete the theory.
Preface ix
Some more specific remarks on the requirements needed for the individual
chapters might be helpful for using this book for lectures or seminars. Of course
Chap. 1 only surveys things to come and requires only curiosity from the readers’
side.
Chapter2isalmostcompletelyelementaryandrequiresnotmuchmorethanthe
skill to manipulate fractions, to follow a “proof by induction” and to compute a
derivativeofpolynomialsorexponentialfunctionsonceinawhile.Thepricetobe
paidforthatisSect.2.6whichdealswithalgebraicnumbersandonlyremainsvery
superficial. Here, it may be useful to have a look at [45] where, for example, also
the transcendence of p and e are proved in part even by using continued fractions.
For a deeper connection between math and music that could make a seminarof its
own, the first choice is still [5], but also [82] and [69] are highly recommended.
Chapter 3 makes use of some basic concepts of algebra, especially Euclidean
rings. Any standard literature on algebraic structures is more than sufficient here
and some acquaintance with these concepts is not really necessary, only helpful.
Due to the algorithmic nature of the chapter, [31] is a good choice for further
reading in case some things are not clear.
Chapter 4 is mostly driven by explaining “Gauss’ quadrature”, i.e., the original
approach by Gauss for developing what is nowadays known as Gaussian quadra-
ture. The quadrature problem is treated in all classics in Numerical Analysis, for
example[54].Forfurtherbackground,Iwouldrecommend[33]whichwaswritten
byanexpertinquadratureandorthogonalpolynomials.Althoughthingsbecomea
bit more intricate and tricky in that chapter, everything is still mostly elementary
and should be understandable from the presentation.
Chapter 5 uses the Dirac distribution which is popular in the engineering liter-
ature and on Wikipedia pages where it often is used like a function which fre-
quently leads to incorrect proofs. Basic facts on distributions are pointed out in
[114] in a very condensed way; a more substantial presentation to understand
generalized functions isto view functions asfunctionalsas provided in [34], but it
isreallynotnecessaryhere.Whatisimportantisanawarenessthatdistributionsare
beautifulbuttreacherous.Again,therestofthechapterisfairlyelementary;infinite
matrices and operators on sequence spaces work in a straightforward extension
ofthefinitecase.Somebasicknowledge,however,inFunctionalAnalysis,see,for
example [63], could be useful.
Chapter 6 introduces basic concepts of mathematical signal processing, still
basedontheclassicslike[44].Usefuladditionalreading,especiallyonallpassand
wavelets would be [106]; and of course [71]. This additional seconding will be
helpful to obtain a more complete image of the underlying theory than what is
brieflysurveyedinChap.6;buttheknowledgewillbeespeciallyusefulforworking
out seminar talks based on this chapter.
Chapter7usesafundamentalconceptinFunctionTheory,namelytheso-called
argument principle which will be recalled in Theorem 7.4. It is a consequence
of the residue theorem, hence it is located at the end of a one-semester lecture on
Function Theory, and can be found, e.g., in [25, 50, 100]. Its main application is
identity (7.11) and if one is willing to “believe” this identity, one can restrict the
x Preface
needed Complex Analysis to elementary manipulations of complex numbers that
should be known from calculus/analysis. The other concepts in this chapter are
provided and nicely introduced in the first chapters of [27].
Literature on Continued Fractions
The literature on continued fractions is not overly huge, but definitely existent.
Besides the classic, Perron’s two volumes [78, 79], one has to mention Khinchin
[60]astheotherextreme;whilePerronistrulyencyclopedic,coversalotofissues,
he is not so easy to read (even besides the fact that the books are in German),
Khinchin’sbookisthin,butaverywell-writtenintroductiontocontinuedfractions.
Even if some material has been added meanwhile and some of the proofs were
modified, it still may be recognized as the guideline for Chap. 2. Another classic
bookwherebasicsoncontinuedfractionscanbefound,ofcourseinthecontextof
Number Theory, is the one by Hardy and Wright [45]. The special application to
musicfromSect.2.7iscoveredinalmostallbooksonMathematicalMusicTheory,
from the highly recommended and mathematically substantial masterpiece by
Benson [5] to [23, 82]. Their connection is also mentioned more from the per-
spectiveofComputerAlgebrain[31]and[61].ApologiestoallIforgottomention
here. Applications in Analysis similar to what we consider in later chapters were
already treated in [107] which is even older than Perron’s work; and in [68].
Connectionstoorthogonalpolynomialscanbefoundin[16].Thestatisticalcontext
is presented in greater detail in [10].
There are also quite recent books on the issue: Hensley [48] covers Number
Theory, Ergodic Theory and complex numbers, and a very careful overview
especially about number-theoretic aspects is also provided in [64]; geometric
aspectsandevengeometry-basedextensionstothemultivariatecasearepointedout
in[57].TheroleofcontinuedfractionsinNumberTheoryisonceagainconsidered
in quite some depth in [9] which combines classical with more modern aspects.
Finally, a more educational approach from a somewhat recreational perspective is
provided in [101].
Some Personal Remarks
Myfirstpersonalencounterwithcontinuedfractionshappenedbyaccidentandwas
totallyunintendedasIlookback.WorkingasateachingassistantattheUniversity
ofErlangen,I was responsiblefor theexercisesinthe Numerical Analysisclass of
my boss and thesis supervisor Hubert Berens. When he arrived to teach Gaussian
quadrature,hetoldmethathehadread,probablyinChihara’sbook[16],thatGauss
himselfhadnotusedorthogonalpolynomials(whichwereunknownatthistimeas