Table Of ContentFractal Functions, Fractal Surfaces, and Wavelets
Fractal Functions, Fractal
Surfaces, and Wavelets
Second Edition
Peter R. Massopust
Centre of Mathematics Technische Universität
München München, Germany
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ThisbookisdedicatedtomywifeMaritza,
mysonJonathan,
andmydaughterSarah.
About the author
Peter R. Massopust is a Privatdozent in the Center of Mathematics at the Technical
University of Munich, Germany. He received his PhD degree in Mathematics from
GeorgiaInstituteofTechnologyinAtlanta,Georgia,UnitedStates,andhishabilitation
from the Technical University of Munich. He worked at several universities in the
UnitedStates,atSandiaNationalLaboratoriesinAlbuquerque(UnitedStates),andas
a senior research scientist in industry before returning to the academic environment.
He has written more than 60 peer-reviewed articles in the mathematical areas of
Fourier analysis, approximation theory, fractals, splines, and harmonic analysis and
wrotemorethan20technicalreportswhileworkinginthenonacademicenvironment.
He has authored or coauthored two textbooks and two monographs and has coedited
twocontemporary mathematics volumes and several special issuesfor peer-reviewed
journals. He is on the editorial board of several mathematics journals, and has
given more than 100 invited presentations at national and international conferences,
workshops,andseminars.
Preface to first edition
This monograph gives an introduction to the theory of fractal functions and fractal
surfaces with an application to wavelet theory. The study of fractal functions goes
back to Weierstraß’s nowhere differentiable function and beyond. However, it was
not until the publication of B. Mandelbrot’s book (cf. [1]) in which the concept of a
fractalsetwasintroducedandcommoncharacteristicsofthesesets,suchasnonintegral
dimension and geometric self-similarity, identified that the theory of functions with
fractalgraphsdevelopedintoanareaofitsown.Seeminglydifferenttypesofnowhere
differentiable functions, such as those investigated by Besicovitch, Ursell, Knopp,
and Kiesswetter, to only mention a few, were unified under the fractal point of view
and this unification led to new mathematical methods and applications in areas that
include:dimensiontheory,dynamicalsystemsandchaoticdynamics,imageanalysis,
andwavelettheory.
The objective of this monograph is to provide essential results from the theory of
fractal functions and surfaces for those interested in this fascinating area, to present
newandexcitingapplicationsandtoindicateinwhichinterestingdirectionsthetheory
canbeextended.Thebookisessentiallyself-containedandcoversthebasictheoryand
the different types of fractal constructions as well as some specialized and advanced
topicssuchasdimensioncalculationsandfunctionspacetheory.
Thematerialispresentedinthefollowingway.Thefirstpartofthebookcontains
basically background material and consists of four chapters. The first chapter intro-
duces the relevant notation and terminology and gives a brief review of some of the
basicconceptsfromclassicalanalysis,abstractalgebra,andprobabilitytheorythatare
necessary for the remainder of the book. The reader who is not quite familiar with
someofthematerialpresentedinthisfirstchapterisreferredtoalistofreferencesin
the bibliography where most of these concepts are defined and motivated. However,
effortsweremadetokeepthemathematicalrequirementsatalevelwhereagraduate
student with a solid background in the above-mentioned areas will be able to work
throughmostofthebook.
The second chapter introduces same basic constructions of fractal sets. The first
is based upon the approach by Hutchinson [2] and Barnsley and Demko [3] using
what is now called an iterated function system. This method is then generalized and
compared to Dekking’s [4] construction of so-called recurrent sets associated with
certainsemigroupendomorphismsandBandt’sapproach[5,6]viatopologicalMarkov
chains.Finally,agraph-directedfractalconstructionduetoMauldinandWilliams[7]
is presented. The emphasis is on iterated function systems and their generalizations;
however. In this chapter the foundations for the rigorous treatment of univariate and
multivariatefractalfunctionsarelaid.
Next,theconceptofdimensionofasetisintroduced.Thisisdonebyfirstreviewing
thedifferentnotionsofdimensionthatareusedtocharacterizeanddescribesets.The
xii Prefacetofirstedition
lasttwosectionsinthischapteraredevotedtothepresentationofdimensionresultsfor
self-affinefractalsets.
Ashortchapterdealingwiththefascinatingtheoryofdynamicalsystemsfollows.
Theemphasisisonthegeometricaspectsofthetheoryanditisshownhowtheycan
beusedtodescribeattractorsofiteratedfunctionsystems.
In the second part of this book, univariate and multivariate fractal functions are
discussed. Chapter 5 introduces fractal functions as the fixed points of a Read-
Bajraktarevic´ operator. This approach differs from that undertaken by Barnsley [8]
who introduced fractal functions for interpolation and approximation purposes. It is
alsoshownhowDekking’sapproachtofractalscanbeusedtodefinefractalfunctions
andtheiterativeinterpolationprocessofDubucandhisco-workersispresented.The
remainderofthechapterdealswithdifferentclassesoffractalfunctionsanddiscusses
severaloftheirproperties.
Chapter 6 is devoted to dimension calculations. Formulae for the box dimension
of the graphs of most of the fractal functions introduced in the previous chapter are
presented here. The second part of the chapter deals with an interesting relationship
betweencertainclassesofsmoothnessspacesandtheboxdimensionofthegraphsof
affinefractalfunctions.
In Chapter 7 the basic concepts and notions from wavelet theory are introduced,
and it is demonstrated how a certain class of fractal functions generated by iterated
function systems can be used to generate a multiresolution analysis of L2(R). This
class of fractal functions then provides a new construction of continuous, compactly
supported,andorthogonalscalingfunctionsandwavelets.
The next chapter introduces multivariate fractal functions. The graphs of these
functions are called fractal surfaces. Properties of fractal surfaces are then discussed
andformulaefortheboxdimensionderived.
Inordertoconstructmultiresolutionanalysesbasedonthefractalsurfacesdefined
in Chapter 8, the theory of Coxeter groups needs to be employed. This is done in
Chapter9,aftersomerudimentaryconceptsfromthistheoryareintroduced.
Becauseofthelimitedscopeofthismonograph,certaintopicscouldnotbecovered.
These include, among others, a more in-depth presentation of the geometric theory
of dynamical systems and the role fractals play in this theory. Furthermore, some of
theworkofT.Lindstrømonnonstandardanalysis,iteratedfunctionsystems,fractals,
andespeciallyBrownianmotiononfractals,turnedouttobebeyondthelimitsofthe
present book. The interesting work of J. Harrison dealing with geometric integration
theoryandfractalscouldalsonotbedescribed.However,referencespertainingtothese
as well as other topics are listed in the bibliography. The bibliography also contains
research papers and books not explicitly used or mentioned in this monograph. They
wereincludedtogivethereaderamorewell-roundedperspectiveofthesubject.
This book grew out of the work of many mathematicians from several areas of
mathematics, and the author has greatly benefited from numerous conversations and
discussionswithmycolleagues.SpecialthanksgotoDougHardinandJeffGeronimo,
who have influenced and shaped some of my thoughts and ideas. In particular, I am
grateful to Doug Hardin for allowing me to use his Mathematica packages to make
Prefacetofirstedition xiii
some of the figures in this monograph. I also wish to thank Patrick Van Fleet for
introducingmetothetheoryofDirichletsplinesandspecialfunctions.
WorkingwithAcademicPresswasapleasure.Iwouldliketoespeciallyexpressmy
gratitude to Christina Wipf, who gave me the idea of writing this monograph, and to
PeterRenz,whoguidedmethroughthefinalstages.
Last but not least I wish to thank my wife Maritza and my family for their
continuoussupportandencouragementduringthepreparationofthismonograph.
PeterR.Massopust
Houston,Texas,USA
December1993
Preface to second edition
Since the publication of the original monograph in 1995, the fields of fractals and
wavelets have experienced phenomenal growth. New concepts and ideas have been
introducedandarenowattheheartofeachtheory.Giventhesecircumstances,itwas
natural to change the perspective of the original monograph and to incorporate some
of the new ideas and to remove those that are no longer at the core of either theory.
Someofthenewconceptsintroducedinthesecondeditionarefractaltransformations
between fractal sets, local iterated function systems, local fractal functions and
surfaces,fractels(ie,fractalelements),andself-referentialfunctions.
The overall structure of the original edition remains intact. Some mathematical
preliminaries were moved from the main text to the introductory chapter, Chapter 1.
Twonewsections,onlocaliteratedfunctionsystemsandfractaltransformations,have
beenaddedtoChapter2.Thenewconceptofalocalfractalfunctionisintroducedin
Chapter 5, and several recent results have been included. A new chapter, Chapter 6,
on fractels and self-referential functions has been added. Both concepts have been
quite recently introduced into the theory, and they deserve to be presented as they
clarify, generalize, and extend some of the fundamental properties of fractal sets and
fractalfunctions.Theintroductiontowavelets,Chapter8,hasbeenupdatedandsome
resultsthatwerenotfullydevelopedwhentheoriginalmonographwaspublishedhave
beenincluded.Finally,Chapter10introducestheconceptofwaveletsetandshowsthe
connectionsbetweenfractalsurfaces,affineWeylgroups,andwaveletsets.
The purpose of this second edition is to correct and extend some of the results in
thefirsteditionandtoreducethenumberofmisprintstherein.Iwouldliketothankall
mycolleagueswhopointedoutthesemisprintsanderrors.
PeterR.Massopust
Munich,Germany
February2016
List of symbols
((cid:2),F,μ) measurespace
2S powerset
∗ convolution
(cid:2) semiconvolutionproduct
(cid:3)S setcomplement
(cid:3)M forwarddifferenceoperator
h
dim(cid:4) Lyapunovdimension
dimP packingdimension
dimB Billingsleydimension
dimb boxdimension
dimM Minkowskidimension
dimH Hausdorffdimension
B˙s homogeneousBesovspace
(cid:2)p,q
˙
disjointunion
F˙s homogeneousTriebel-Lizorkinspace
p,q
C complexnumbers
E Euclideanspace
N naturalnumbers
Nn {1,...,n}
R realnumbers
Z integers
Ps s-dimensionalpackingmeasure
S(Rn) Schwartzspace
D(Rn) spaceofinfinitelydifferentiablefunctionsonRnofcompactsupport
B(X) σ-algebraofallμ-measurablesubsetsofX⊂Rn
Bn Borelσ-algebraofRn
Cs Zygmundspace
Cw addressstructure
F(X,Y) classoffractalfunctionsX→Y
Fk[0,1] classofreal-valuedCkfractalfunctionson[0,1]
Hh h-Hausdorffmeasure
KK(X) linearspaceoffunctionsX→Kwithcompactsupport
L(X,Y) spaceofcontinuouslinearoperators
Mh h-netmeasure
MK(X) K-vectorspaceofmeasuresonXwithvaluesinK
Pd classorreal-valuedpolynomialsoforderd
RF(X,Y) classofrecurrentfractalfunctionsX→Y
S(cid:5)(Rn) spaceoftempereddistributions
d Hutchinsonmetric
h Hausdorffmetric
Twv fractaltransformation
Vk approximationspacegeneratedbyfractalfunctions
