Sidon used b21 sets, which are the sidon sets mentioned earlier, as a tool in his study of fourier series. Some new thin sets of integers in harmonic analysis. The advantage of this group, however, is how close it is to classical fourier space and for this reason the tools of fourier analysis developed in chapters 3 and 4 are used so successfully. Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of fourier series and fourier transforms i. We establish the equivalence between several different characterizations of sidon sets for.
The point here is that the subject of harmonic analysis is a point of view and a collection of tools, and harmonic analysts continually seek new venues in which to ply their wares. The theory of abstract fourier series and fourier integrals cf. Numerous exercises and problems make the text suitable for selfstudy and the classroom alike. Sidon sets in uniformly bounded orthonormal systems. Geometry of banach spaces and harmonic analysis citeseerx. It sounds like a very technical idea, but basic harmonic analysis just means understanding how a chord is related to the key and to the other chords in a piece of music. The proof that these sets are sidon sets requires a brief study of 4norms in varopoulos algebras.
Thin sets of integers in harmonic analysis and pstable. A complete annotated bibliography of work related to sidon. Apostolosdimitromanolakis department of computer science. A major step towards the modern approach to harmonic analysis was made by the work of hugo riemann around the end of the 19th century, whose most important contribution was to introduce the idea of the tonal function of. The harmonic analysis methods and inequalities among various lp norms we use to derive lower bounds for dx also provide uniform upper. Sets of integers with this property are now called sidon sets, b2 or b21 sets. Much of the material in these notes is taken from the books of stein singular integrals and di erentiability properties of functions, and harmonic analysis and the book of stein and weiss, fourier analysis on euclidean spaces. Journal of functional analysis 15, 404427 1974 p sidon sets r. Thin sets of integers in harmonic analysis and pstable random fourier series p. Finite unions of uniformly approximable sidon sets are uniformly approximable sidon.
Sidon sets or b2 sequences is a related problem from combinatorial number theory. Examples of sidon sets that are not sidon sets are given in section 5. They are small since they are very close, in various meanings, to sidon sets. Pdf some problems in harmonic analysis on quantum groups. Ross university of oregon, eugene, oregon 97403 presented by the editor received may 15, 1973 let g be a compact abelian group with character group x. Linear programming in harmonic analysis 1 since k is a compact set and n h l k. Kyle hambrook san jose state university and itay londner tel aviv university.
Coordinatewise summability, inclusion theorems and psidon sets. Sidon sets are named after fourier analyst simon sidon who defined these sets in order to solve a problem in harmonic analysis. Automatic harmonic analysis of jazz chord progressions. This twovolume text provides a complete overview of the theory of banach spaces, emphasising its interplay with classical and harmonic analysis particularly sidon sets and probability. We obtain that the union of two lustpiquard sets is a lustpiquard set, provided that one of these two sets is closed for the bohr topology.
Numerous and frequentlyupdated resource results are available from this search. Some new thin sets of integers in harmonic analysis daniel li, herv. Much of the material in these notes is taken from the books of stein singular integrals and di erentiability properties of functions, 19 and harmonic analysis 20 and the book of stein and weiss, fourier analysis on euclidean spaces 21. Modern harmonic analysis encompasses areas as diverse as group representation theory, functionalanalysis and applications in signal processing, machine learning and data analysis. It was erdos who gave the name sidon sets to these constructions. We call a subset s of an abelian group g a sidon set if for any x, y, z, we s of. Edwards australian national university, canberra, a. A linear programming problem in harmonic analysis charles f. Some problems in harmonic analysis on quantum groups. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. Modern harmonic analysis and applications institute for. Incidentally, while completing this manuscript, the author learned about. These techniquesdeliver the answers desired by engineers and scientists working with big data sets or searching for novelmethods to connect experiments with theory.
The effect of translating a singer set on its fourier transform. Finally, in section 7, we discuss the optimality of our results. Rodrguezpiazza dedicated to the memory of daniel rider november 3, 2018 abstract. We warn the reader that another type of lacunary subsets of groups are commonly called sidon sets in harmonic analysis cf. Indeed, one must step exclusively into the realm of in.
Apr 30, 2015 modern harmonic analysis encompasses areas as diverse as group representation theory, functionalanalysis and applications in signal processing, machine learning and data analysis. Other sessions contained contributions to a variety of tools of harmonic analysis such as cohomology, beurling algebras, sidon sets, multiplier algebras, hypergroups, representation theory of lie groups, group calgebras, quantum groups, spectral theory, and compacti cations. Interpolation and sidon sets for compact groups colin. Sidon communicated the problem to erdos who, together with paul turan, made the first publication on the topic in 1934.
Harmonic analysis and its connections to geometric measure theory and combinatorics org. A sidon set e is said to be uniformly approximable if the characteristic section i e belongs to mgy. The main goal of the school was to introduce graduate students and young mathematicians to three broad and interrelated areas in the theory of automorphic forms. Harmonic analysis, abstract encyclopedia of mathematics. It is largely selfcontained and is intended for graduates and researchers in pure and applied analysis. Unfortunately but this is quite natural since sidon was mostly interested in the subject, the other sidon sets are also related with harmonic analysis. It was held at the fields institute in toronto, canada, from june 2 to june 27, 2003. It is a unique book, aimed at both new and experienced researchers. Classical and multilinear harmonic analysis by camil muscalu. Numerous exercises and problems make the text suitable for selfstudy and the. Riesz formulated harmonic analysis as an independent.
Some new thin sets of integers in harmonic analysis univartois. Pdf this thesis studies some problems in the theory of harmonic analysis on compact quantum groups. Classical harmonic analysis the theory of fourier series and fourier integrals underwent a rapid development, stimulated by physical problems, in the 18th century and 19th century. Abstract harmonic analysis is the study of locally. Understanding special sets of integers was classically of interest to hadamard, zygmund and others, and continues to be of interest today. All known constructions proceed by first finding a sidon set modulo a prime and then.
Ii, structure and analysis for compact groups, analysis on locally. Outline introduction to abstract harmonic analysis on quantum groups convolutions of states and lpimproving operators sidon sets and p sets 2 of 41 simeng wang some problems in harmonic analysis on quantum groups. A journal of mathematics, 2019 a fatou equation for randomly stopped variables sudderth, william d. Phd thesis defence pure mathematics university of waterloo. The harmonic analysis methods and inequalities among various lp norms we use to derive lower bounds for. Harmonic analysis overlaps and interacts quite fruitfully with many other. For instance, the study of decomposing a function on a.
Walsh analysis is a common tool in the study of the distributional properties of digital nets 18, 22 this sample convincingly demonstrates the impact of harmonic analysis on discrepancy theory. This can be such useful information that you will find many musicians who have not studied much music theory, and even some who dont read music, but who can tell you what the i one or the v five chord are in a certain. This book is a modern treatment of the subject of interpolation and sidon sets. Cayley graphs let g be a group and c a subset of g. In 3, progress on the theory of sidon sets and the dichotomy problem for restriction algebras is described, starting from the remarkable discoveries of g. Sidon sets of the second kind are closely related to the problem of embedding graphs as induced subgraphs in cayley graphs section 2.
It is largely selfcontained and will be useful to graduate students and researchers in both pure and applied analysis. Some arithmetical conditions are given, leading to the nonstationarity of prime numbers. Continuous ramsey theory and sidon sets internet archive. Stein and his school introduced the idea of studying classical harmonic analysisfractional. A handbook of harmonic analysis yoshihiro sawano contents preface 10 acknowledgement 10 orientation of this book 10 notations in this book part 1. The link between this question and harmonic analysis is via the study of lacunary. The first class seem to have some kind of pitch associated with them and can be. Sidon sets in groups and induced subgraphs of cayley graphs. A closed subset e of r is a helson set if for every fcoe there exists a fourier transform gear such that gef. They are small since they are very close, in various senses, to sidon sets. Rodrguezpiazza dedicated to the memory of daniel rider february 14, 2009 abstract.
Golomb rulers and sidon sets dimitromanolakis apostolos. Computational excursions in analysis and number theory. These sequences are subsets of f1ng having distinct pair wise sums between the elements. This paper is devoted to the study of sidon sets and some related objects for compact quantum groups. This is reasonable enough because they were rst studied by sidon in relation to a question about trigonometric series, but may lead to confusion because there is a quite di erent type of object in harmonic analysis which is also called a sidon set.
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