The book aims at giving a monographic presentation of the abstract harmonic analysis of hypergroups, while combining it with applied topics of spectral analysis, approximation by orthogonal expansions and stochastic sequences. Hypergroups are locally compact Hausdorff spaces equipped with a convolution, an involution and a unit element. Related algebraic structures had already been studied by Frobenius around 1900. Their axiomatic characterisation in harmonic analysis was later developed in the 1970s. Hypergoups naturally emerge in seemingly different application areas as time series analysis, probability theory and theoretical physics.
The book presents harmonic analysis on commutative and polynomial hypergroups as well as weakly stationary random fields and sequences thereon. For polynomial hypergroups also difference equations and stationary sequences are considered. At greater extent than in the existing literature, the book compiles a rather comprehensive list of hypergroups, in particular of polynomial hypergroups. With an eye on readers at advanced undergraduate and graduate level, the proofs are generally worked out in careful detail. The bibliography is extensive.
Contents:
- Hypergroups
- Basics of Harmonic Analysis of Hypergroups
- Harmonic Analysis on Commutative Hypergroups
- Fourier Analysis on Polynomial Hypergroups
- Weakly Stationary Random Fields on a Commutative Hypergroup
- Weakly Stationary Random Sequences on a Polynomial Hypergroup
- Difference Equations and Stationary Sequences on Polynomial Hypergroups
- Further Hypergroup Examples
Readership: Advanced undergraduate and graduate students, researchers in harmonic analysis and fields of application.
Key Features:
- The book combines abstract harmonic analysis with applied areas of approximation, difference equations and stochastic sequences. It is among the first monographs in the field bridging the gap between the abstract foundations of hypergroup theory and their mathematical application
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