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Analysis on Locally Compact Groups
Published in Hugo D. Junghenn, Principles of Analysis, 2018
Lebesgue measure on R $ \mathbb R $ and counting measure on Z $ \mathbb Z $ are examples of measures μ $ \mu $ that are translation invariant, that is, μ(B+x)=μ(B) $ \mu (B + \boldsymbol{x}) = \mu (B) $ for all Borel sets B. These are special cases of a general construct called Haar measure. As we shall see, the existence Haar measure leads to a unification and generalization of Fourier analysis, the basic aspects of which are presented in this chapter.
Space-time fractional diffusion equation associated with Jacobi expansions
Published in Applicable Analysis, 2023
The polynomials satisfy the following product formula: where the coefficients are nonnegative for any n and m, furthermore we have The polynomials yield a hypergroup structure on by means of the convolution of two Dirac measures The Haar measure of this discrete polynomial hypergroup is defined by Note that the coefficients satisfy the identity So that this quantity is invariant under permutation of the variables n, m, k and we have It is worthy to note that being a positive constant depending on .
A dynamical proof of the van der Corput inequality
Published in Dynamical Systems, 2022
Nikolai Edeko, Henrik Kreidler, Rainer Nagel
It would be interesting to also apply our approach to closed subsemigroups of a locally compact group with right Haar measure m having a topological right Følner net (e.g. and ), i.e. is compact with positive measure for every and for every (cf. [24, Examples 1.2 (e)]). In particular, this could lead to a new proof of [2, Theorem 2.12].
A review on some classes of algebraic systems
Published in International Journal of Control, 2020
Víctor Ayala, Heriberto Román-Flores
We just show the main ideas of the proof of 2. Since G is compact, we note that the Haar measure is finite. On the other hand, since the group is semisimple any derivation is inner, which implies that the linear vector field has the form for some and any . In this case, for any constant control the dynamic of the system combines conjugation with invariant, both preserving the Haar measure. Since G is compact, it is closed. So for more arbitrary control, the argument works for the limit. Furthermore, according to a well-known fact, the system is controllable if and only if satisfy the LARC condition. The invariance argument together with the Poincaré recurrence theorem, finish the proof (Lobry, 1974).