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Author:Ojalammi, Antti
Title:Fourier-analyysia Heisenbergin ryhmällä
Fourier Analysis on the Heisenberg Group
Publication type:Master's thesis
Publication year:2012
Pages:[4] + 50      Language:   fin
Department/School:Matematiikan ja systeemianalyysin laitos
Main subject:Matematiikka   (Mat-1)
Supervisor:Nevanlinna, Olavi
Instructor:Turunen, Ville
OEVS:
Electronic archive copy is available via Aalto Thesis Database.
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Location:P1 Ark Aalto  157   | Archive
Keywords:Stone-von Neumann theorem
Heisenberg group
Fourier analysis
group transform
Wigner distribution
Stone-von Neumannin lause
Heisenbergin ryhmä
Fourier-analyysi
ryhmämuunnos
Wigner-distribuutio
Abstract (eng): In this thesis we derive the Stone-von Neumann theorem which can be used to characterize the strongly continuous unitary representations of the Heisenberg group.
We then study the group Fourier transform on the Heisenberg group.

By defining the group transform on the Schwartz space of the Heisenberg group we derive an inversion theorem as well as an equivalent of Plancherel's theorem.
We then extend these results to the L2-space.

In addition, the Wigner transform and Wigner distribution are studied.
Aside from proving some important properties of these transforms, special attention is paid to the connection between the Wigner distribution and the windowed Fourier transform.
Abstract (fin): Tässä työssä johdetaan Heisenbergin ryhmän unitaariset vahvasti jatkuvat esitykset karakterisoiva Stone-von Neumannin lause, sekä tutkitaan esitysten kautta määritellyn Heisenbergin ryhmän Fourier-muunnoksen ominaisuuksia.

Ryhmän Fourier-muunnos määritellään Schwartzin testifunktioille, ja sille saadaan käänteismuunnos sekä Plancherelin lausetta vastaava tulos, jonka avulla muunnos saadaan laajennettua L2-avaruudelle.

Lisäksi tutkitaan Wigner-muunnosta sekä -distribuutiota, joiden tärkeimmät ominaisuudet johdetaan.
Erityisesti kiinnitetään huomiota Wigner-distribuution yhteyteen ikkunoituun Fourier-muunnokseen.
ED:2012-09-24
INSSI record number: 45292
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