Convolution-Dominated Matrices in Groups of Polynomial Growth

Convolution-Dominated Matrices in Groups of Polynomial Growth

Kitsios, Christos (TU Delft Electrical Engineering, Mathematics and Computer Science)

Caspers, M.P.T. (mentor)
van Velthoven, J.T. (mentor)
Gijswijt, D.C. (graduation committee)

Delft University of Technology

Applied Mathematics

2022-07-20

In this thesis, we use a variation of a commutator technique to prove that l^p-stability is independent of p, for p greater than or equal to one, and for convolution-dominated matrices indexed by relatively separated sets in groups of polynomial growth. Moreover, from the inverse-closedness of the Schur matrices we deduce a Wiener type Lemma for the matrices in the intersection of the weighted convolution-dominated matrices, over all polynomial weights.
Finally, applications of the convolution-dominated matrices are presented. We prove the inverse-closedness of a non-commutative space generated by a discrete series representation restricted to a lattice in a nilpotent Lie group. In addition, we apply the aforementioned result on l^p-stability to show that if \pi(\Lambda)g is a p-frame for the coorbit space Co(L^p) for some p in [1,\infty], then \pi(\Lambda)g is a q-frame for the coorbit space Co(L^q) for each q in [1,\infty], where (\pi, H) is a discrete series representation of a group G of polynomial growth, \Lambda is a relatively separated set in G, and g is a vector in H such that the matrix coefficient V_g g is in the Amalgam space W_{w_a}(G). Moreover, we prove that the frame operator of the frame \pi(\Lambda)g is invertible on the coorbit spaces Co(L^p) for each p in [1,\infty], under the assumption that g is a vector in H, such that V_g g belongs in W_{w_a}(G) for each polynomial weight w_a.

Convolution operators
Wiener's lemma
Functional analysis
Twisted Group Algebras
Frame Theory

http://resolver.tudelft.nl/uuid:66af57f7-21db-40c3-8c9e-af0c74199e4e

Student theses

master thesis

© 2022 Christos Kitsios