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SUMMARY:Rebricking frames and bases - Brigitte Forster-Heinlein (Universit
 ät Passau)
DTSTART:20240716T150000Z
DTEND:20240716T154000Z
UID:TALK218158@talks.cam.ac.uk
DESCRIPTION:In 1946\, Dennis Gabor introduced the analytic signal $f+iHf$ 
 for real-valued signals $f$. Here\, $H$ is the Hilbert transform. This com
 plexification of functions allows for an analysis of their amplitude and p
 hase information and has ever since given well-interpretable insight into 
 the properties of the signals over time. The idea of complexification has 
 been reconsidered with regard to many aspects: Examples are the dual tree 
 complex wavelet transform\, or via the Riesz transform and the monogenic s
 ignal\, i.e. a multi-dimensional version of the Hilbert transform\, which 
 in combination with multi-resolution approaches leads to Riesz wavelets\, 
 and others. In this context\, we ask two questions:- Which pairs of real o
 rthonormal bases\, Riesz bases\, frames and Parseval frames $\\{f_{n}\\}_{
 n\\in\\mathbb{N}}$ and $\\{g_{n}\\}_{n\\in\\mathbb{N}}$ can be ``rebricked
 '' to complex-valued ones $\\{f_{n}+ig_{n}\\}_{n\\in\\mathbb{N}}$?\n- And 
 which real operators $A$ allow for rebricking via the ansatz $\\{f_{n}+iAf
 _{n}\\}_{n\\in\\mathbb{N}}$?In this talk\, we give answers to these questi
 ons with regard to a characterization which linear operators $A$ are suita
 ble for rebricking while maintaining the structure of the original real va
 lued family. Surprisingly\, the Hilbert transform is not among them.&nbsp\
 ;\nThis is joint work with Thomas Fink\, Florian Heinrich\, and Moritz Pro
 ell.
LOCATION:Seminar Room 1\, Newton Institute
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