In this thesis we discuss Petermichls characterization of the Hilbert transform as an average of dyadic shift operators following the presentation by Thomas Hytonen [Hyt]. A linear and bounded operator T in L2(R) that commutes with translations, dilations and anticommutes with reflections must be a constant multiple of Hilbert transform; T = cH. Using this principle Stefanie Petermichl showed that we can write H as a suitable average of dyadic operators [Pet]. Each Dyadic Shift Operator does not have the symmetries that characterize the Hilbert transform, but an average over all random dyadic grids do.
Level of Degree
Mathematics & Statistics
Pereyra, Maria Cristina
First Committee Member (Chair)
Second Committee Member
Hilbert Transform, Dyadic Shift Operators, Random Dyadic Grids, Singular Integrals, Fourier Multipliers
Atasever, Nuriye. "The Hilbert Transform as an Average of Dyadic Shift Operators." (2015). http://digitalrepository.unm.edu/math_etds/3