The Square Variation of Rearranged Fourier Series
Allison Lewko Mark Lewko
Columbia University Institute forAdvanced Study
Background on Orthonormal Systems
Background on Orthonormal Systems
Sensitivity to Ordering
Would imply “Yes” above
Known Results For Reorderings
Variation Operators
Comparing Maximal and Variation Operators
Variation Results for the Trigonometric System
What Tools Do We Have to Analyze Variation?
Dyadic IntervalsArbitrary subinterval is contained in dyadic interval of comparable length (approx.)Arbitrary subinterval can be decomposed into dyadic pieces
How Do We Reorder?
From Selectors to Fixed Size Subsets
Structure of the Proof
Reducing to a Sub-Level of Intervals
Tool for Controlling Smaller Intervals: Orlicz Space Norms
Orlicz Space Norms
Proof of Decomposition Property
Proof of Decomposition Continued
Deriving Lp, L2 bounds for Decomposition
Deriving Lp, L2 bounds from ¡K (contd.)
Getting from ¡K Bounds to V2 Bounds
Controlling ¡K Norms by Probabilistic Estimates
Controlling the Supremum of a Random Process
Generic Chaining
Covering Numbers
Strategy for our Base Estimates
Further Improving the Bounds
High-Level Recap of Proof
Lots of detailsswept under the rug!
Remaining Questions
Other Implications of Variational Quantities
Other Implications of Variational Quantities
Implications of Variational Quantities (contd.)
Thanks!
Questions?
