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?