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The Square Variation of Rearranged Fourier Series

The Square Variation of Rearranged Fourier Series. Allison Lewko Mark Lewko. Columbia University. Institute for Advanced Study. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A. Background on Orthonormal Systems.

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The Square Variation of Rearranged Fourier Series

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  1. The Square Variation of Rearranged Fourier Series Allison Lewko Mark Lewko Columbia University Institute for Advanced Study TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAA

  2. Background on Orthonormal Systems

  3. Background on Orthonormal Systems

  4. Sensitivity to Ordering Would imply “Yes” above

  5. Known Results For Reorderings

  6. Variation Operators

  7. Comparing Maximal and Variation Operators

  8. Variation Results for the Trigonometric System

  9. What Tools Do We Have to Analyze Variation?

  10. Dyadic Intervals Arbitrary subinterval can be decomposed into dyadic pieces Arbitrary subinterval is contained in dyadic interval of comparable length (approx.)

  11. How Do We Reorder?

  12. From Selectors to Fixed Size Subsets

  13. Structure of the Proof

  14. Reducing to a Sub-Level of Intervals

  15. Tool for Controlling Smaller Intervals: Orlicz Space Norms

  16. Orlicz Space Norms

  17. Proof of Decomposition Property  

  18. Proof of Decomposition Continued   

  19. Deriving Lp, L2 bounds for Decomposition

  20. Deriving Lp, L2 bounds from ¡K (contd.)

  21. Getting from ¡K Bounds to V2 Bounds

  22. Controlling ¡K Norms by Probabilistic Estimates

  23. Controlling the Supremum of a Random Process

  24. Generic Chaining

  25. Covering Numbers

  26. Strategy for our Base Estimates

  27. Further Improving the Bounds

  28. High-Level Recap of Proof Lots of details swept under the rug!

  29. Remaining Questions

  30. Other Implications of Variational Quantities

  31. Other Implications of Variational Quantities

  32. Implications of Variational Quantities (contd.)

  33. Thanks! Questions?

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