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HARMONIC ANALYSIS OF BV

HARMONIC ANALYSIS OF BV. Ronald A. DeVore. Ronald A. DeVore. HARMONIC ANALYSIS OF BV. Industrial Mathematics Institute Department of Mathematics University of South Carolina. BV ( ) SPACE OF FUNCTIONS OF BOUNDED VARIATION. WHAT IS BV?. WHY BV?. • BV USED AS A MODEL FOR REAL IMAGES

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HARMONIC ANALYSIS OF BV

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  1. HARMONIC ANALYSIS OF BV Ronald A. DeVore Ronald A. DeVore HARMONIC ANALYSIS OF BV Industrial Mathematics Institute Department of Mathematics University of South Carolina

  2. BV ( ) SPACE OF FUNCTIONS OF BOUNDED VARIATION WHAT IS BV? WHY BV? •BV USED AS A MODEL FOR REAL IMAGES • BV PLAYS AN IMPORTANT ROLE IN PDES

  3. •REPLACE BV BY BESOV SPACE EXTREMAL PROBLEM FOR BV •DENOISING •STATISTICAL ESTIMATION •EQUIVALENT TO MUMFORD-SHAH •EXAMPLE OF K-FUNCTIONAL •LIONS-OSHER-RUDIN PDE APPROACH

  4. WAVLET ANALYSIS -DYADIC CUBES -DYADIC CUBES OF LENGTH IS A COS

  5. = 0 1 HAAR FUNCTION +1 0 1 -1

  6. DAUBECHIES WAVELET

  7. WAVELET COEFFICIENTS DEFINE

  8. THIS MAKES IT EASY TO SOLVE This decouples and the solution is given by (soft) thresholding. Coefficients larger than in others into .

  9. CAN WE REPLACE BY BV • BV HAS NO UNCONDITIONAL BASIS THEOREM(Cohen-DeV-Petrushev-Xu) SANDWICH THEOREM-AMER J. 1999 • IS WEAK THE PROBLEM ALSO SOLVED BY THRESHOLDING AT . SIMPLE NON PDE SOLUTION TO OUR ORIGINAL PROBLEM

  10. POINCARÉ INEQUALITIES Simplest case nice domain; • Does not scale correctly for modulation • Replace THEOREM (Cohen-Meyer) • Scales correctly for both modulation and dilation SPECIAL CASE MEYER’S CONJECTURE: ABOVE HOLDS FOR ALL

  11. (1,1) - BV (1/2,0) L2 B-1  (0,-1) Smoothness (1/q, )  Lq 1/q Lq Space

  12. THEOREM (Cohen-Dahmen- Daubechies-DeVore) FOR ALL •Gagliardo-Nirenberg

  13. THESE THEOREMS REQUIRE FINER STRUCTURE OF BV LET New space THIS IS EQUIVALENT TO

  14. THEOREM (Cohen-Dahmen- Daubechies-DeVore) i. If , then implies ii. Counterexamples for • is original weak result. • solves Meyer conjecture.

  15. DYADIC CUBES BAD CUBES BAD CUBES GOOD CUBES

  16. IDEA OF PROOF GOOD CUBE: COLLECTION OF GOOD CUBES THE COLLECTION OF BAD CUBES IF BV, THEN IS IN

  17. CONCLUDING REMARKS • FINE STRUCTURE OF BV • NEW SPACES • NEW INTERPOLATION THEORY • CARLESON MEASURE • Sandwich Theorem • spaces • RELATED PAPERS: DEVORE-PETROVA: AVERAGING LEMMAS-JAMS 2001 COHEN-DEVORE-HOCHMUTH-RESTRICTED APPROXIMATION -ACHA 2001 COHEN-DEVORE-KERKYACHARIAN-PICARD: MAXIMAL SPACES FOR THRESHOLDING ALGORITHMS -ACHA 2001

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