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

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Presentation transcript:

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

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

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

IMI WAVLET ANALYSIS -DYADIC CUBES -DYADIC CUBES OF LENGTH IS A COS

IMI HAAR FUNCTION =

IMI DAUBECHIES WAVELET

IMI WAVELET COEFFICIENTS DEFINE

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

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

IMI 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

IMI LqLq 1/q  Smoothness L q Space (1/q, ) (1/q,  ) L2L2 (1,1) - BV (1/2,0 ) (0,-1) B -1 

IMI THEOREM (Cohen-Dahmen- Daubechies-DeVore) Gagliardo-Nirenberg FOR ALL

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

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

IMI DYADIC CUBES BAD CUBES GOOD CUBES BAD CUBES

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

IMI CONCLUDING REMARKS FINE STRUCTURE OF BV NEW SPACES NEW INTERPOLATION THEORY CARLESON MEASURE 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 Sandwich Theorem spaces