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HARMONIC ANALYSIS OF BVHARMONIC ANALYSIS OF BV

Ronald A. DeVoreRonald A. DeVoreRonald A. DeVoreRonald A. DeVore

Industrial Mathematics InstituteDepartment of Mathematics

University of South Carolina

HARMONIC ANALYSIS OF BVHARMONIC ANALYSIS OF BV

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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

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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

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WAVLET ANALYSIS

-DYADIC CUBES -DYADIC CUBES OF LENGTH

IS A COS

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HAAR FUNCTION

0 1

0 1

+1

-1

=

Ψ = H

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DAUBECHIES WAVELET

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WAVELET COEFFICIENTS

DEFINE

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THIS MAKES IT EASY TO SOLVE

This decouples and the solution is given by (soft) thresholding. Coefficients larger than in others into .

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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

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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

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•

Lq 1/q1/q

•

SmoothnessSmoothness

Lq Space

(1/q, (1/q, ))

• L2

(1,1) (1,1) - - BVBV •

(1/2,0(1/2,0))•

• (0,-1)(0,-1) B-1

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THEOREM (Cohen-Dahmen- Daubechies-DeVore)

• Gagliardo-Nirenberg

FOR ALL

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THESE THEOREMS REQUIRE FINER STRUCTURE OF BV

LET

New space

THIS IS EQUIVALENT TO

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THEOREM (Cohen-Dahmen- Daubechies-DeVore)

i. If , then implies

ii. Counterexamples for

• is original weak result.

• solves Meyer conjecture.

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DYADIC CUBESBAD CUBESGOOD CUBES BAD CUBES

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IDEA OF PROOF

GOOD CUBE:

COLLECTION OF GOOD CUBES

THE COLLECTION OF BAD CUBES

IF BV, THEN IS IN

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CONCLUDING REMARKS

• FINE STRUCTURE OF BV

• NEW SPACES • NEW INTERPOLATION THEORY • CARLESON MEASURE

• RELATED PAPERS: DEVORE-PETROVA: AVERAGING LEMMAS-JAMS 2001

COHEN-DEVORE-HOCHMUTH-RESTRICTEDAPPROXIMATION -ACHA 2001

COHEN-DEVORE-KERKYACHARIAN-PICARD:MAXIMAL SPACES FOR THRESHOLDING ALGORITHMS -ACHA 2001

• Sandwich Theorem• spaces