1. IMI 1 Approximation Theory Metric: Complicated Function Signal Image Solution to PDE Simple Function Polynomials Splines Rational Func

2. IMI 2 1. Linear space of dimension n. 2. Nonlinear manifold of dimension n. 3. Highly nonlinear: Highly redundant dictionary. Functions g chosen from:

3. IMI 3 Examples: (i) -- Alg. poly. of degree. (ii) -- Trig. poly. of degree. (iii) Splines -- piecewise poly. of degree r, pieces. (iv) span, CONS 0 1 Linear: 1, 2,..., n,

4. IMI 4 Nonlinear : n dimensional manifold (i) : Rational function. (ii) Splines with (iii) - term approximation CONS free knots. 0 1 pieces I N

5. IMI 5 Highly Nonlinear, arbitrary, Bases B 1, B 2,... B m,... BjBj best n-term choose best basis choose n-term approximation

6. IMI 6 Main Question Characterize We shall restrict ourselves to approximation by piecewise constants in what follows.

7. IMI 7 Linear Theorem (DeVore-Richards) Fix Piecewise Constants 0 1 1/n. close to

8. IMI 8 Theorem (DeVore-Richards),, for.

9. IMI 9 Noninear Theorem (Kahane). Linear Nonlinear Know (Petrushev).

10. IMI 10 n - term Haar Basis 0 1 1 Dyadic Interval I 0 1

11. IMI 11 CONS

12. IMI 12 Theorem (DeVore-Jawerth-Popov) known. Simple strategy: Choose n terms where largest.

13. IMI 13 Linear Nonlinear

14. IMI 14 Application Image Compression Piecewise constant function (Haar) Threshold Problem: Need to encode positions. Dominate Bits Image

15. IMI 15 Tree Approximation Cohen-Dahmen-Daubechies-DeVore: are almost the same requirements.

16. IMI 16 Generate tree as follows: 1) Threshold: 2) Complete to Tree: 3) Encode the subtree: 1 0 0 0 0 0 0 0 0 1 1 1 1 1 (Each bit tells whether the child is in the tree.)

17. IMI 17 Progressive Universal Optimal Burn In Features of Tree Encoder

18. IMI 18 Encoder P B B B BBB P P 0 00 1 10 11 2 202122... P k = Position Bits of B { bit b jk = j of, }

19. IMI 19 Cohen-Dahmen-DeVore Elliptic Equation Wavelet transform gives - positive definite. - has decay properties. CDD gives an adaptive algorithm

20. IMI 20 Theorem If, then using n computations the adaptive algorithm produces : Theorem If, then the adaptive algorithm produces :

21. IMI 21 Error: “ Error Indicators ”: Refinement: Let be the smallest set of indices such that residual Define new set.