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Approximation Theory

Simple Function. Complicated Function. Signal Image Solution to PDE. Polynomials Splines Rational Func. Metric:. Approximation Theory. 1. Linear space of dimension n. 2. Nonlinear manifold of dimension n. 3. Highly nonlinear: Highly

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Approximation Theory

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

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

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

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

  5. arbitrary, , Bases B1, B2, . . . Bm, . . . best n-term Bj Highly Nonlinear choose best basis choose n-term approximation

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

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

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

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

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

  11. CONS

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

  13. Nonlinear Linear

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

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

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

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

  18. Position Bits of = P k B B P B B B P P B . . . 00 10 0 11 20 21 22 1 2 Encoder } B { bit b of = , jk j

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

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

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

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