Approximation Theory

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Approximation Theory

Metric:

Complicated Function

SignalImage

Solution to PDE

Simple Function

PolynomialsSplines

Rational Func

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1. Linear space of

dimension n.

2. Nonlinear manifold of

dimension n.

3. Highly nonlinear: Highly

redundant dictionary.

Functions g chosen from:

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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,. . .

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Nonlinear: n dimensional manifold

(i) : Rational function .

(ii) Splines with

(iii) - term approximation

CONS

free knots.

0 1

pieces

IN

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Highly Nonlinear

, arbitrary,

Bases B1, B2, . . . Bm, . . .

Bj best n-term

choose best basis choose n-term approximation

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Main Question

Characterize

We shall restrict ourselves toapproximation by piecewise constants in what follows.

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Linear

Theorem (DeVore-Richards) Fix

Piecewise Constants

0 11/n

.

close to

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Theorem (DeVore-Richards)

, ,

for .

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Noninear

Theorem (Kahane)

.

Linear Nonlinear

Know (Petrushev)

.

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n - term

Haar Basis0 1

1

-1

Dyadic Interval

I0 1

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CONS

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Theorem (DeVore-Jawerth-Popov)

known.

Simple strategy:

Choose n terms where

largest.

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Lin

ear

Nonlin

ear

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ApplicationImage Compression

Piecewise constant function

(Haar)

Threshold

Problem: Need to encode positions.

Dominate Bits

Image

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Tree Approximation

Cohen-Dahmen-Daubechies-DeVore:

are almost the same requirements.

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Generate tree as follows:

1) Threshold:

2) Complete to Tree:

3) Encode the subtree:

1

00 0 0

0

0

0

0 1

1 1

1 1

(Each bit tells whether the child is in the tree.)

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• Progressive

• Universal

• Optimal

• Burn In

Features of Tree Encoder

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Encoder

P BB B B B BP P0 00 1 10 11 2 20 21 22 . . .

Pk = Position Bits of

B { bit bjk = j of , }

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Cohen-Dahmen-DeVore

Elliptic Equation

Wavelet transform gives

- positive definite.

- has decay properties.

CDD gives an adaptive algorithm

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Theorem If , then

using n computations the adaptive algorithm produces :

Theorem If , then the

adaptive algorithm produces :

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Error:

“Error Indicators”:

Refinement: Let be the smallest

set of indices such that

residual

Define new set

.