I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 1 Calculational Linear Algebra.
 2 Wavelet Analysis.
 3 Finite element method.
 4 Construction of approximation spaces.
 A. Finite element.
 B. Averaged Taylor polynomial.
 C. Stable space splittings.
 D. Frames.
 E. Tensor product splitting.
 F. Sparse tensor product. Cure for curse of dimensionality.
 a. Definition of sparse tensor product.
 b. Wavelet estimates in Sobolev spaces.
 c. Stability of wavelet splitting.
 d. Stable splitting for tensor product of Sobolev spaces.
 e. Approximation by sparse tensor product.
 5 Time discretization.
 6 Variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## Wavelet estimates in Sobolev spaces.

e proceed to extend results of the sections ( Vanishing moments for biorthogonal wavelets ) and ( Vanishing moments of wavelet ) to Sobolev spaces , see the chapter ( Sobolev spaces ). The section ( Construction of approximation spaces ) is important prerequisite.

Proposition

(Jackson inequality for wavelets) Assume the condition ( Sparse tensor product setup ). Then for .

Proof

According to the proposition ( Bramble-Hilbert lemma ), for some choice of and . The function is an -degree polynomial of . Thus, it is contained in .

Because have finite support, if we increase the scale then, perhaps starting from some scale , we receive enough freedom to replicate polynomials on subdivisions separately. Hence, the result applies with , for some constant . With substitution into , we obtain .

Corollary

(Jackson inequality for wavelets 2) Assume the condition ( Sparse tensor product setup ). Then for any and .

Proof

For any we have for some . Thus the proposition ( Jackson inequality for wavelets ) in such context reads and we have a freedom of choice for a pair . Thus the above should hold for any so

Proposition

(Vanishing moments vs approximation 2) For assume that

1. ,

2. the derivative is bounded on : for some ,

3. a function has compact support,

4. , ,

5.

then there exists a constant such that

Proof

Observe that by compactness of support of and for we have (again by compactness of support) The proposition ( Vanishing moments vs approximation ) applies with the substitutions , and .

Proposition

(Vanishing moments vs approximation 3) For and assume that

1. a function ,

2. the derivative is bounded on : for some .

3. a function has compact support,

4. , ,

5.

then there exists a constant such that

Proof

is a simple extension of the proof of the previous proposition ( Vanishing moments vs approximation 2 ).

Note that

 (Derivative vs scale)

Proposition

(Bernstein inequality for wavelets) Assume the condition ( Sparse tensor product setup ). Then for any for .

Proof

First, we prove the result for on :

The method of the prove is assume that does not hold and arrive to contradiction using the proposition ( Vanishing moments vs approximation 3 ). If does not hold then there exists an increasing sequence , such that

Note that the scale operation does not alter the -norm, see the formula ( Property of scale and transport 2 ), hence we only need to estimate the numerator of to show that, in fact, LHS of cannot blow up.

In context of the proposition ( Vanishing moments vs approximation 3 ) we take the sequence , then Note that Thus, for to -approximate the the max-norm has to grow like : We also use the formula ( Derivative vs scale ): or This estimate is in contradiction with , thus, is proven.

We extend the estimate to as follows. Let then We apply . The is a -constant: . We use the proposition ( Frame property 2 ).

Next, we extend the result to . Let , then

We have proven the estimate in case of .

To extend the result to we note that the procedure in the section ( Construction of MRA and wavelets on half line or an interval ) is a finite linear combination taken within .

 Notation. Index. Contents.