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.
 A. Elementary definitions of wavelet analysis.
 B. Haar functions.
 C. Multiresolution analysis.
 D. Orthonormal wavelet bases.
 a. Auxiliary function of OST.
 b. Scaling equation for wavelet.
 c. Existence of orthonormal wavelet bases.
 E. Discrete wavelet transform.
 F. Construction of MRA from scaling filter or auxiliary function.
 G. Consequences and conditions for vanishing moments of wavelets.
 H. Existence of smooth compactly supported wavelets. Daubechies polynomials.
 I. Semi-orthogonal wavelet bases.
 J. Construction of (G)MRA and wavelets on an interval.
 3 Finite element method.
 4 Construction of approximation spaces.
 5 Time discretization.
 6 Variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## Existence of orthonormal wavelet bases.

roposition

(Existence of orthonormal wavelet bases 1) Let be an MRA and a function satisfies the following conditions

a. .

b. is OST.

c. in .

d. , .

Then is an orthogonal basis in .

See the definition ( Approximation and detail operators ) for the notation .

Proof

Orthogonality with respect to the index follows trivially from (b). To show orthogonality with respect to the index we apply the operation to (a),(c) and use the definition ( Multiresolution analysis )-4 to conclude Thus for any pair , , we have hence

We use (d) and the definition ( Multiresolution analysis )-2 to prove that is a basis. Indeed, we have and

Proposition

(Existence of orthonormal wavelet bases 2) Let be an MRA (see the definition ( Multiresolution analysis )) with the scaling function and is the sequence defined in the proposition ( Scaling equation ). Set Then is an orthonormal wavelet basis.

Proof

We prove the statement by verifying the conditions of the proposition ( Existence of orthonormal wavelet bases 1 ). The condition (a) is trivial.

We verify the condition (b) directly: Note that the coefficients have to obey orthogonality of : Therefore

We verify the requirement ( Existence of orthonormal wavelet bases 1 )-(c) next. According to the proposition ( Basic properties of Fourier transform ), We use the formula ( Property of scale and transport 4 ). We use the propositions ( Scaling equation ) and ( Scaling equation 2 ). We make a change . We separate even and odd terms. We use the properties , . We use the proposition ( OST property 1 ). We use the proposition ( Scaling equation 2 ), .

Finally, we verify ( Existence of orthonormal wavelet bases 1 )-(d) using the proposition ( OST property 2 ). We need to show that we have According to the proposition ( OST property 2 ) it suffices to show that where and We put all together and obtain the requirement where we introduced the convenience notations We simplify the equation: and introduce the functions Thus The requirement may be restated as Therefore, we restate the target of the proof as finding a function such that We rewrite the above relationships in matrix form: and substitute the definition of from the proposition ( Scaling equation 2 ): We introduce convenience notations and . We have and we utilize the proposition ( Scaling equation 3 ), , We multiply by and obtain an equivalent equation This proves existence of needed .

 Notation. Index. Contents.