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.
 E. Discrete wavelet transform.
 F. Construction of MRA from scaling filter or auxiliary function.
 a. Quadrature mirror filter (QMF) conditions.
 b. Recovering scaling function from auxiliary function. Cascade algorithm.
 c. Recovering MRA from 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.

## Recovering scaling function from auxiliary function. Cascade algorithm.

iven satisfying the definition ( QMF conditions ) on may construct the auxiliary function (see the proposition ( Scaling equation )) If the scaling function exists then it would satisfy Then, assuming convergence of the product,

Proposition

(Convergence of product) If the series converge absolutely then the product converges absolutely.

Proof

We calculate Since starting from some and for some .

Proposition

(Finite QMF convergence) Let be a finite QMF. Then for any the product converges absolutely and in .

Proof

We estimate and utilize the proposition ( QMF property 2 )-a: Therefore Since is finite, the sum has some finite value . We use the proposition ( Convergence of product ), convergence of is investigated by looking at and the statement follows.

Proposition

(Cascade algorithm) Let be a finite QMF. Define operator : and let be a sequence such that Then

Proof

Note that by proposition ( QMF property 2 )-d and formulas ( Property of scale and transport 2 ),( Property of scale and transport 3 ), Thus

By calculations of the proof of the proposition ( Scaling equation ), thus and by the proposition ( Finite QMF convergence ) Combine this with , proposition ( Basic properties of Fourier transform ) and the desired conclusion follows.

Proposition

A QMF is finite if and only if has finite support.

Proof

( ). is obtained by iterations of the proposition ( Cascade algorithm ). The scaling within the operator halves the support of and the summation increases it linearly. Such operation, when applied repeatedly, leads to compact support.

( ) If has compact support then is finite by the defining formula of the proposition ( Scaling equation ).

 Notation. Index. Contents.