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.
 a. Scaling equation.
 b. Support of scaling function.
 c. Piecewise linear MRA, part 1.
 d. Orthonormal system of translates.
 e. Approximation by system of translates.
 f. Orthogonalization of system of translates.
 g. Piecewise linear MRA, part 2.
 h. Construction of MRA summary.
 D. 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.

## Orthonormal system of translates.

efinition

(Orthonormal system of translates) For a function we form . If has the property then we call it "orthonormal system of translates" (OST).

Note that and the calculation goes in either direction. Thus the OST may be redefined with the requirement

 (OST property 0)

Proposition

(Property of transport 1) For any function we have where is the Fourier transform of and the integral is a -th Fourier coefficient of a 1-periodic function .

Proof

Fourier transform preserves geometry, see ( Parseval equality ) and ( Basic properties of Fourier transform )-3. Hence, We use the formula ( Property of scale and transport 4 ). Make the change . We use the proposition ( Fubini theorem ), .

Proposition

(Property of transport 2) If has compact support then the function has the form for some numbers and finite .

Proof

According to the proposition ( Property of transport 1 ), Since has compact support, for a finite number of .

Proposition

(OST property 1) The set is OST iff

Proof

According to the proposition ( Property of transport 1 ), The Fourier transformation is 1-1 in and the collection is orthonormal in . Hence, if and only if and since the summation in is to , the above is true for .

 Notation. Index. Contents.