I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 1 Real Variable.
 2 Laws of large numbers.
 3 Characteristic function.
 4 Central limit theorem (CLT) II.
 5 Random walk.
 6 Conditional probability II.
 7 Martingales and stopping times.
 8 Markov process.
 9 Levy process.
 10 Weak derivative. Fundamental solution. Calculus of distributions.
 11 Functional Analysis.
 12 Fourier analysis.
 A. Fourier series in L2.
 B. Fourier transform.
 C. Fourier transform of delta function.
 D. Poisson formula for delta function and Whittaker sampling theorem.
 13 Sobolev spaces.
 14 Elliptic PDE.
 15 Parabolic PDE.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Fourier series in L2.

et be the interval with point and point made identical. In other words, a function is well defined on if and only if it has the property

Proposition

(Fourier series on unit interval) The family is an orthogonal basis in .

Definition

( ) We define the class of sequences and the product The range of summation may be or or otherwise (but always countable infinite) depending on context.

Notation

For a sequences we use the notations

Proposition

For we have

Using results of the section ( Fourier analysis in Hilbert space section ) we state that for there is

Such result expands to :

The version is

 Notation. Index. Contents.