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 transform.

e introduce the class of functions

Definition

(Fourier transform) We introduce the operations (Fourier transform) and (Inverse Fourier transform):

Remark

There are several more forms of Fourier transform used in the literature. One can do a change of variables . Then the Fourier transforms take the form We can also change conventions to impose symmetry: In these notes the term "Fourier transform" and notations and always refer to the definition ( Fourier transform ).

Proposition

(Basic properties of Fourier transform)

1. The Fourier transform maps onto itself.

2. .

3. .

4. .

5. .

6. .

7. .

 Notation. Index. Contents.