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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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 MATH

Definition

(Fourier transform) We introduce the operations $^{\wedge}$ (Fourier transform) and $^{\vee}$ (Inverse Fourier transform): MATH

Remark

There are several more forms of Fourier transform used in the literature. One can do a change of variables $\tau=2\pi z$ . Then the Fourier transforms take the form MATH We can also change conventions to impose symmetry: MATH In these notes the term "Fourier transform" and notations $^{\vee}$ and $^{\wedge}$ always refer to the definition ( Fourier transform ).

Proposition

(Basic properties of Fourier transform)

1. The Fourier transform $^{\wedge}$ maps MATH onto itself.

2. MATH .

3. MATH .

4. MATH .

5. MATH .

6. MATH .

7. MATH .





Notation. Index. Contents.


















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