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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 series in L2.


et $Q$ be the interval $\left[ 0,1\right] $ with point $0$ and point $1$ made identical. In other words, a function MATH is well defined on $Q$ if and only if it has the property MATH

Proposition

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

Definition

( $l^{2}$ ) We define the class of sequences MATH and the product MATH The range of summation may be MATH or MATH or otherwise (but always countable infinite) depending on context.

Notation

For a sequences $a,b\in l^{2}$ we use the notations MATH

Proposition

For $a,b\in l^{2}$ we have MATH

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

Such result expands to $\left[ 0,a\right] $ : MATH

The MATH version is MATH





Notation. Index. Contents.


















Copyright 2007