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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 analysis.


et $H$ be a Hilbert space and $\dim H=\infty$ .

Definition

(Basis in Hilbert space) The countable collection of elements MATH is a "basis" in Hilbert space $H$ if any element of $H$ may be approximated with arbitrary precision by a finite linear combination of elements from MATH . A Hilbert space is called "separable" if it has a basis.

Proposition

The countable collection of elements MATH is a basis in Hilbert space $H$ iff MATH

Definition

(Fourier decomposition) The "Fourier decomposition" of $f\in H$ with respect to the basis MATH is the series MATH The quantities MATH are called "Fourier coefficients".

Proposition

(Main property of Fourier decomposition) For a family MATH with the property MATH and any $f\in H$ the minimum MATH is attained at MATH . If MATH is a basis then MATH

Proposition

(Bessel equality) For a family MATH with the property MATH and any $f\in H$ , MATH

Proposition

(Parseval equality) Assume that MATH is a basis in $H$ . For any $f,g\in H$ we have MATH




A. Fourier series in L2.
B. Fourier transform.
C. Fourier transform of delta function.
D. Poisson formula for delta function and Whittaker sampling theorem.

Notation. Index. Contents.


















Copyright 2007