Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
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.

Poisson formula for delta function and Whittaker sampling theorem.


hat happens if one takes Fourier series of non-periodic function? To put the question in exact terms, suppose the function MATH is such that the restriction of it to the interval MATH belongs to MATH . We form the function MATH with the functions MATH being the Fourier exponentials normalized to MATH . Such function will be the $a$ -periodic repetition of the restriction MATH to the intervals MATH , $k\in\QTR{cal}{Z}$ : MATH

This is so because the function MATH is $a$ -periodic and the sum MATH is the Fourier series of MATH . Then by uniqueness of Fourier series for $a$ -periodic functions we must have MATH

Proposition

(Poisson formula for delta function) For any $a>0$ we have MATH in MATH -sense. See the section ( Weak derivative section ) for $\delta$ notation.

Proof

We use the above remark to formally take Fourier series of the function MATH on the interval MATH . We have MATH Thus MATH

Proposition

(Whittaker sampling theorem) For a continuous function MATH and $a>0$ we form MATH Then MATH

Proof

We have MATH MATH Then, by the proposition ( Basic properties of Fourier transform )-6, MATH where MATH and by the proposition ( Poisson formula for delta function ) MATH Therefore MATH

Proposition

(Fourier transform of projection on span of translates) Let $u$ be a function MATH . We introduce the notation MATH Suppose MATH is such that the operation MATH , $P:v\rightarrow Pv$ MATH is well defined. (For example, the $P_{u}$ is well defined if MATH are orthonormal or if $u$ has compact support.)

Then MATH

Proof

We introduce the notation MATH Let $f$ be such that MATH Then MATH where MATH . We have MATH We apply the proposition ( Whittaker sampling theorem ). MATH where MATH MATH





Notation. Index. Contents.


















Copyright 2007