Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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 be a Hilbert space and .

Definition

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

Proposition

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

Definition

(Fourier decomposition) The "Fourier decomposition" of with respect to the basis is the series The quantities are called "Fourier coefficients".

Proposition

(Main property of Fourier decomposition) For a family with the property and any the minimum is attained at . If is a basis then

Proposition

(Bessel equality) For a family with the property and any ,

Proposition

(Parseval equality) Assume that is a basis in . For any we have

 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.