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.
 A. Space of distributions. Weak derivative.
 B. Fundamental solution.
 C. Fundamental solution for the heat equation.
 11 Functional Analysis.
 12 Fourier analysis.
 13 Sobolev spaces.
 14 Elliptic PDE.
 15 Parabolic PDE.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Space of distributions. Weak derivative.

efinition

(Weak derivative). Let be a collection of infinitely smooth functions with compact support. We define topology by stating that iff all supports are included in a compact set and for any all derivatives of order up to converge uniformly to 0 on . We call "distribution on " any linear continuous functional on . We use the notation The is included in ( =space of distributions) by ( acts on according to this rule, thus ). The derivative of the distribution is the distribution that acts according to the rule

One could think at this point that the last definition may be extended to the functions acting for some and . Even though such extension is formally possible it would be of little use because the main tools of the following analysis are convolution and Fourier transform.

Example

Consider the given by the step function

 (step function)
The derivative is called Dirac's delta function:
 (Dirac delta function)
We will use the notation

The Fourier transform and the convolution extend to similarly to the differentiation.

 Notation. Index. Contents.