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.
 13 Sobolev spaces.
 A. Convolution and smoothing.
 B. Approximation by smooth functions.
 C. Extensions of Sobolev spaces.
 D. Traces of Sobolev spaces.
 E. Sobolev inequalities.
 F. Compact embedding of Sobolev spaces.
 G. Dual Sobolev spaces.
 H. Sobolev spaces involving time.
 I. Poincare inequality and Friedrich lemma.
 14 Elliptic PDE.
 15 Parabolic PDE.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Approximation by smooth functions.

roposition

(Local approximation by smooth functions). Let for and . Then for every subset .

Proof

The statement is a direct consequence of the proposition ( Properties of mollifiers ) and the definition ( Weak derivative ).

Proposition

(Global approximation by smooth functions). Let be a bounded set and for . There exist a sequence , such that

Proof

Let Let be a partition of unity subordinated to . Fix . Let for such that

Let Then

Proposition

(Approximation by smooth functions). Let be a bounded set, admits a locally continuously differentiable parametrization and for . There exist a sequence , such that

Proof

1. For any point there exists a radius and a function such that

2. Let where the is the -th coordinate vector. Let . Observe that

3. We introduce and claim

4. Let are such that and is such that Fix . According to the proposition ( Local approximation by smooth functions ) there exists a function , such that Let be a smooth partition of unity subordinated to . We set We claim for some function such that

 Notation. Index. Contents.