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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.
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 MATH for $1\leq p<\infty,$ MATH and MATH . Then MATH for every subset $V\subset U$ .

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 MATH be a bounded set and MATH for $1\leq p<\infty$ . There exist a sequence MATH , MATH such that MATH

Proof

Let MATH MATH Let MATH be a partition of unity subordinated to $V_{i}$ . Fix $\delta>0$ . Let MATH for $\varepsilon_{i}$ such that MATH

Let MATH Then MATH

Proposition

(Approximation by smooth functions). Let MATH be a bounded set, $\partial U$ admits a locally continuously differentiable parametrization and MATH for $1\leq p<\infty$ . There exist a sequence MATH , MATH such that MATH

Proof

1. For any point $x_{0}\in\partial U$ there exists a radius $r$ and a function MATH such that MATH

2. Let MATH where the $e_{n}$ is the $n$ -th coordinate vector. Let MATH . Observe that MATH

3. We introduce MATH and claim MATH

4. Let MATH are such that MATH and $V_{0}$ is such that MATH Fix $\delta>0$ . According to the proposition ( Local approximation by smooth functions ) there exists a function MATH , $i=0,...,N$ such that MATH Let MATH be a smooth partition of unity subordinated to MATH . We set MATH We claim MATH for some function MATH such that MATH





Notation. Index. Contents.


















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