Content of present website is being moved to . Registration of will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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.
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.

Extensions of Sobolev spaces.


(Extension theorem) Let MATH are bounded and $U\subset V$ . Assume that $\partial U$ admits a locally continuously differentiable parametrization. Then there exists a bounded linear "extension" operator MATH such that for any MATH $Eu=u$ a.s. in $U$ and the support of $Eu$ lies within $V$ .


The $u$ may be $C^{1}$ -expanded across a flat boundary MATH by the reflection of the form MATH If the boundary is not flat then there exists a change of variables that makes it locally flat. Then such procedure extends globally using the partition of unity (see the proof of the proposition ( Global approximation by smooth functions ) for an example of the technique). The partition of unity insures that the support of the result is localized.

Notation. Index. Contents.

Copyright 2007