Quantitative Analysis
Parallel Processing
Numerical Analysis
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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.

Sobolev spaces.


ome of the notation was introduced in the section ( Function spaces section ). The references are [Evans] and [Brenner] . A fundamentally different tool of Sobolev space analysis may be found in the section ( Averaged Taylor polynomial ).

Definition

(Sobolev space) For a subset MATH , we define MATH where the $\alpha$ is the multi-index defined in the section ( Function spaces section ) and the derivative is understood in the weak sense, see the definition ( Weak derivative ).

The norm for MATH is defined by MATH

The space MATH is the closure of MATH with respect to MATH .

We introduce the notation MATH MATH

Proposition

The space MATH equipped with the operation MATH is a Banach space.

If $p=2$ then MATH is a Hilbert space.




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.

Notation. Index. Contents.


















Copyright 2007