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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.

Dual Sobolev spaces.


et $U$ be a bounded subset of $\QTR{cal}{R}^{n}$ .

Definition

We denote MATH the dual space of MATH (with respect to the MATH topology, see the definition ( Dual space )). MATH where MATH represents the action of $f\in H^{-1}$ on MATH .

Remark

It will be clear from the proposition ( Embedding of dual Sobolev space ) of this section that the operation MATH is the extension of the scalar product MATH from MATH to MATH .

Proposition

(Representation of dual Sobolev space) For an MATH there exist functions MATH such that MATH

Proof

According to the proposition ( Riesz representation theorem ), for MATH there exists a MATH such that MATH We set MATH We calculate MATH MATH

Proposition

(Embedding of dual Sobolev space) For a bounded domain $U$ we have MATH where the last embedding is understood as MATH MATH

Proof

The first embedding is trivial. The equality follows from the proposition ( Riesz representation theorem ) and MATH being a Hilbert space. For the last embedding observe that MATH thus MATH





Notation. Index. Contents.


















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