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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.

Compact embedding of Sobolev spaces.


efinition

Let $X$ and $Y$ be Banach spaces and $X\subset Y$ . The $X$ is "compactly embedded" in $Y$ (notation $X\subset\subset Y$ ) if any $X-$ bounded sequence has a $Y$ -convergent subsequence.

Proposition

(Uniformly smooth approximation lemma) Suppose $U$ is an open bounded subset of $\QTR{cal}{R}^{n}$ and $\partial U\,$ admits a locally continuously differentiable parametrization. For any MATH -bounded sequence MATH there is a family of MATH functions MATH such that MATH for each MATH . Furthermore, the sequence MATH is uniformly bounded and equicontinuous for every $\varepsilon$ .

Proof

We seek such family using the standard mollifiers, see the definition ( Standard mollifier definition ). We introduce MATH : MATH we substitute MATH , MATH : MATH Note that the argument $x-\varepsilon z$ lies outside of $U$ for $x$ close to the boundary $\partial U$ . We solve this problem via the proposition ( Extension theorem ): we extend the $u_{m}$ to a slightly larger set $V$ and perform the convolution using the extended functions.

We initially assume that the functions $u_{m}$ are smooth and estimate MATH note that MATH , MATH Hence, MATH MATH Since the support of the extended function $u_{m}$ lies strictly within $V$ , we can drop the small shift $\varepsilon tz$ in the integral MATH without changing the integral: MATH Hence, MATH According to the proposition ( Approximation by smooth functions ) this estimate extends to the functions MATH .

Finally, we use the boundedness of $U$ (and $V$ ) and the formula ( Holder inequality ) with $v=1$ : MATH Hence, MATH Thus we established that MATH

Next we intend to the other values of MATH using the formula ( Lp interpolation ): MATH

We aim to establish that MATH dominates MATH . Hence, we match $r=q$ , $s=1$ and $t=p^{\ast}$ . Then MATH Hence, MATH The sequences MATH are bounded in MATH . Hence, by the proposition ( Gagliardo-Nirenberg-Sobolev inequality ) these are bounded in MATH . Therefore, the MATH term in the last estimate is bounded and, consequently, MATH

To see the second part of the statement we verify that MATH and MATH are bounded for every $\varepsilon$ : MATH MATH

Proposition

(Rellich-Kondrachov compactness theorem). Let $U$ be a bounded open subset of $\QTR{cal}{R}^{n}$ and $\partial U\,$ admits a locally continuously differentiable parametrization. Suppose $1\leq p<n$ . Then MATH

Proof

Let MATH be a MATH -bounded sequence. We utilize the sequence MATH from the proposition ( Uniformly smooth approximation lemma ): MATH and, according to the proposition ( Arzela-Ascoli compactness criterion ), MATH

Since $U$ is bounded, it follows that MATH Let MATH , $k=1,2,...$ . For $\delta_{1}$ we find $\varepsilon_{1}$ such that MATH and for such $\varepsilon_{1}$ we find the subindexing MATH such that MATH Then for $\delta_{2}$ we find $\varepsilon_{2}$ such that MATH and for such $\varepsilon_{2}$ we find the subindexing MATH of MATH such that MATH and we continue so indefinitely.

Then MATH Hence, the MATH is the desired subindexing that turns MATH into an $L^{q}$ -convergent sequence.

Proposition

(W1p embedding). Let $U$ be a bounded open subset of $\QTR{cal}{R}^{n}$ and $\partial U\,$ admits a locally continuously differentiable parametrization ( $\equiv\partial U$ is $C_{1}$ ). Then for MATH MATH

Proof

For $p\in\lbrack1,n)$ and MATH we have $p^{\ast}>p$ . Hence, by the proposition ( Rellich-Kondrachov compactness theorem ) MATH

For $p\in(n,\infty]$ we use the proposition ( C0gamma vs W1p estimate ) and note that the structure of $C^{0,\gamma}$ norm implies that a $C^{0,\gamma}$ -bounded sequence satisfies conditions of the proposition ( Arzela-Ascoli compactness criterion ). We then follow the proof of the proposition ( Rellich-Kondrachov compactness theorem ) and arrive to MATH To see the embedding for $p=n$ note from the proposition ( Rellich-Kondrachov compactness theorem ) that MATH if $p\uparrow n$ . The $U$ is bounded, hence for $p<p_{1}$ MATH Therefore, if MATH is a MATH -bounded sequence then it is also a MATH -bounded sequence and we can choose $\varepsilon>0$ so that MATH . Then the proposition ( Rellich-Kondrachov compactness theorem ) provides existence of MATH -convergent subsequence.





Notation. Index. Contents.


















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