Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
1. Calculational Linear Algebra.
2. Wavelet Analysis.
3. Finite element method.
4. Construction of approximation spaces.
A. Finite element.
B. Averaged Taylor polynomial.
a. Properties of averaged Taylor polynomial.
b. Remainder of averaged Taylor decomposition.
c. Estimates for remainder of averaged Taylor polynomial. Bramble-Hilbert lemma.
d. Bounds for interpolation error. Homogeneity argument.
C. Stable space splittings.
D. Frames.
E. Tensor product splitting.
F. Sparse tensor product. Cure for curse of dimensionality.
5. Time discretization.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Properties of averaged Taylor polynomial.


(Properties of averaged Taylor polynomial)

1. Let $\Omega$ be a bounded domain in $\QTR{cal}{R}^{n}$ . For any $k,x_{0},\rho$ we have MATH

2. For MATH we have MATH

3. For $\forall\alpha$ s.t. MATH and MATH we have MATH


Only the statement $\left( 3\right) $ needs a proof.

First, we verify that for MATH MATH By the properties of derivative and structure of the formula MATH it suffices to establish it for MATH . We calculate MATH At this point we make a change $i_{1}-1=j_{1}$ combined with renaming MATH then the rules $i_{1}>0$ and MATH transform into $i_{1}\geq0$ and MATH thus MATH

Next, we verify $\left( 3\right) $ for MATH MATH Finally, we extend the proof to $u\in L^{1}$ by noting that $C^{\infty}$ is dense in $L^{1}$ and the part $\left( 1\right) $ of this statement insures that the $C^{\infty}$ case extends to $L^{1}$ by taking an $L^{1}$ -convergent sequence of $C^{\infty}$ functions.

Notation. Index. Contents.

Copyright 2007