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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.
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.

Remainder of averaged Taylor decomposition.


e introduce the remainder

Definition

(Remainder of averaged Taylor polynomial) MATH

According to the proposition ( Properties of averaged Taylor polynomial )-2 and formula ( Mollifier for a ball 2 ), for a function MATH we have MATH We introduce the function $f\left( s\right) $ : MATH According to the proposition ( Integral form of Taylor decomposition ), MATH and MATH Consequently, MATH

We continue calculation of $R^{m}$ : MATH Our goal is to transform the integral to the form MATH .We make the change $1-\tau=s$ in the $\tau $ -integral, MATH , $d\tau=-ds$ , MATH We aim to make a change of variables MATH where MATH . Hence, we change the order of integration (see ( Fubini theorem )) MATH and proceed with the change MATH , MATH , $dz=s^{n}dy$ , MATH MATH We would like to put result in the form MATH , hence, we reverse the order of integration again. The set of integration is MATH Note that the line MATH for a fixed MATH connects $x$ with a point $y$ in MATH . Thus, the $z$ -plane projection of the set $\Omega$ is the convex hull of $x$ and MATH : MATH Let MATH be the section of $\Omega$ with a plane MATH : MATH We continue the calculation: MATH where the functions $k_{\alpha}$ are given by MATH We estimate (see the figure ( Estimation of averaged Taylor polynomial )) MATH
Estimation of averaged Taylor polynomial
Estimation of averaged Taylor polynomial

Thus MATH We state MATH and estimate the $k$ : MATH Note that MATH , MATH hence MATH By the formulas ( Mollifier for a ball 1 ),( Mollifier for a ball 2 ) and the definition ( Standard mollifier definition )-2 MATH where the $C$ depends on $n$ , so we drop $\frac{1}{n}$ and substitute for $s$ : MATH We summarize our findings in the following proposition.

Proposition

(Remainder of averaged Taylor polynomial 2) For MATH we have MATH where MATH MATH and MATH

Definition

(Chunkiness parameter) Let $\Omega$ be a bounded subset of $\QTR{cal}{R}^{n}$ and $x_{0}\in\Omega$ . Let MATH The the "chunkiness parameter" $\gamma$ is MATH

Proposition

(Remainder of averaged Taylor polynomial 3) In context of the proposition ( Remainder of averaged Taylor polynomial 2 ) there is a choice of the radius $\rho$ such that MATH

Proof

We have MATH It suffices to take MATH then MATH Thus MATH





Notation. Index. Contents.


















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