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)

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

We continue calculation of : Our goal is to transform the integral to the form .We make the change in the -integral, , , We aim to make a change of variables where . Hence, we change the order of integration (see ( Fubini theorem )) and proceed with the change , , , We would like to put result in the form , hence, we reverse the order of integration again. The set of integration is Note that the line for a fixed connects with a point in . Thus, the -plane projection of the set is the convex hull of and : Let be the section of with a plane : We continue the calculation: where the functions are given by We estimate (see the figure ( Estimation of averaged Taylor polynomial ))
Estimation of averaged Taylor polynomial

Thus We state and estimate the : Note that , hence By the formulas ( Mollifier for a ball 1 ),( Mollifier for a ball 2 ) and the definition ( Standard mollifier definition )-2 where the depends on , so we drop and substitute for : We summarize our findings in the following proposition.

Proposition

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

Definition

(Chunkiness parameter) Let be a bounded subset of and . Let The the "chunkiness parameter" is

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 such that

Proof

We have It suffices to take then Thus

 Notation. Index. Contents.