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.

roposition

(Properties of averaged Taylor polynomial)

1. Let be a bounded domain in . For any we have

2. For we have

3. For s.t. and we have

Proof

Only the statement needs a proof.

First, we verify that for By the properties of derivative and structure of the formula it suffices to establish it for . We calculate At this point we make a change combined with renaming then the rules and transform into and thus

Next, we verify for Finally, we extend the proof to by noting that is dense in and the part of this statement insures that the case extends to by taking an -convergent sequence of functions.

 Notation. Index. Contents.