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.

## Averaged Taylor polynomial.

n the section ( Convolution and smoothing ) we introduced and stated properties of the standard mollifier . We use to extend Taylor expansion to locally integrable functions.

Definition

(Taylor polynomial) For a function , the Taylor polynomial evaluated at a point is where we used the multi-index notation of the section ( Function spaces ) in the second line of the above definition.

Note that we do not include -order derivatives. The reason for it will become apparent when we discuss the residue term of Taylor decomposition.

To extend the above definition to integrable functions without derivatives we mollify (see the section ( Convolution and smoothing )) over a small ball and shift all derivatives to the mollifier via integration by parts. Thus, let be the mollifier over a ball :

 (Mollifier for a ball 1)
where the constant comes from the requirement
 (Mollifier for a ball 2)
We define the averaged Taylor polynomial We introduce the coefficients by postulating
 (Coefficients a)
and write

Definition

(Averaged Taylor polynomial) For a ball and a function we define a polynomial where the function is defined by the formulas ( Mollifier for a ball 1 ),( Mollifier for a ball 2 ) and the coefficients are defined by the formula ( Coefficients a ).

 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.
 Notation. Index. Contents.