Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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.

## Estimates for remainder of averaged Taylor polynomial. Bramble-Hilbert lemma.

e continue investigation of the operations and introduced in the definitions ( Averaged Taylor polynomial ) and ( Remainder of averaged Taylor polynomial ). We consider a bounded set and use the notation

 (Parameter d)

Proposition

(Riesz potential bound 1) Let and either and or and then

Proof

First, we consider the case and . According to the formula ( Holder inequality ) We make the change to polar coordinates in the first integral: for some function and estimate We need the expression to be greater then for the last integral to exist. We require and transform the above into a condition for : We continue estimation of thus In case and we take more direct route:

Proposition

(Remainder bound 1) Let and either and or and then

Proof

Let then we apply the propositions ( Remainder of averaged Taylor polynomial 2 ),( Remainder of averaged Taylor polynomial 3 ): We apply the proposition ( Riesz potential bound 1 ): The proof extends to because is dense in .

Definition

(Star shaped set) The set is "star shaped" with respect to the ball if

Proposition

(Sobolev inequality) Suppose is star shaped with respect to the ball . Let and either or then is continuous in and

Proof

We have We apply the proposition ( Remainder bound 1 ) to the first term and the proposition ( Properties of averaged Taylor polynomial )-1 to the second term:

Proposition

(Sobolev inequality 2) Suppose is star shaped with respect to the ball . Let and either , or , then

Proof

We apply the proposition ( Sobolev inequality ) in the following form for all s.t. . Thus we need either or . We set and arrive to either , or , as required in the proposition. Therefore, we arrive to

Proposition

(Riesz potential bound 2) Let , , and then

Proof

For we estimate We apply the formula ( Holder inequality 3 ) with and to the internal integral. We estimate , . We change the order of integration (see the proposition ( Fubini theorem )). Note that .

Thus

For we estimate We change the order of integration. For we estimate

Proposition

(Bramble-Hilbert lemma) Suppose is star shaped with respect to and (see the definitions ( Star shaped set ),( Chunkiness parameter ) and the formula ( Parameter d )). Let , . Then

Proof

It suffices to prove the statement for .

If then (see the proposition ( Properties of averaged Taylor polynomial )-2), thus

If then and We apply the proposition ( Remainder of averaged Taylor polynomial 2 ). We apply the proposition ( Remainder of averaged Taylor polynomial 3 ). We apply the proposition ( Riesz potential bound 2 ).

If then We apply the proposition ( Properties of averaged Taylor polynomial )-3. We apply the case of this proof.

 Notation. Index. Contents.