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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.

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


e continue investigation of the operations $Q^{m}$ and $R^{m}$ introduced in the definitions ( Averaged Taylor polynomial ) and ( Remainder of averaged Taylor polynomial ). We consider a bounded set MATH and use the notation

MATH (Parameter d)

Proposition

(Riesz potential bound 1) Let MATH and either $1<p<\infty$ and $m>n/p$ or $p=1$ and $m\geq n$ then MATH

Proof

First, we consider the case $1<p<\infty$ and $m>n/p$ . According to the formula ( Holder inequality ) MATH We make the change to polar coordinates in the first integral: MATH for some function MATH and estimate MATH We need the expression MATH to be greater then $-1$ for the last integral to exist. We require MATH and transform the above into a condition for $m$ : MATH We continue estimation of MATH thus MATH In case $p=1$ and $m\geq n$ we take more direct route: MATH

Proposition

(Remainder bound 1) Let MATH and either $1<p<\infty$ and $m>n/p$ or $p=1$ and $m\geq n$ then MATH

Proof

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

Definition

(Star shaped set) The set $\Omega$ is "star shaped" with respect to the ball $B$ if MATH

Proposition

(Sobolev inequality) Suppose $\Omega$ is star shaped with respect to the ball MATH . Let MATH and either $1<p<\infty,$ $m>n/p$ or $p=1,$ $m\geq n$ then $u$ is continuous in $\Omega$ and MATH

Proof

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

Proposition

(Sobolev inequality 2) Suppose $\Omega$ is star shaped with respect to the ball MATH . Let MATH and either $1<p<\infty$ , $m-l-\frac{n}{p}>0$ or $p=1$ , $m-l-n\geq0$ then MATH

Proof

We apply the proposition ( Sobolev inequality ) in the following form MATH for all $\alpha$ s.t. MATH . Thus we need either $1<p<\infty,M>n/p$ or $p=1,M\geq n$ . We set $M=m-l$ and arrive to either $1<p<\infty$ , $m-l>\frac{n}{p}$ or $p=1$ , $m-l\geq n$ as required in the proposition. Therefore, we arrive to MATH

Proposition

(Riesz potential bound 2) Let MATH , $p\geq1$ , $m\geq1$ and MATH then MATH

Proof

For $1<p<\infty$ we estimate MATH We apply the formula ( Holder inequality 3 ) with MATH and $v=1$ to the internal integral. MATH We estimate MATH , MATH . MATH We change the order of integration (see the proposition ( Fubini theorem )). MATH Note that MATH . MATH

Thus MATH

For $p=1$ we estimate MATH We change the order of integration. MATH For $p=\infty$ we estimate MATH

Proposition

(Bramble-Hilbert lemma) Suppose $\Omega$ is star shaped with respect to MATH and MATH (see the definitions ( Star shaped set ),( Chunkiness parameter ) and the formula ( Parameter d )). Let MATH , $p\geq1$ . Then MATH

Proof

It suffices to prove the statement for MATH .

If $k=m$ then MATH (see the proposition ( Properties of averaged Taylor polynomial )-2), thus MATH

If $k=0$ then MATH and MATH We apply the proposition ( Remainder of averaged Taylor polynomial 2 ). MATH We apply the proposition ( Remainder of averaged Taylor polynomial 3 ). MATH We apply the proposition ( Riesz potential bound 2 ). MATH

If $0<k\leq m$ then MATH We apply the proposition ( Properties of averaged Taylor polynomial )-3. MATH We apply the case $k=0$ of this proof. MATH





Notation. Index. Contents.


















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