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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.
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.

Frames.


efinition

(Frame system) Let $H$ be a separable Hilbert space. A finite or countable subset $\Phi$ , $\Phi\subset H$ , is called a "frame system" if there are constants $A,B$ : $0<A\leq B<+\infty$ such that for any MATH MATH $\Phi$ is a called "frame" if its linear span is dense in $H$ : MATH Let MATH MATH , MATH are called "lower and upper frame bounds". The ratio MATH is called "condition number" of $\Phi$ . The system $\Phi$ is called "minimal" if removing any element from it changes the linear span of $\Phi$ . The system $\Phi$ is called "tight" if MATH .

Remark

Analysis of frames and frame systems is similar. Frame systems have properties in MATH . Frames have the same properties in $H$ . For this reason we study frames in this section.

Definition

(Frame operator) Given a frame MATH the operator MATH is called "synthesis operator". The adjoint operator MATH is called "analysis operator". The operator MATH is called "frame operator".

Proposition

(Properties of frame operator) Let MATH be a frame.

1. Analysis operator $R^{\ast}$ takes the form MATH

2. The frame operator $\QTR{cal}{P}$ is a symmetric positive definite invertible operator,

2a. MATH ,

2b. MATH ,

2c. MATH ,

2d. MATH ,

2e. MATH .

Proof

By definition of adjoint operator MATH We set MATH for every $p$ s.t. $f_{p}\in\Phi$ : MATH where, by definition of $R$ , MATH Thus MATH or MATH We have proven the claim 1.

2a is a direct verification.

To verify 2b we calculate, by definitions, MATH and 2b follows by the definition ( Frame system ). Hence, $\QTR{cal}{P}$ is invertible and the rest follows directly.

Definition

(Dual frame) Given a frame $\Phi$ and associated frame operator $\QTR{cal}{P}$ the system MATH is called the "dual frame". We introduce the notation MATH

Proposition

(Properties of dual frame) Let $\Phi$ and $\QTR{cal}{P}$ are frame and frame operator.

1. For any $g\in H$ MATH

2. $\tilde{\Phi}$ is a frame with a frame operator $\QTR{cal}{P}^{-1}$ .

3. If $\Phi$ is tight then $\Phi=\tilde{\Phi}$ .

Proof

We have MATH MATH and MATH follows from the last result because $\QTR{cal}{P}$ (and thus $\QTR{cal}{P}^{-1}$ ) is symmetric. We have proven the claim 1.

The rest is a consequence of the claim 1 and the proposition ( Properties of frame operator ).

Proposition

(Frame norm) Let $\Phi$ be a frame. For any $g\in H$ MATH

Proof

Note that MATH We intend to derive the statement from the proposition ( Properties of Schwarz operator ). Thus, we consider the frame $\Phi$ in context of the section ( Stable space splittings ). The subspaces MATH are one-dimensional. $\tilde{H}$ is a space of sequences. The operator $R$ acts MATH and is surjective by denseness of $span~\Phi$ . The operator $A$ of the section ( Stable space splittings ) is identity. The scalar product in $\tilde{H}$ is given by MATH Hence, we take MATH : MATH and achieve MATH The existence of $\QTR{cal}{P}^{-1}$ has already been established in the proposition ( Properties of frame operator ). In the same proposition, part 2b, stability of splitting was established. Thus MATH





Notation. Index. Contents.


















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