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 be a separable Hilbert space. A finite or countable subset , , is called a "frame system" if there are constants : such that for any is a called "frame" if its linear span is dense in : Let , are called "lower and upper frame bounds". The ratio is called "condition number" of . The system is called "minimal" if removing any element from it changes the linear span of . The system is called "tight" if .

Remark

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

Definition

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

Proposition

(Properties of frame operator) Let be a frame.

1. Analysis operator takes the form

2. The frame operator is a symmetric positive definite invertible operator,

2a. ,

2b. ,

2c. ,

2d. ,

2e. .

Proof

By definition of adjoint operator We set for every s.t. : where, by definition of , Thus or We have proven the claim 1.

2a is a direct verification.

To verify 2b we calculate, by definitions, and 2b follows by the definition ( Frame system ). Hence, is invertible and the rest follows directly.

Definition

(Dual frame) Given a frame and associated frame operator the system is called the "dual frame". We introduce the notation

Proposition

(Properties of dual frame) Let and are frame and frame operator.

1. For any

2. is a frame with a frame operator .

3. If is tight then .

Proof

We have and follows from the last result because (and thus ) 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 be a frame. For any

Proof

Note that We intend to derive the statement from the proposition ( Properties of Schwarz operator ). Thus, we consider the frame in context of the section ( Stable space splittings ). The subspaces are one-dimensional. is a space of sequences. The operator acts and is surjective by denseness of . The operator of the section ( Stable space splittings ) is identity. The scalar product in is given by Hence, we take : and achieve The existence of has already been established in the proposition ( Properties of frame operator ). In the same proposition, part 2b, stability of splitting was established. Thus

 Notation. Index. Contents.