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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
1. Real Variable.
2. Laws of large numbers.
3. Characteristic function.
4. Central limit theorem (CLT) II.
5. Random walk.
6. Conditional probability II.
7. Martingales and stopping times.
8. Markov process.
9. Levy process.
10. Weak derivative. Fundamental solution. Calculus of distributions.
11. Functional Analysis.
A. Weak convergence in Banach space.
B. Representation theorems in Hilbert space.
C. Fredholm alternative.
D. Spectrum of compact and symmetric operator.
E. Fixed point theorem.
F. Interpolation of Hilbert spaces.
G. Tensor product of Hilbert spaces.
12. Fourier analysis.
13. Sobolev spaces.
14. Elliptic PDE.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Interpolation of Hilbert spaces.


et $X$ and $Y$ are separable Hilbert spaces, $X\subset Y$ , $X$ is dense in $Y$ and the natural injection operator $I:X\rightarrow Y$ , $Ix=x$ is continuous ( $X$ topology is stronger: $X$ -convergence implies $Y$ -convergence). We denote such relationship MATH Consider the form MATH Let $D\left( S\right) $ be the set of $u$ such that the form MATH is continuous in $Y$ topology. $D\left( S\right) $ is dense in $X$ with respect to $Y$ topology. By denseness of $X$ in $Y$ the form extends to $Y$ . Therefore, by the proposition ( Riesz representation theorem ) for any $u\in X$ there is an element $Su\in Y$ such that MATH The $S$ is a linear operator $S:X\rightarrow Y$ unbounded in $Y$ topology. Furthermore, MATH By MATH , continuous functions of $S$ are well defined. In particular, the operator $\Lambda$ MATH is well defined. We introduce MATH

Proposition

(Interpolation inequality) Given MATH we have MATH

Proposition

(Interpolated continuity) Suppose we have the pairs MATH and MATH . For a linear operator $A$ we have MATH





Notation. Index. Contents.


















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