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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
1. Finite differences.
2. Gauss-Hermite Integration.
A. Gram-Schmidt orthogonalization.
B. Definition and existence of orthogonal polynomials.
C. Three-term recurrence relation for orthogonal polynomials.
D. Orthogonal polynomials and quadrature rules.
E. Extremal properties of orthogonal polynomials.
F. Chebyshev polynomials.
3. Asymptotic expansions.
4. Monte-Carlo.
5. Convex Analysis.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Chebyshev polynomials.


efinition

(Chebyshev polynomials) Chebyshev polynomials MATH are deduced from the rule MATH Note that MATH . The particular normalization $t^{n}+...$ is denoted $T_{n}^{0}$ , MATH .

Proposition

(Trigonometry primer) MATH MATH MATH

Proposition

(Chebyshev polynomials calculation) We have MATH MATH In particular, MATH

Proposition

(Chebyshev polynomials orthogonality) Chebyshev polynomials are orthogonal with respect to the measure MATH : MATH

Proof

We verify orthogonality directly: MATH We make the change $t=\cos\theta$ , MATH , MATH , MATH for MATH , MATH . MATH Note that MATH Thus MATH

Proposition

(Minimum norm optimality of Chebyshev polynomials) We have MATH

Proof

Because MATH the polynomial MATH alternates between its minimal value $-C_{n}$ and maximal value $C_{n}$ on the interval MATH and achieves each extremum $n$ times on MATH . Assume that there exists a MATH such that MATH . Then MATH changes sign $n$ times and, thus, has $n$ zeros. But MATH and cannot have $n$ zeros.





Notation. Index. Contents.


















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