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 are deduced from the rule Note that . The particular normalization is denoted , .

Proposition

(Trigonometry primer)

Proposition

(Chebyshev polynomials calculation) We have In particular,

Proposition

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

Proof

We verify orthogonality directly: We make the change , , , for , . Note that Thus

Proposition

(Minimum norm optimality of Chebyshev polynomials) We have

Proof

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

 Notation. Index. Contents.