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.

## Extremal properties of orthogonal polynomials.

roposition

(Extremal properties of orthogonal polynomials) Let be the orthogonal polynomials with respect to the measure . Then

Proof

By the proposition ( Basic property of orthogonal polynomials ), for any we have for some numbers . The leading term is such because By orthogonality of , Thus is achieved at

 Notation. Index. Contents.