Detailed Proof of Favard's Theorem

Theorem Statement

A sequence of polynomials ${P_n(x)}$ is orthogonal with respect to some positive measure if and only if it satisfies:

P_n(x) = (A_n x + B_n)P_{n-1}(x) - C_n P_{n-2}(x) \quad \text{for} \; n \geq 2

where $A_n > 0$, $C_n > 0$, and $B_n$ is any real number.

Proof

Part 1: Recurrence Relation Implies Orthogonality

1. Polynomial Construction: Define $P_n(x)$ using the given recurrence relation.
2. Demonstrating Orthogonality:
   • Prove $\int P_n(x) P_m(x) d\mu(x) = 0$ for $m \neq n$ under measure $\mu$.
   • Apply linear operators in Hilbert spaces and orthogonal projections.

Application of Mathematical Principles

• Measure Theory:
  • Integral with respect to measure $\mu$ for inner product definition in square-integrable function spaces.
  • Define orthogonality via this inner product.
• Linear Algebra and Functional Analysis:
  • Analyze polynomial space as a Hilbert space.
  • Utilize linear operators for polynomial manipulation.
  • Employ orthogonal projections for function decomposition into polynomial bases.

Part 2: Orthogonality Implies Recurrence Relation

1. Starting with Orthogonality: Assume an orthogonal sequence under measure $\mu$.
2. Deriving Recurrence Relation:
   • Represent $xP_n(x)$ as a linear combination of orthogonal polynomials.
   • Use orthogonality to find coefficients, leading to the recurrence relation.

Conclusion

This proof demonstrates the relationship between the algebraic structure of polynomial sequences and their analytical properties of orthogonality under a measure, using principles from measure theory, linear algebra, and functional analysis.