An Introduction to Orthogonal Polynomials
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In this introductory talk, we first revisit with proof for illustration purposes some basic properties of a specific system of orthogonal polynomials, namely the Chebyshev polynomials of the first kind. Then we define the notion of orthogonal polynomials and provide with proof some basic properties such as: The uniqueness of a family of orthogonal polynomials with respect to a weight (up to a multiplicative factor), the matrix representation, the three-term recurrence relation, the Christoffel-Darboux formula and some of its consequences such as the separation of zeros and the Gauss quadrature rules.
KeywordsOrthogonal polynomials Differential equations Chebyshev polynomials Zeros
Mathematics Subject Classification (2000)Primary 33C45; Secondary 42C05
The author would like to thank the anonymous reviewer for his careful reading of the manuscript and his many insightful comments and suggestions.
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