Orthogonality of Chebyshev Polynomials

Theorem

First Kind

The Chebyshev polynomials of the first kind form a set of orthogonal polynomials with respect to:

the closed real interval $\closedint {-1} 1$
the weight function $\map w x := \dfrac 1 {\sqrt {1 - x^2} }$ on $\closedint {-1} 1$


Second Kind

The Chebyshev polynomials of the second kind form a set of orthogonal polynomials with respect to:

the closed real interval $\closedint {-1} 1$
the weight function $\map w x := \sqrt {1 - x^2}$ on $\closedint {-1} 1$


Sources

  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Chebyshev polynomials
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