Orthogonality of Chebyshev Polynomials of the Second Kind

Theorem

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 $\closedint {-1} 1$


That is:

Inequality

$\ds \int_{-1}^1 \sqrt {1 - x^2} \, \map {U_m} x \map {U_n} x \rd x = 0$

when $m \ne n$.


Equality

$\ds \int_{-1}^1 \sqrt {1 - x^2} \, \paren {\map {U_n} x}^2 \rd x = \dfrac \pi 2$


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 30$: Chebyshev Polynomials: Orthogonality: $30.38$, $30.39$
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 31$: Chebyshev Polynomials: Orthogonality: $31.38.$, $31.39.$
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Chebyshev polynomials
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