Orthogonality of Chebyshev Polynomials of the First Kind

Theorem

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 $\closedint {-1} 1$


That is:

Inequality

$\ds \int_{-1}^1 \dfrac {\map {T_m} x \map {T_n} x} {\sqrt {1 - x^2} } \rd x = 0$

when $m \ne n$.


Equality

$\ds \int_{-1}^1 \dfrac {\paren {\map {T_n} x}^2} {\sqrt {1 - x^2} } \rd x = \begin {cases} \pi & : n = 0 \\ \\ \dfrac \pi 2 & : n = 1, 2, \ldots \end {cases}$


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 30$: Chebyshev Polynomials: Orthogonality: $30.18$, $30.19$
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): orthogonal polynomials
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Tchebyshev polynomials
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): orthogonal polynomials
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Tchebyshev polynomials
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 31$: Chebyshev Polynomials: Orthogonality: $31.18.$, $31.19.$
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Chebyshev polynomials
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