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==Rational functions==
[[File:ChebychevRational1.png|thumb|Plot of the Chebyshev rational functions of order n=0,1,2,3 and 4 between x=0.01 and 100.]]
Legendre and Chebyshev polynomials provide orthogonal families for the interval {{nowrap|[−1, 1]}} while occasionally orthogonal families are required on {{nowrap|[0, ∞)}}. In this case it is convenient to apply the [[Cayley transform#Real homography|Cayley transform]] first, to bring the argument into {{nowrap|[−1, 1]}}. This procedure results in families of [[rational function|rational]] orthogonal functions called [[Legendre rational functions]] and [[Chebyshev rational functions]].
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