In  Olver and Townsend presented a fast spectral method for solving linear differential equations using bases of ultraspherical polynomials
, which has subsequently been exploited in Chebfun  and ApproxFun .
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uncertainty principle, self-adjoint operators, symmetric operators, normal operators, periodic functions, ultraspherical polynomials, sphere.
Ultraspherical polynomials are solutions of the differential equations [L.