Chebyshev interpolation and quadrature formulas of very high degree

All the zeros x2m,i, i = 1(1)2m, of the Chebyshev polynomials T2m(x), m = 0(1)n, are found recursively just by taking n2n-1 real square roots. For interpolation or integration of ƒ(x), given ƒ(x2m,i), only x2m,i is needed to calculate (a) the (2m - 1)-th degree Lagrange interpolation polynomial, and (b) the definite integral over [-1, 1], either with or without the weight function (1 - x2)-1/2, the former being exact for ƒ(x) of degree 2m+1 - 1.

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