gram_polynomial, a Python code which evaluates the Gram polynomials, also known as the discrete Chebyshev polyomials.
The Gram polynomial P(n,m,x) can be evaluated at a point x by:
P(0,m,x) = 1 P(1,m,x) = x P(n+1,m,x) = x * P(n,m,x) - beta(n,m) * P(n-1,m,x)where beta(n,m) = (m^2-n^2)*n^2/m^2/(4*n^2-1).
The polynomials are orthogonal with respect to a discrete inner product
(f,g) = sum ( 1 <= i <= m ) f(x(i)) * g(x(i))where
x(i) = - 1 + ( 2*i-1)/m, 1 <= i <= m.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
gram_polynomial is available in a MATLAB version and a Python version.
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