Fast algorithms for discrete polynomial transforms

Consider the Vandermonde-like matrix P := (P-k(cos j pi/N))(j,k=0)(N), where the polynomials Pk satisfy a three-term recurrence relation. If P-k are the Chebyshev polynomials T-k, then P coincides with CN+1 := (cos jk pi/N)(j,k=0)(N). This paper presents a new fast algorithm for the computation of the matrix-vector product Pa in O(N log(2) N) arithmetical operations. The algorithm divides into a fast transform which replaces Pa with CN+1 (a) over tilde and a subsequent fast cosine transform. The first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications. Numerical tests demonstrate that our fast polynomial transform realizes Pa with almost the same precision as the Clenshaw algorithm, but is much faster for N greater than or equal to 128.