From: rwg@best.com Message-Id: <200003260524.VAA21784@shell12.ba.best.com> To: math-fun@optima.CS.Arizona.EDU

A&S 9.1.44 with z=r, cos(theta) = t is

                    inf                     ====                     \          k        cos(r t) = 2  >    (- 1)  J   (r) T   (t) + J (r),                     /             2 k     2 k       0                     ====                     k = 1

whose truncations give optimal (in the Tchebychev sense) approximations to cos(r t) in [-1,1].  Thus, taking only terms 0,1, with r=pi/2, gives

--RWG may be confused, anyhow this sentence was wrong. Specifically, Chebyshev series do NOT when truncated yield optimum i.e. equiripple approximations, though there are theorems and experience both saying that Chebyshev truncations usually are not too bad. To get the true equiripple there are proceudres such as Remez iteration that converge to it starting from good enough initial guess.