1. In addition to their applications for data fitting, the Chebyshev polynomials Tn(x) are useful for approximating func

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1 In Addition To Their Applications For Data Fitting The Chebyshev Polynomials Tn X Are Useful For Approximating Func 1 (113.56 KiB) Viewed 75 times

1. In addition to their applications for data fitting, the Chebyshev polynomials Tn(x) are useful for approximating functions. This problem is concerned with approximations of the type f(x) – Sn(x), where Sn(x) knCnTn(x). (*) N n=0 if n = 0, Here {cn} is a set of constants that depend on f, and kn = -{ otherwise. (a) Let m and n be nonnegative integers. Use the substitution x = cos 8 to show that the Chebyshev polynomials Tn (2) satisfy the orthogonality relationship 1 if m= n, | Tm(z)Ty() dc = TJ-1 1 - 22 kin 0 otherwise. = COS Hint: 2 cos(m6) cos(no) (b) Consider the residual cos((m+n)e) + cos((m – n)o). т 2 RN [f(x) – Sn(x)]? d.x. V1 - x2 TJ-1 aRN Find an expression for ac; and hence deduce that the residual is minimised if Cj - L1,(dr, j = 0,1.... 2 TJ-x2 z)f(x ,