I am having trouble with this question (gap in my notes, always happens)

The nth Chebyshev polynomial Tn(x) is defined by Tn(x) = cos(n cos^-1(x)) for -1<x<1 and n = 0,1,2,3,...

Assume these polynomials satisfy the recurrence relation

xTn(x) = 1/2Tn+1(x) + 1/2Tn-1(x) for n =1,2,3,4,...

a) Express the polynomial p(x) = -2x^4 +3x^3 -2x^2 +x -2 in the form

c0T0(x) + c1T1(x) +...+c4T4(x) for suitable constants c0,c1,...,c4.

(Hint: x^k = x.x^k-1)

b) Econimise p(x) first to a degree three polynomial and then to a degree two polynomial.

c) For each of the approximations in part (b), give an upper bound for the total error and state the range of values of x for which this bound is valid.

I dont know where to start with part a) but if I can get help with it I should have no problems with parts b) & c)

Thanks for any help...

The nth Chebyshev polynomial Tn(x) is defined by Tn(x) = cos(n cos^-1(x)) for -1<x<1 and n = 0,1,2,3,...

Assume these polynomials satisfy the recurrence relation

xTn(x) = 1/2Tn+1(x) + 1/2Tn-1(x) for n =1,2,3,4,...

a) Express the polynomial p(x) = -2x^4 +3x^3 -2x^2 +x -2 in the form

c0T0(x) + c1T1(x) +...+c4T4(x) for suitable constants c0,c1,...,c4.

(Hint: x^k = x.x^k-1)

b) Econimise p(x) first to a degree three polynomial and then to a degree two polynomial.

c) For each of the approximations in part (b), give an upper bound for the total error and state the range of values of x for which this bound is valid.

I dont know where to start with part a) but if I can get help with it I should have no problems with parts b) & c)

Thanks for any help...

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