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Consider the vector space C[-1,1] with an inner product defined by

<f,g> = the integral from 1 to -1 of f(x)g(x) dx

a)

Show that

u1(x)= 1/(2^.5) u2(x)= ((6^.5)/2)x

form an orthonormal set of vectors

b)

Use the result from a) to find the best least squates approximation to

h(x)= x^(1/3) + x^(2/3)

by a linear function.

For part a) I have shown that u1 and u2 each have an inner product of zero and a length of one.

I've been trying to find a solution to b) in the form

p(x) = (c1)(u1(x)) + (c2)(u2(x))

where ci = the integral from -1 to 1 of (ui(x))h(x)

evaluating this integral for c1 produces

(1/(2^.5))[(3/4)x^(4/3) + (3/5)x^(5/3)] evaluated from -1 to 1

Now I've finally got to the problem

Putting -1 in would produce complex numbers and I don't know how to proceed.

I'm not sure weather the problem here is that I'm not aproaching the least squares problem correctly or weather I'm not approaching the integral correctly. Any help would be appreciated, thanks.

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# Least squares and integration problem

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