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Inner Product, Linear Space, Constant Polynomial and Mean Square Deivation

Let C[1,3] be the (real) linear space of all real continuous functions on the closed interval [1,3], equipped with the inner product defined by setting

<f,g> := 1&#8747;3f(t)g(t)dt, f,g E C[1,3].

Let f(t) = 1/t, t E [1,3].

(i). Show that the constant polynomial g which best approximates f on [1,3] (in the sense of least squares) is given by
g(t) = ½ ln3, t E [1,3].

Find the mean square deviation ||f-g||2.

(ii) Find the best linear polynomial approximate to f on [1,3] and calculate the corresponding mean square deviation.
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(See attached file for full problem description and accurate equations)

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Inner Product, Linear Space, Constant Polynomial and Mean Square Deivation are investigated. The solution is detailed and well presented.

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