What can you say about the convergence of the previously

Question

Question (1) Expand the functions f, g, h : [0, Π] → R; f(t) = t, g(t) = t², h(t) = 2 in terms of the cosine family previously introduced. Expand also the function F: [0, Π] → R, F(t) = at² + bt + c where a, b, c are nonzero constants.

Question (2) Are the series that you have found in (2) converging uniformly and pointwisely to the functions f, g and h respectively? Why? Justify your answers!

Question (3) What can you say about the L² convergence of the previously defined series? Justify your answer!

Question (4) Is it possible to expand the function sin(kt) in terms of the cosine family from (1) on the interval [0, Π] (where k ∈ N)? Why or why not? Is yes, then perform the expansion. Justify your answers!