please see the picture below. Exercise 2 is just for definition.

This exercise uses the definition of the previous Exercise 2. Consideru : [−π, π] → R piecewise continuous with finitely many singularities, and its corresponding partial Fourier series for N ∈ N:

a0(f) N

SN f(x) = 2 +

where the ak's and bk's are the Fourier coefficients of f.

1. (4pts) Show that for any j = 0, ..., N:π

⟨f−SNf,cos(jx)⟩= (f(x)−SNf(x))cos(jx)=0 and ⟨f−SNf,sin(jx)⟩=0.−π

ak(f) cos(kx) + bk(f) sin(kx).

2. (4pts) Consider for some numbers c0 , c1 , ..., cN , and d1 , ..., dNpolynomial of degree N:

c0 N

P(x) = 2 +Show ⟨f − SN f, P ⟩ = 0.

another trigonometric

k=1

ck cos(kx) + dk sin(kx)

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