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# Exercise 3 ( 16 pts ) . This exercise uses the definition of the previous Exercise 2. Consider 2 : 1 - IT , IT ] -&gt; KR piecewise continuous with...

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)   Exercise 3 ( 16 pts ) . This exercise uses the definition of the previous Exercise 2. Consider
2 : 1 - IT , IT ] -&gt; KR piecewise continuous with finitely many singularities , and it's corresponding
partial Fourier series for NEN :`
SN f ( ac ) = 20(&gt; &gt; &gt; &gt; ak ( f ) cos ( Kal ) + 6x ( f ) sin ( Kac ) .
where the ar's and be's are the Fourier coefficients of f
1 . ( 4 pts ) Show that for any j = O , .. , N :`
( f - SNS , cos ( jac ) ) = / ( f ( DC ) - S'N f ( 20 ) cos ( jac ) = 0 and (f - SNS , sin ( joc ) ) = 0 .
2 . ( 4pts ) Consider for some numbers co , CI . .. . CN , and di . .., AN another trigonometric
polynomial of degree N :
P ( 20 ) =`
co _
NN
+ &gt; CK cos ( kx ) + d* sin ( Knee )
Show ( f - SNS , P ) = 0.
3 . ( 4pts ) Show that if g , h : [ - IT , IT ) ~ R are two piecewise continuous functions with
finitely many discontinuities such that ( g , h ) = O , then Ilg + hillZz = 1/8/13,2 + 11 /2/13,2\
4. ( 4pts ) Use the two previous Questions 2. and 3. to show that , for any trigonometric
polynomial P of degree N :`
|1.8 - SN filEz &lt; If - PlLIZ.
Note : this is interpreted as the fact that Sif is the best trigonometric polynomial of
degree N approximative f , because the distance from &amp; to Sif is the smallest one
among all trigonometric polynomials of degree N . Exercise 2 ( The pre- Hilbert space [ _ ( 10pts) ) . Assume f, g: [- IT, IT ] - IR are both contin-
vous functions . We define the IZ norm of f , and the scalar product between { and }:`
&quot; IT
If ( *c ) 1 2 doc
and
( 5 . 9 ) =
f ( 20 ) 9 ( ac ) doc .
- IT
- IT

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