Please see the picture below

This exercise uses the definition of the previous Exercise 2. Consideru: [??, ?]?Rpiecewise continuous with finitely many singularities, and its corresponding partial Fourier series forN?N:

a0(f)N

SNf(x) =2+

where theak's andbk's are the Fourier coefficients off.

1. (4pts) Show that for anyj= 0, ..., N:?

?f?SNf,cos(jx)?= (f(x)?SNf(x))cos(jx)=0and?f?SNf,sin(jx)?=0.??

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

2. (4pts) Consider for some numbersc0, c1, ..., cN, andd1, ..., dNpolynomial of degreeN:

c0N

P(x) =2+Show?f?SNf, P?= 0.

another trigonometric

k=1

ckcos(kx) +dksin(kx)

Exercise 3 ( 1 6 pts) . This exercise uses the definition of the previous Exercise 2. Consider U : \\ - TT , TT - R piecewise continuous with finitely many singularities , and it's corresponding partial Fourier series for NEN:" Golf ) N SN f ( 20 ) =` 2 + [ ax ( f ) cos ( Knee ) + 6* ( f ) sin ( K`) . K = ] 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 ( joc ) ) = PIT ( of ( 20 ) - SN f ( 20 ) cos ( joc) = 0 and (f - SN.f , sin ( joc ) ) = O. - IT 2 . ( 4 pts ) Consider for some numbers co , CI . .., CN , and d. . .., AN another trigonometric polynomial of degree N :` P ( 20 ) = CO N . + > CK cos ( Kac ) + d* sin ( Kac ) Show ( f - SN f , P ) = O.Exercise 2 ( The pre- Hilbert space [ _ ( 10pts) ) . Assume fig : [ - IT, IT] - TR are both contin- vous functions . We define the IZ norm of f , and the scalar product between { and }: " IT 1.81/ 12 =\\ VL` If (*c ) 12 dac and ( f . 9 ) = f ( 20 ) 9 ( 20 ) doc