Question

A Trigonometric Polynomial of order n is a function of the form: ?(?) = ?0 + ?1 cos ? + ?1 sin ? + ?3 cos(2?) + ?2sin(2?) + ⋯ + ?ncos(??) + ?nsin (??)

1) Show that the set {1, cos ? , sin ? , cos(2?) , sin(2?)} is a basis for the vector space

?2 = {?(?) | ?(?)?? ? ????????????? ?????????? ?? ????? ≤ 2}

< ?, ? > = ∫ ?(?)?(?)?? defines an inner-product on T

2) Use Gram-Schmidt to show an ONB for T is:

?0 = 1 √2? , ?1 = 1/ √? cos(?) , ?2 = 1 /√? cos(2?) ?3 = 1/√? sin(?) , ?4 = 1/ √? sin(2?)

Given any ?(?) ∈ ?2

????r? = < ?, ?0 > ?0+ < ?, ?1 > ?1+ . . + < ?, ?4 > ?4

3) Show that in ?m

????r ? = ?0 + ∑ [?n cos(??) + ?nsin (??)] from n to m, when  n=1

Where: ?0 = 1/2pi ∫ ?(?)?? from 0 to 2pi

?n = 1/pi∫ ?(?) cos(??) ?? from 0 to 2pi , ? ≥ 1

?n = 1/pi ∫ ?(?) sin(??) ?? from 0 to 2pi , ? ≥ 1

We call the ?/? and ?/? the Fourier Coefficients of ?(?)

We call the Projection the Fourier Series for ?(?)

4) Compute the ?2 series for ?(?) = ?^2 using − pi/2 < ? < pi/2. Plot your series and f(x) together

5) Compute the series for ?(?) = ?^3 using − pi/2 < ? < pi/2 . Plot your series and g(x) together

Answer 1