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**Unformatted text preview: **Math 602 Homework 8 1. (i) Check that the functions 1, sin(nx), cos(nx), (n = 1, 2, 3 . . .) form
an orthogonal system in L2 [−π, π ].
(ii) Normalize them to obtain an orthonormal system.
(iii) Assuming that f ∈ L2 [−π, π ] has its Fourier series expansion
∞ a0
[an cos(nx) + bn sin(nx)]
+
f=
2
n=1 (1) verify that the Fourier coeﬃcients an and bn are given by the formulas
an = 1
π π f (x) cos(nx) dx , bn =
−π 1
π π f (x) sin(nx) dx,
−π 2. (i) Check that the functions einx , n ∈ Z form an orthogonal system
in the complex valued functions L2 [−π, π ].
(ii) Normalize them to obtain an orthonormal system.
(iii) Assuming that f ∈ L2 [−π, π ] has its Fourier series
∞ ˆ
fn einx f= (2) n=−∞ verify that the Fourier coeﬃcients are given by
1
ˆ
fn =
2π π f (x)e−inx dx
−π ˆ
(iv) Verify that the Fourier coeﬃcients an , bn , fn of a function f (x) are
related by the formulas
ˆ
ˆ
ˆ
ˆ
an = fn + f−n for n = 0, 1, 2, . . . , bn = i(fn − f−n ) for n = 1, 2, . . .
ˆ
(v) Find the conditions on fn that ensure that the function f (x) is real
valued.
1 4. Use a change of the variable x in (2) to show that the Fourier series
of a function g ∈ L2 [a, b] has the form
∞ gn e2πinx/(b−a)
ˆ g=
n=−∞ and ﬁnd the formula that expresses gn in terms of g (x).
ˆ
5. The Legendre orthogonal polynomials are orthogonal in L2 [−1, 1].
Use a Gram-Schmidt process on the polynomials 1, x, x2 , x3 to obtain an
orthonormal set; these are the ﬁrst four Legendre polynomials.
6. The Laguerre orthogonal polynomials are orthogonal in the weighted
L2 ([0, +∞), e−x dx).
Use a Gram-Schmidt process on the polynomials 1, x, x2 , x3 to obtain an
orthonormal set; these are the ﬁrst four Laguerre polynomials. 2 ...

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