HW8_2011 copy

# HW8_2011 copy - Math 602 Homework 8 1(i Check that the...

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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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HW8_2011 copy - Math 602 Homework 8 1(i Check that the...

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