Homework 4

Due: Friday, 22 March 2013.
Homework 4
MATH 504, Spring 2013
In the following (1. - 3.) let {φ n } be a sequence of orthogonal polynomials
in L 2 [a, b] with respect to some weight w. Assume that φ n has degree
exactly n. P n is the space of polynomials of degree n or less.
1. Prove that the φ n s are unique in the following sense: If {ψ n } is another
sequence of orthogonal polynomials such that ψ n has degree exactly n,
then ψ n = α n φ n for some α n ≠ 0.
2. Given w > 0, f ∈ C[a, b], and n ≥ 1, show that p ∗ ∈ P n−1 is the best
(or least squares) approximation to f out of P n−1 (with respect to w)
if and only if 〈f − p ∗ , p〉 = 0 for every p ∈ P n−1 .
3. Let {φ n } be a complete family in L 2 [a, b]. Define a function A from
L 2 [a, b] into l 2 as
A(f) = {c k } ∞ k=1
where c k = 〈f, φ k 〉.
(a) Show that A is one-to-one.
(b) Show that A maps L 2 [a, b] onto l 2 .
4. Given the partition x 0 = 0, x 1 = 0.05, x 2 = 0.1 of [0, 0.1] and f(x) =
e 2x .
(a) Find the cubic spline s with natural boundary conditions that
interpolates f.
(b) Find an approximation for ∫ 0.1
e 2x dx by evaluating ∫ 0.1
s(x) dx.
0 0 (c) Estimate ∣ ∫ 0.1
f(x) dx − ∫ 0.1
s(x) dx
0 0
∣
5. Given the partition x 0 = 0, x 1 = 0.05, x 2 = 0.1 of [0, 0.1], find the
∫
piecewise linear interpolating function F for f(x) = e 2x . Approximate
0.1
e 2x dx with ∫ 0.1
F (x) dx. Compare the results to those of the previous
0 0
problem.
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6. (Algebraic properties of the Fourier transform) Let f ∈ L 1 (R) and
ˆf = F. Prove the following properties.
(a) ˆF (γ) = f(−γ)
(b) (Translation or time shifting) For any u ∈ R,
̂(τ u f)(γ) = e −u (γ) ˆf(γ)
(c) (Modulation or frequency shifting) For any λ ∈ R,
ê λ f(γ) = ˆf(γ − λ)
(d) (Time dilation) For any λ ∈ R \ {0},
ˆf λ (γ) = λ
|λ| ˆf( γ λ )
7. Show that the Fourier transform of the triangle function f(t) = max(1−
(
|t|, 0) is equal to
sin πγ
πγ
) 2.
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