MATH 421
Assignment # 4
Fall 2024
Date Due: 2024/12/03
Exercise 5.2.9: Answer the following questions.
(a) Is the function
in L1(R>0;R)?
(b) Show that
Hint: Use Example 5.2.19–2.
Exercise 5.3.5: Consider four functions f1 , f2 , f3 , f4 : R → R satisfying
Answer the following questions.
(a) For each of the transforms FCC(f1 ), FCC(f2 ), FCC(f3 ), and FCC(f4 ), indicate whether it exists in the sense that
for a ∈ {1, 2, 3, 4}.
(b) For each of the transforms FCC(f1 ), FCC(f2 ), FCC(f3 ), and FCC(f4 ), indicate whether it is continuous.
(c) For each of the transforms FCC(f1 ), FCC(f2 ), FCC(f3 ), and FCC(f4 ), indicate whether it is differentiable.
(d) For each of the transforms FCC(f1 ), FCC(f2 ), FCC(f3 ), and FCC(f4 ), indicate whether it is in L1(R;R).
(e) For each of the transforms FCC(f1 ), FCC(f2 ), FCC(f3 ), and FCC(f4 ), indicate whether it is in L2(R;R).
Table 1.
Exercise 6.1.1: In Table 1 are given plots of three discrete-time functions defined on Z and their DCFT’s. You are not told which function goes with which DCFT. Without doing any computations, indicate which function in the left column goes with which DCFT in the right column.
Exercise 6.1.2: Prove Proposition 6.1.9.
Exercise 6.1.5: Find a function f ∈ ℓ2 (Z(∆);C) such that the function FDC(f) is not continuous.
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