Let x ( t ) be an arbitrary function in L 2 () and define the sequence
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Question:
Let x(t) be an arbitrary function in L2(ℝ) and define the sequence of functions xn(t) = x(t + nT) for n ∈ ℤ.
a. Show that ⟨x(t + mT), x(t + nT)⟩ = ⟨x(t), x(t + (n − m)T)) for all m ∈ ℤ, where the inner product is the standard inner product on L2.
b. Use the previous result to show that the functions {xn(t)} form an orthonormal set if and only if ⟨x(t), x(t + nT)⟩ = δ0n.
Related Book For
An Introduction to Measure Theoretic Probability
ISBN: 978-0128000427
2nd edition
Authors: George G. Roussas
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