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Unformatted text preview: k xy k H ∞ ≤ . 01? (d) Show that k x k ∞ ≤ k x k H ∞ ≤ n k x k ∞ . Show that both inequalities are taken on for some (diﬀerent) choice of x . Hint: for the ﬁrst inequality, use x m = 1 2 π Z 2 π ˆ n X k =1 x k ejkθ ! e jmθ dθ . 7. Show that the norms k u k 1 = Z 1  u ( t )  dt k u k ∞ = sup { u ( t )  : 0 ≤ t ≤ 1 } are not equivalent on the vector space of continuous functions from [0 , 1] into IR . 8. Let V be the following vector space of sequences: V = { x [ k ] : x [ k ] ∈ IR, k ≥ ,  x [ k ]  < ∞} . Deﬁne k u k ∞ = sup k  u [ k ]  and k u k w = sup k w [ k ]  u [ k ]  for some given “weighting” sequence w [ k ]. (a) What conditions on w make k · k w a norm on V ? (b) What conditions on w make k · k w a norm on V equivalent to k · k ∞ ?...
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 Fall '09
 Swindlehurst
 Digital Signal Processing, Signal Processing, Banach space, robust control theory, xk e−jkθ, xk e−jkθ ejmθ

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