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A convolution inequality

  1. Jun 1, 2012 #1
    Dear friends,

    I am interesting to find some functions g satisfying the following convolution inequality

    (g[itex]\ast[/itex]v)(t)[itex]\leq[/itex]v(t)

    for any positive function v[itex]\in[/itex]L[itex]^{1}[/itex][0,T] and * denotes the convolution between g and v.
     
  2. jcsd
  3. Jun 4, 2012 #2
    The way you've worded the statement, it's not possible. Suppose that [itex] v \in L^1[0, T] [/itex] satisfies 0 < (g*v)(0) < v(0). Let v'(t) = v(t) for all t other than 0 and v'(0) = .5(g*v)(0). Then v = v' in the sense of L1, but (g*v)(0) > v'(0).
     
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