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Math Help - Range of integral operator

  1. #1
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    Range of integral operator

    Let n be a positive integer. Let T_n:C[a,b]\longrightarrow C[a,b] be defined by:

    T_nv(x)=\int^b_a(x-t)^nv(t)dt

    For a\leq x\leq b, v\in C[a,b]. Determine the range of T_n and decide if it is injective.
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  2. #2
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    Quote Originally Posted by mathematicalbagpiper View Post
    Let n be a positive integer. Let T_n:C[a,b]\longrightarrow C[a,b] be defined by:

    T_nv(x)=\int^b_a(x-t)^nv(t)dt

    For a\leq x\leq b, v\in C[a,b]. Determine the range of T_n and decide if it is injective.
    By the binomial theorem we know that

    \displaystyle (x-t)^n=\sum_{k=0}^{n}(-1)^kx^{n-k}t^{k}

    Using this we get

    \displaystyle T_n(v(x))=\int_{a}^{b}\sum_{k=0}^{n}(-1)^kx^{n-k}t^{k}v(t)dt=\sum_{k=0}^{n}(-1)^kx^{n-k}\int_{a}^{b}t^{k}dt

    Remember that \int_{a}^{b}t^{k}v(t)dt is a constant for each value of k.

    This should get you started
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  3. #3
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    Oh, that makes things work out beautifully. Thanks!
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