# Thread: Range of integral operator

1. ## Range of integral operator

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

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

For $\displaystyle a\leq x\leq b, v\in C[a,b]$. Determine the range of $\displaystyle T_n$ and decide if it is injective.

2. Originally Posted by mathematicalbagpiper
Let $\displaystyle n$ be a positive integer. Let $\displaystyle T_n:C[a,b]\longrightarrow C[a,b]$ be defined by:

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

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

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

Using this we get

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

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

This should get you started

3. Oh, that makes things work out beautifully. Thanks!