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How to find a set of polynomials $P_j(x)$ of degree $j$, for $j=0,1,2$, that are orthonormal with respect to the inner product $$\langle f,g\rangle =\int_0^{+\infty}e^{-t}f(t)\overline{g}(t)\,dt\,?$$

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What is the domain and range of these polynomials? –  Elias Jan 4 '13 at 10:10

2 Answers 2

You should use the Gram–Schmidt process: Take any three linearly independent polynomials, for example $1,x,x^2$. Now apply the process to this set. $$\|1\|^2=\langle1,1\rangle=\int_0^\alpha e^{-t}dt=1-e^{-\alpha}\hspace{5pt}\Rightarrow\hspace{5pt} p_0(x)=\frac{1}{\|1\|}=\frac{1}{\sqrt{1-e^{-\alpha}}}$$ $$proj_{p_0}(v_1)=\langle v_1,p_0 \rangle p_0=p_0\cdot\langle x,p_0 \rangle=\frac{1}{1-e^{-\alpha}}\int_0^\alpha e^{-t}tdt$$ and then $u_1=v_1-proj_{p_0}(v_1)$ and $p_1=\frac{u_1}{\|u_1\|}$ and so on.

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the second step is not clear for me! i am confusing how you wrote p 0 ⋅⟨x,p 0 ⟩. Would be great if you give more details. Thanks a lot. –  Iya Jan 4 '13 at 10:34
    
Following the Gram-Schmidt process, we want to subtract the projection of the second vector (i.e. $x$) on $p_0(x)$ from $x$. The projection is calculated from the geometric interpretation of inner product: $proj_v(u)=\frac{\langle u,v\rangle}{\|v\|}\frac{v}{\|v\|}$. Since $\|p_0\|=1$, I omitted the denumerator. –  Dennis Gulko Jan 4 '13 at 11:11

If $\alpha=\infty$ you will obtain the Laguerre-polynomials.

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