completeness of Laguerre polynomials
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Can you help me to prove that system of Laguerre polynomials
$$ L_n = dfrac{e^t}{n!}dfrac{d^n}{dt^n} (t^n e^{-t})$$ is completeness in space $L_2((0, infty),e^{-t}dt)$ ?
i have idea of proof:
system is complete if
for $xin H$ fair equalities $(x,e_n)=0$ $forall n$, then $x=0$
i.e.
$int_0^infty x(t)L_ndt =0$, I can not integrate this integral, can you help me with this?
functional-analysis orthogonal-polynomials
asked Jan 11 at 15:13
Gera SlanovaGera Slanova
453
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add a comment |
$begingroup$
Can you help me to prove that system of Laguerre polynomials
$$ L_n = dfrac{e^t}{n!}dfrac{d^n}{dt^n} (t^n e^{-t})$$ is completeness in space $L_2((0, infty),e^{-t}dt)$ ?
i have idea of proof:
system is complete if
for $xin H$ fair equalities $(x,e_n)=0$ $forall n$, then $x=0$
i.e.
$int_0^infty x(t)L_ndt =0$, I can not integrate this integral, can you help me with this?
functional-analysis orthogonal-polynomials
asked Jan 11 at 15:13
Gera SlanovaGera Slanova
453
$endgroup$
$begingroup$
Take $alpha=0$ in math.stackexchange.com/q/417870 which is devoted to generalized Laguerre polynomials
$endgroup$
– Jean Marie
Jan 11 at 16:36
add a comment |
$begingroup$
Can you help me to prove that system of Laguerre polynomials
$$ L_n = dfrac{e^t}{n!}dfrac{d^n}{dt^n} (t^n e^{-t})$$ is completeness in space $L_2((0, infty),e^{-t}dt)$ ?
i have idea of proof:
system is complete if
for $xin H$ fair equalities $(x,e_n)=0$ $forall n$, then $x=0$
i.e.
$int_0^infty x(t)L_ndt =0$, I can not integrate this integral, can you help me with this?
functional-analysis orthogonal-polynomials
asked Jan 11 at 15:13
Gera SlanovaGera Slanova
453
$endgroup$
Can you help me to prove that system of Laguerre polynomials
$$ L_n = dfrac{e^t}{n!}dfrac{d^n}{dt^n} (t^n e^{-t})$$ is completeness in space $L_2((0, infty),e^{-t}dt)$ ?
i have idea of proof:
system is complete if
for $xin H$ fair equalities $(x,e_n)=0$ $forall n$, then $x=0$
i.e.
$int_0^infty x(t)L_ndt =0$, I can not integrate this integral, can you help me with this?
functional-analysis orthogonal-polynomials
functional-analysis orthogonal-polynomials
asked Jan 11 at 15:13
Gera SlanovaGera Slanova
453
asked Jan 11 at 15:13
Gera SlanovaGera Slanova
453
asked Jan 11 at 15:13
Gera SlanovaGera Slanova
453
asked Jan 11 at 15:13
Gera SlanovaGera Slanova
453
asked Jan 11 at 15:13
Gera SlanovaGera Slanova
453
453
$begingroup$
Take $alpha=0$ in math.stackexchange.com/q/417870 which is devoted to generalized Laguerre polynomials
$endgroup$
– Jean Marie
Jan 11 at 16:36
add a comment |
$begingroup$
Take $alpha=0$ in math.stackexchange.com/q/417870 which is devoted to generalized Laguerre polynomials
$endgroup$
– Jean Marie
Jan 11 at 16:36
$begingroup$
Take $alpha=0$ in math.stackexchange.com/q/417870 which is devoted to generalized Laguerre polynomials
$endgroup$
– Jean Marie
Jan 11 at 16:36
$begingroup$
Take $alpha=0$ in math.stackexchange.com/q/417870 which is devoted to generalized Laguerre polynomials
$endgroup$
– Jean Marie
Jan 11 at 16:36
add a comment |
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$begingroup$
Take $alpha=0$ in math.stackexchange.com/q/417870 which is devoted to generalized Laguerre polynomials
$endgroup$
– Jean Marie
Jan 11 at 16:36