completeness of Laguerre polynomials












1












$begingroup$


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?










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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
















1












$begingroup$


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?










share|cite|improve this question









$endgroup$












  • $begingroup$
    Take $alpha=0$ in math.stackexchange.com/q/417870 which is devoted to generalized Laguerre polynomials
    $endgroup$
    – Jean Marie
    Jan 11 at 16:36














1












1








1





$begingroup$


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?










share|cite|improve this question









$endgroup$




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






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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


















  • $begingroup$
    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










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