Timeline for Asymptotic behavior of the integral of Hermite functions/polynomials on half-lines

Current License: CC BY-SA 4.0

13 events

when toggle format	what		by	license	comment
Nov 15 at 16:13	vote	accept	Darius
Nov 15 at 16:14
Nov 15 at 16:13	vote	accept	Darius
Nov 15 at 16:13
Nov 15 at 16:12	comment	added	Darius		I actually wanted to ask if $\frac{J_m}{{2^m m!}}=O(m^p)$ but you already answered implicitly to my question. Thank you very much !
Nov 15 at 11:17	comment	added	Hjalmar Rosengren		@IosifPinelis The only problem here is that $\rightarrow$ should be replaced with $\sim$. The stated asymptotic formula for Hermite polynomials holds pointwise and in fact locally uniformly on $\mathbb R$. This is enough to deduce the stated asymptotics of the integrals.
Nov 14 at 19:26	history	edited	Carlo Beenakker	CC BY-SA 4.0	added 79 characters in body
Nov 14 at 18:28	comment	added	Iosif Pinelis		(i) Again, the limit for $m\to\infty$ cannot depend on $m$, because $m$ is then a dummy variable. (ii) In the linked Wikipedia article, the asymptotic formula you are apparently trying to use here is given without any specifications on $x$, and it cannot possibly make sense when the value of the cosine term is $0$. (iii) Even if the asymptotic formula were true pointwise, the convergence of the corresponding integrals must be proved/justified.
Nov 14 at 15:56	history	edited	Carlo Beenakker	CC BY-SA 4.0	added 296 characters in body
Nov 14 at 15:52	comment	added	Iosif Pinelis		(I) The limit for $m\to\infty$ cannot depend on $m$. (ii) how do you get this asymptotic?
Nov 14 at 15:50	history	edited	Carlo Beenakker	CC BY-SA 4.0	added 2 characters in body
Nov 14 at 15:44	history	edited	Carlo Beenakker	CC BY-SA 4.0	added 245 characters in body
Nov 14 at 15:37	history	edited	Carlo Beenakker	CC BY-SA 4.0	added 245 characters in body
Nov 14 at 15:21	history	edited	Carlo Beenakker	CC BY-SA 4.0	added 112 characters in body
Nov 14 at 15:13	history	answered	Carlo Beenakker	CC BY-SA 4.0