View source for Hermite transform

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The integral

$$ 
f ( n)  =  H \{ F ( x) \}  =  \int\limits _ {- \infty } ^  \infty  
e ^ {- x  ^ {2} } H _ {n} ( x) F ( x)  d x ,\ \ 
n = 0 , 1 \dots
$$

where  $  H _ {n} ( x) $
are the [[Hermite polynomials|Hermite polynomials]]. The inversion formula is

$$ 
F ( x)  =  \sum _ { n= } 0 ^  \infty   
\frac{1}{\sqrt \pi }

\frac{f ( n) }{2  ^ {n} n ! }
 H _ {n} ( x)  = \ 
H  ^ {-} 1 \{ f ( n) \} ,\  - \infty < x < \infty ,
$$

provided that the series converges. The Hermite transform reduces the operator

$$ 
R [ F ( x) ]  =  e ^ {x  ^ {2} } 
\frac{d}{dx}

\left [ e ^ {x  ^ {2} } 
\frac{d}{dx}
 F ( x) \right ]
$$

to an algebraic one by the formula

$$ 
H \{ R [ F ( x) ] \}  =  - 2 n f ( n) .
$$

If  $  F $
and all its derivatives up to and including the  $  p $-
th order are bounded, then

$$ 
H \{ F ^ { ( p) } ( x) \}  =  f ( n + p ) .
$$

The Hermite transform has also been introduced for a special class of generalized functions (see [[#References|[2]]]). They are used to solve differential equations containing the operator  $  R $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Debnath,   "On the Hermite transform"  ''Mat. Vesnik'' , '''1'''  (1964)  pp. 285–292</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Zemanian,   "Generalized integral transforms" , Wiley  (1968)</TD></TR></table>