Hermite transform

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. 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 [2]). They are used to solve differential equations containing the operator $ R $.

References

[1]	L. Debnath, "On the Hermite transform" Mat. Vesnik , 1 (1964) pp. 285–292
[2]	A.G. Zemanian, "Generalized integral transforms" , Wiley (1968)