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Hermite transform

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The integral

where are the Hermite polynomials. The inversion formula is

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

to an algebraic one by the formula

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

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 .

References

[1] L. Debnath, "On the Hermite transform" Mat. Vesnik , 1 (1964) pp. 285–292
[2] A.G. Zemanian, "Generalized integral transforms" , Wiley (1968)
How to Cite This Entry:
Hermite transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermite_transform&oldid=47217
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article