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Gauss transform

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The linear functional transform of a function defined by the integral

If , then ; for real values , the operator is self-adjoint [1]. The inversion formula for the Gauss transform is

If , the Gauss transform is known as the Weierstrass transform.

References

[1] E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[2] V.A. Ditkin, A.P. Prudnikov, "Integral transforms" Itogi Nauk. Ser. Mat. Mat. Anal. (1966) pp. 7–82 (In Russian)


Comments

The above inversion formula can be interpreted in terms of semi-groups. Another way to invert the Gauss transform is to write in the first equation , from which substitution a double-sided Laplace transform results. Then the inversion formula follows from well-known Laplace-transform techniques.

References

[a1] I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972)
How to Cite This Entry:
Gauss transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_transform&oldid=13024
This article was adapted from an original article by A.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article