The Weierstrass transform f(x) of a function ϕ(y) is defined by
1.1![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X0001347X/resource/name/S0008414X0001347X_eqn01.gif?pub-status=live)
where
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X0001347X/resource/name/S0008414X0001347X_eqn02.gif?pub-status=live)
whenever this integral exists (7, p. 174). It is also known as the Gauss transform (11; 12). Its basic properties have been developed and studied in (7) and in particular it has been shown that the symbolic operator
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X0001347X/resource/name/S0008414X0001347X_eqn03.gif?pub-status=live)
will invert this transform under suitable assumptions and with certain definitions of this operator. We propose to study the definition
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X0001347X/resource/name/S0008414X0001347X_eqn04.gif?pub-status=live)
for f(x) in C∞. This formula seems to have been first examined by Pollard (9) and later by Rooney (12). In so far as convergence of (1.2) is concerned, we will considerably improve the results (12).