Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-20T00:48:15.243Z Has data issue: false hasContentIssue false
  • English
  • Français

An Inversion Formula for the Weierstrass Transform

Published online by Cambridge University Press:  20 November 2018

G. G. Bilodeau
G. G. Bilodeau*
Affiliation:
Boston College
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Weierstrass transform f(x) of a function ϕ(y) is defined by

1.1

where

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

will invert this transform under suitable assumptions and with certain definitions of this operator. We propose to study the definition

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).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 593 - 601
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Amerio, L., Un metode di sommazione per le série di potenze, Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Mat., 2, 8 (1939), 167–180.Google Scholar
2. Bailey, W. N., An integral representation for the product of two Hermite polynomials, J. London Math. Soc, 13 (1948), 202–203.Google Scholar
3. Bernstein, S., Sur le convergence de certaines suites des polynômes, J. Math. Pures et Appl., 9, 15 (1936), 345–358.Google Scholar
4. Bilodeau, G. G., Inversion and representation theory for the Weierstrass transform (Thesis, Harvard, March 1959).Google Scholar
5. Erdelyi, A., Asymptotic expansions (Dover, 1956).Google Scholar
6. Hardy, G. H., Divergent series (Oxford, 1949).Google Scholar
7. Hirschman, I. I. and Widder, D. V., The convolution transform (Princeton, N.J., 1955).Google Scholar
8. Kogbetliantz, E., Recherches sur la sommabilité des séries d'Hermite, Ann. Sci. de L'Ecole Norm. Sup., 49, 3 (1932), 147–221.Google Scholar
9. Pollard, H., Integral transforms, Duke Math. J., 13 (1946), 307–330.Google Scholar
10. Rey-Pastor, J., Un metodo de sumacion de series, Rend. Circ. Mat. Palermo, 55 (1931), 450–455.Google Scholar
11. Rooney, P. G., On the inversion of the Gauss transformation, Can. J. Math., 9 (1957), 459–464.Google Scholar
12. Rooney, P. G., On the inversion of the Gauss transformation II, Can. J. Math., 10 (1958), 613–616.Google Scholar
13. Szegö, G., Orthogonal polynomials (New York, 1939).Google Scholar