The dual Weierstrass-Laguerre transform

Haimo, Deborah Tepper

doi:10.1090/S0002-9947-1985-0792814-X

The dual Weierstrass-Laguerre transform
HTML articles powered by AMS MathViewer

by Deborah Tepper Haimo PDF

Trans. Amer. Math. Soc. 290 (1985), 597-613 Request permission

Abstract:

An inversion algorithm is derived for the dual Weierstrass-Laguerre transform $\int _0^\infty {{g_\alpha }(x,y;1)\varphi (y){y^\alpha }{e^{ - y}}/(\alpha + 1)dy}$, where the function ${g_\alpha }(x,y,t)$ is associated with the source solution of the Laguerre differential heat equation $x{u_{xx}}(x,t) = (\alpha + 1 - x){u_x}(x,t) = {u_t}(x,t)$. Correspondingly, sufficient conditions are established for a function to be represented by a Weierstrass-Laguerre Stieltjes transform $\int _0^\infty {{g_\alpha }(x,y;1)\;d\beta (y)}$ of a nondecreasing function $\beta$.

References

Frank M. Cholewinski and Deborah Tepper Haimo, The Weierstrass-Hankel convolution transform, J. Analyse Math. 17 (1966), 1–58. MR 215021, DOI 10.1007/BF02788651
Deborah Tepper Haimo (ed.), Orthogonal expansions and their continuous analogues, Southern Illinois University Press, Carbondale, Ill.; Feffer & Simons, Inc., London-Amsterdam, 1968. MR 0230581
Deborah Tepper Haimo and Frank M. Cholewinski, The Poisson-Laguerre transform, Bull. Amer. Math. Soc. 74 (1968), 137–139. MR 218844, DOI 10.1090/S0002-9904-1968-11909-3
Deborah Tepper Haimo and Frank M. Cholewinski, The Weierstrass-Laguerre transform, J. Math. Anal. Appl. 23 (1968), 41–57. MR 227701, DOI 10.1016/0022-247X(68)90114-5
Frank M. Cholewinski and Deborah Tepper Haimo, The dual Poisson-Laguerre transform, Trans. Amer. Math. Soc. 144 (1969), 271–300. MR 252975, DOI 10.1090/S0002-9947-1969-0252975-9
Deborah Tepper Haimo and Frank M. Cholewinski, Expansions in terms of Laguerre heat polynomials and of their temperature transforms, J. Analyse Math. 24 (1971), 285–322. MR 283265, DOI 10.1007/BF02790378

Tables of integral transforms

Deborah Tepper Haimo, Generalized temperature functions, Duke Math. J. 33 (1966), 305–322. MR 201924
I. I. Hirschman and D. V. Widder, The convolution transform, Princeton University Press, Princeton, N. J., 1955. MR 0073746