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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Principal local ideals in weighted spaces of entire functions


Author: James J. Metzger
Journal: Trans. Amer. Math. Soc. 165 (1972), 149-158
MSC: Primary 46J15; Secondary 30A66
DOI: https://doi.org/10.1090/S0002-9947-1972-0296700-4
MathSciNet review: 0296700
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Abstract: This paper deals with principal local ideals in a class of weighted spaces of entire functions of one variable. Let $ \rho > 1$ and $ q > 1$, and define $ {E_I}[\rho ,q]$ (respectively, $ {E_P}[\rho ,q]$) to be the space of all entire functions f of one variable which satisfy $ \vert f(x + iy)\vert = O(\exp (A\vert x{\vert^\rho } + A\vert y{\vert^q}))$ for some (respectively, all) $ A > 0$. It is shown that in each of the spaces $ {E_I}[\rho ,q]$ and $ {E_P}[\rho ,q]$, the local ideal generated by any one function coincides with the closed ideal generated by the function. This result yields consequences for convolution on these spaces. It is also proved that when $ \rho \ne q$ a division theorem fails to hold for either space $ {E_I}[\rho ,q]$ or $ {E_P}[\rho ,q]$.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0296700-4
Keywords: Local ideals, locally convex spaces, weighted spaces of entire functions, convolution
Article copyright: © Copyright 1972 American Mathematical Society