Principal local ideals in weighted spaces of entire functions
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- by James J. Metzger PDF
- Trans. Amer. Math. Soc. 165 (1972), 149-158 Request permission
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 $|f(x + iy)| = O(\exp (A|x{|^\rho } + A|y{|^q}))$ 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 ${E_P}[\rho ,q]$.References
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Additional Information
- © Copyright 1972 American Mathematical Society
- 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