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A comparison between the closed modular ideals in $$\ell ^ 1(w)$$ and $$L^ 1(w)$$. (English) Zbl 0654.46056
Let $$\omega$$ be a submultiplicative weight on $$R^+$$ and let $$\ell^ 1(\omega)$$ and $$L^ 1(\omega)$$ be the associated Beurling algebras on N and $$R^+$$. The author establishes the existence of a bijection $$\theta$$ from a large class of closed modular ideals in $$L^ 1(\omega)$$ and a corresponding class of closed ideals in $$\ell^ 1(\omega)$$. He exhibits an isomorphism between the annihilator of an ideal I in $$L^ 1(\omega)$$ and the annihilator of $$\theta$$ (I). Of interest in themselves are explicit formulas for an analytic extension of the Laplace transform, and for the Borel transform, of a function in the annihilator of one of these ideals.
Reviewer: H.Hedenmalm

##### MSC:
 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46A45 Sequence spaces (including Köthe sequence spaces)
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