Density in spaces of interpolation by Hankel translates of a basis function. (English) Zbl 1280.46014
Summary: The function spaces \(Y_m\) \((m \in \mathbb Z_+)\), arising in the theory of interpolation by Hankel translates of a basis function, as developed by the authors elsewhere, are defined through a seminorm which is expressed in terms of the Hankel transform of each function and involves a weight \(w\). At least two special classes of weights allow to write these indirect seminorms in direct form, that is, in terms of the function itself rather than its Hankel transform. In this paper, we give fairly general conditions on \(w\) which ensure that the Zemanian spaces \(\mathfrak{B}_{\mu}\) and \(\mathfrak{H}_{\mu}\) \((\mu > -1/2)\) are dense in \(Y_m\) \((m \in \mathbb Z_+)\). These conditions are shown to be satisfied by the weights giving rise to direct seminorms of the so-called type II.
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Citations:
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