Abstract
In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximation, for any element in a Banach tensor space, by means of a tensor given in a typical representation format (Tucker, hierarchical, or tensor train). We show that this result holds in a tensor Banach space with a norm stronger than the injective norm and in an intersection of finitely many Banach tensor spaces satisfying some additional conditions. Examples using topological tensor products of standard Sobolev spaces are given.
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Notes
Note that the meaning of id [j] and id [k] may differ: in the second line of (2.7b), (id [k]⊗A k )∈L(V,V [k]⊗ a W k ) and (id [j]⊗A j )∈L(V [k]⊗ a W k ,V [j,k]⊗ a W j ⊗ a W k ) (cf. (2.5b)), whereas in the third one (id [j]⊗A j )∈L(V,V [j]⊗ a W j ) and (id [k]⊗A k )∈L(V [j]⊗ a W j ,V [j,k]⊗ a W k ⊗ a W j ).
Recall that an elementary tensor is a tensor of the form v 1⊗⋯⊗v d .
In (3.1a) it suffices to have the terms for n=0 and n=N. The derivatives are to be understood as weak derivatives.
We recall that the definition of \(U_{j}^{\mathrm{IV}}(\mathbf{v})\) requires the definition of a norm on V [j]. The following arguments will be based on \(U_{j}^{\mathrm{III}}(\mathbf{v})\).
Here, infinite dimensions are identified and not considered as possibly different infinite cardinalities.
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This work is partially supported by the PRCEU-UCH30/10 grant of the Universidad CEU Cardenal Herrera.
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Communicated by Wolfgang Dahmen.
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Falcó, A., Hackbusch, W. On Minimal Subspaces in Tensor Representations. Found Comput Math 12, 765–803 (2012). https://doi.org/10.1007/s10208-012-9136-6
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DOI: https://doi.org/10.1007/s10208-012-9136-6