Therefore, the coincidence of two of these topologies is equivalent to the coincidence of the corresponding completion of H p( T) ⊗ X. Of course, H p( T) ⊗ X is also dense in H p ( T ) ⊗ ˜ ∊ X and H p ( T ) ⊗ ˜ π X. ) which are in H p( T) ⊗ X, the Fejér kernel K l being a trigonometric polynomial.The space H P( T) ⊗ X is dense in H p( T, X) as, for instance, every F ∈ H p( T, X) can be approximated by its Fejér means σ l(F, In the space H P( T) ⊗ X we shall consider the following three norms: the injective norm, the norm which is induced by H P( T, X), actually the L P-norm, and the projective norm. Moreover, in a “norm” was introduced so that the completion of L P( S) ⊗ X is isometric to L P( S, X). For p > 1 the situation is very different in for a class of Banach spaces it is shown to fail that L P( S, X) and L p ( S ) ⊗ ˜ π X coincide. Let us recall the classical result by Grothendieck that the Bochner space L 1( S, X) can be identified with L 1 ( S ) ⊗ ˜ π X for every Banach space X. Let ( S, Σ, σ) be a finite measure space. Let us recall that, given two Banach spaces X and Y, the projective norm in X ⊗ Y generates the finest topology which makes the bilinear map (x, y) ∈ X × Y ↦ x⊗ y ∈ X⊗ Y continuous. Luis Rodríguez-Piazza, in North-Holland Mathematics Studies, 2001 4 Projective Tensor Products The reason for this deficiency in an MRA (say) is that the whole frequency band is covered by the analysis and synthesis filters and hence the PR in such systems includes PR of the accompanying noise.įrancisco J. It should be pointed out that wavelet subband coding, or even more general PRFBs, can not be expected to achieve the simultaneity mentioned above and this simultaneity plays a role in effecting computational efficiency. This is achieved by a preprocessing method associated with the scaling properties of FMRAs. For many other signals, FMRA systems should also yield perfect reconstruction in the case where channel noise is negligible. For the frames of interest, these periodizations have spectral gaps which allow for the suppression of broadband channel noise and give perfect signal reconstruction. Our main theoretical tool is our characterization of frames of translates in terms of periodizations of the spectral content of scaling functions associated with FMRAs. Using this notion of an FMRA, we can construct a corresponding sub-band coding system, which suppresses channel/quantization noise, and yields signal reconstruction.īecause of the importance of narrow-band signals in topics as varied as EEG analysis and spread-spectrum communications, we are using FM-RAs to achieve simultaneous narrow-band signal reconstruction and channel/quantization noise reduction, in a subband coding scheme. In the context of our Hilbert space, if for every sequence $(x_n)$ in $M$ with a limit $x\in\mathcal H$, we have $x\in M$, then $M$ is closed.Īs Michael Hardy pointed out, a subspace of a finite-dimensional vector space is always closed, so this distinction only need be made for infinite-dimensional Hilbert spaces.A frame multiresolution analysis of the space L 2 of finite energy signals on ℝ is an increasing sequence of closed linear subspaces V j ⊆ L 2 and an element φ ϵ V 0, for which the following hold: I.į( t) ∈ V j if and only if f( 2 t)∈ V j 1, III.į ∈ V o implies TKf ∊ V 0 for all integers k, where τ κ f( t) = f( t − K), IV. Therefore $M$ is a "closed subspace" if and only if $M$ is a subspace, and the complement of $M$ is open with respect to the metric topology.Īn equivalent definition for a set $M$ to be closed in a metric space is that if $(x_n)$ is a sequence of elements in $M$ that is convergent, then the limit is in $M$ (i.e. if $u,v\in W$ then $u v\in W$ and if $c$ is a scalar then $cv\in W$) - and Hilbert spaces are vector spaces. Recall that if $V$ is a vector space and $W\subset V$, then $W$ is a subspace if and only if $W$ contains the zero vector and is closed under addition and scalar multiplication (i.e. suppose the $n$th member of a basis of $\ell^2$ is In infinite-dimensional spaces, the space of all finite linear combinations of the members of an infinite linearly independent set is not closed because it fails to contain infinite linear combinations of is members. In finite-dimensional Hilbert spaces, all subspaces are closed. It is a subspace that is closed in the sense in which the word "closed" is usually used in talking about closed subsets of metric spaces.
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