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Spaces \(L_2(\lambda)\) of a positive vector measure \(\lambda\) and generalized Fourier coefficients. (English) Zbl 1088.46023

In this paper, \(L\) ia a Banach lattice with \(L'\) its dual, \(( \Omega, \Sigma)\) a measurable space and \(\lambda: \Sigma \to L^{+}\) a positive vector measure, countably additive in the norm topology of \(L\). The vector space \( L_{2}(\lambda)= \{ f: \Omega \to \mathbb R: f^{2} \in L_{1}(\lambda) \} \) is given the norm \( \| f \| _{\lambda, 2}= \| \int f^{2} d \lambda \|^{1/2}\) (note that \(\lambda\) is positive). Using Rybakov’s theorem, \( R_{L'}^{+} = \{ x' \in (L')^{+}, \lambda\ll x' \circ \lambda \} \) is dense in \( (L')^{+}\). A sequence \( \{ f_{n} \} \subset L_{2}(\lambda) \) is called \(\lambda\)-orthonormal if \(\| f_{n} \| _{\lambda, 2} =1\) for all \(n\) and \( \int f_{n} f_{m} \,d \lambda =0 \) for all \(n\) and \(m\) with \( n \neq m\); this implies that, for every \( x' \in R_{L'}^{+}\), \( \{ f_{n} \} \) is orthogonal in \( L_{2}(x' \circ \lambda) \). Relative to this sequence, for a \( g \in L_{2}(\lambda) \), the generalized Fourier coefficients are defined by \[ \alpha_{i}(x')= \frac{ \int g f_{i} \,d(x' \circ \lambda)}{\int f_{i}^{2} \,d(x' \circ \lambda)} \] for all \(x' \in R_{L'}^{+}.\) \(g\) is said to be projectable with respect to this sequence \( \{ f_{n} \}\) if, for each \(i\), \( \int g f_{i} \,d \lambda \) lies in the subspace generated by the single element \( \int f_{i}^{2} \,d \lambda\). The following are the main results:
I. A \(g \in L_{2}(\lambda) \) is projectable with respect to a \(\lambda\)-orthonormal sequence \( \{ f_{n} \} \subset L_{2}(\lambda) \) if and only if \( \alpha_{i}(x')\) are constant functions on \(R_{L'}^{+}\), for each \(i\).
II. Suppose, in addition, that \(L\) is also weakly sequentially complete and a \(g \in L_{2}(\lambda) \) is projectable with respect to a \(\lambda\)-orthonormal sequence \( \{ f_{n} \} \subset L_{2}(\lambda) \), with \( \alpha_{i}\) its constant Fourier coefficients. Then the series \( \sum_{i=1}^{\infty} \alpha_{i} f_{i}\) converges to a function in \(L_{2}(\lambda)\) and is the unique function of the set \[ S=\{ h \in L_{2}(\lambda): \lim_{n} \| h - \sum_{i=1}^{n} \beta_{i} f_{i} \| _{\lambda, 2}=0, \; (\beta_{i})_{i=1}^{\infty} \in \mathbb R^{\mathbb N} \}, \] which satisfies the condition \[ \inf_{h \in S} \| g-h \| _{\lambda, 2} = \| g - \sum_{i=1}^{\infty} \alpha_{i} f_{i} \| _{\lambda, 2}. \]
III. These results are then applied to some very special Banach lattices which include the \(\ell_{p}\) spaces, \(1 < p < \infty\).

MSC:

46G10 Vector-valued measures and integration
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42A65 Completeness of sets of functions in one variable harmonic analysis
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References:

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