When may two systems of orthonormal functions be interchanged in vector-valued orthogonal sums?
Abstract
Given a finite orthonormal sequence(Error rendering LaTeX formula) in some
and vectors
in some Banach space X we are interested in the norm of the sums(Error rendering LaTeX formula) in
.A constuction in [1] suggests that the system
may be replaced by the set(Error rendering LaTeX formula) of coordinate functions(Error rendering LaTeX formula) on
viewed as un orthonormal system with respect to a suitable measure λ on
.We show by a convolutional argument that after symmetrization the measure λ is uniquely determined.We also discuss related questions.
![L<sub>2</sub>(\mu)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/bbe09169759d01401d8fcc25cb4d57a5.png)
![x1,·s, x<sub>n</sub>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/f4e8a6e78082821c699520a37f05f5bb.png)
![L<sub>2</sub><sup>X</sup>(\mu)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/352b0179449a6a77b1b3c39ac34f9f89.png)
![\phi<sub>n</sub>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/a5b264f0ad58ca3a1fcfd7dd3e117584.png)
![𝕊<sup>n-1</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/b960610c6320f08c7e99ebab54bbdfcb.png)
![𝕊<sup>n-1</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/b960610c6320f08c7e99ebab54bbdfcb.png)
DOI Code:
10.1285/i15900932v16n1p47
Full Text: PDF