When may two systems of orthonormal functions be interchanged in vector-valued orthogonal sums?


Given a finite orthonormal sequence(Error rendering LaTeX formula) in some L<sub>2</sub>(\mu) and vectors x1,·s, x<sub>n</sub> in some Banach space X we are interested in the norm of the sums(Error rendering LaTeX formula) in L<sub>2</sub><sup>X</sup>(\mu).A constuction in [1] suggests that the system \phi<sub>n</sub> may be replaced by the set(Error rendering LaTeX formula) of coordinate functions(Error rendering LaTeX formula) on 𝕊<sup>n-1</sup> viewed as un orthonormal system with respect to a suitable measure λ on 𝕊<sup>n-1</sup>.We show by a convolutional argument that after symmetrization the measure λ is uniquely determined.We also discuss related questions.

DOI Code: 10.1285/i15900932v16n1p47

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