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On the Bochner theorem on orthogonal operators


Author: Zinoviy Grinshpun
Journal: Proc. Amer. Math. Soc. 131 (2003), 1591-1600
MSC (2000): Primary 44A05, 44A15, 46F12
DOI: https://doi.org/10.1090/S0002-9939-02-06707-2
Published electronically: September 20, 2002
MathSciNet review: 1949890
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Abstract: We prove the following theorem. Any isometric operator $U$, that acts from the Hilbert space $H_1(\Omega)$ with nonnegative weight $p(x)$ to the Hilbert space $H_2(\Omega)$ with nonnegative weight $q(x)$, allows for the integral representation

\begin{displaymath}Uf=\frac{1}{q(\xi)} \frac{\partial^n}{\partial\xi_1\ldots\partial\xi_n}\int_{\Omega} \overline{L(\xi,t)}f(t)p(t)dt, \end{displaymath}


\begin{displaymath}U^{-1}f= \frac{1}{p(\xi)}\frac{\partial^n}{\partial\xi_1\ldots\partial\xi_n} \int_{\Omega}\overline{K(\xi,t)}f(t)q(t)dt, \end{displaymath}

where the kernels $L(\xi,t)$ and $K(\xi,t)$ satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.


References [Enhancements On Off] (What's this?)

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Additional Information

Zinoviy Grinshpun
Affiliation: Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: miriam@macs.biu.ac.il

DOI: https://doi.org/10.1090/S0002-9939-02-06707-2
Keywords: Hilbert space, weight function, isometric operator, orthogonal polynomials, Bochner Theorem
Received by editor(s): April 3, 2001
Received by editor(s) in revised form: January 11, 2002
Published electronically: September 20, 2002
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society

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