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On the Bochner theorem on orthogonal operators

Author(s): Zinoviy Grinshpun
Journal: Proc. Amer. Math. Soc. 131 (2003), 1591-1600.
MSC (2000): Primary 44A05, 44A15, 46F12
Posted: September 20, 2002
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Abstract: We prove the following theorem. Any isometric operator , that acts from the Hilbert space $H_1(\Omega)$ with nonnegative weight to the Hilbert space $H_2(\Omega)$ with nonnegative weight , 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:

1.: Achieser N.J. and Glazman I.M., Theory of Linear Operators in Hilbert Space, Vol. 1, Ungar, New York, 1963. MR 41:9015a
2.: Grinshpun Z.S., Analytic form of isometric operators in weighted Hilbert spaces. Izvestiya Akad. Nauk Kazach. SSR, Ser. Phys.-Mat., 1983, No. 3, pp. 17-20 (Russian). MR 85i:47029
3.: Riesz F. and Nagy B.Sz., Functional Analysis, Ungar, New York, 1955. MR 17:175i
4.: Bateman H. and Erdely A., Higher Transcendental Functions, V. 2, New York, 1953. MR 84h:33001b

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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: 10.1090/S0002-9939-02-06707-2
PII: S 0002-9939(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
Posted: September 20, 2002
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2002, American Mathematical Society


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