TY - JOUR
T1 - On the Bochner theorem on orthogonal operators
AU - Grinshpun, Zinoviy
PY - 2003/5
Y1 - 2003/5
N2 - We prove the following theorem. Any isometric operator U, that acts from the Hubert space H1(Ω) with nonnegative weight p(x) to the Hilbert space H2(Ω) with nonnegative weight q(x), allows for the integral representation Uf = 1/q(ζ) ∂n/∂ζ1...∂ζn ∫Ω L(ζ, t)f(t)p(t)dt, U-1f = 1/p(ζ) ∂n/∂ζ1...∂ζn ∫Ω K(ζ, t)f(t)p(t)dt, where the kernels L(ζ,t) and K(ζ,t) satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.
AB - We prove the following theorem. Any isometric operator U, that acts from the Hubert space H1(Ω) with nonnegative weight p(x) to the Hilbert space H2(Ω) with nonnegative weight q(x), allows for the integral representation Uf = 1/q(ζ) ∂n/∂ζ1...∂ζn ∫Ω L(ζ, t)f(t)p(t)dt, U-1f = 1/p(ζ) ∂n/∂ζ1...∂ζn ∫Ω K(ζ, t)f(t)p(t)dt, where the kernels L(ζ,t) and K(ζ,t) satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.
KW - Bochner Theorem
KW - Hilbert space
KW - Isometric operator
KW - Orthogonal polynomials
KW - Weight function
UR - http://www.scopus.com/inward/record.url?scp=0037407310&partnerID=8YFLogxK
U2 - 10.1090/S0002-9939-02-06707-2
DO - 10.1090/S0002-9939-02-06707-2
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AN - SCOPUS:0037407310
SN - 0002-9939
VL - 131
SP - 1591
EP - 1600
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 5
ER -