Applications of operator semigroups to Fourier analysis |
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Authors: | Jerome A Goldstein |
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Institution: | (1) Department of Mathematics, Louisiana State University, 70803 Baton Rouge, Louisiana |
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Abstract: | Of concern are semigroups of linear norm one operators on Hilbert space of the form (discrete case)T={T
n
/n=0,1,2,...} or (continuous case)T={T(t)/t=≥0}. Using ergodic theory and Hilbert-Schmidt operators, the Cesàro limits (asn→∞) of |〈T
n
f,f〉|2, |〈T
(n)f,f〉|2 are computed (withn∈ℤ+ orn∈ℤ+). Specializing the Hilbert space to beL
2(T,μ) (discrete case) orL
2(ℝ,μ) (continuous case) where μ is a Borel probability measure on the circle group or the line, the Cesàro limit of
(asn→±∞, with,n∈ℤ orn∈ℝ) is obtained and interpreted. Extensions toT
M
, and ℝ
M
are given. Finally, we discuss recent operator theoretic extensions from a Hilbert to a Banach space context.
Partially supported by an NSF grant |
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Keywords: | |
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