 On Log Integrability of Radon–Nikodym DerivativesRobert T. W. Martin ()
Chapter 80 in Operator Theory, 2026, pp 2461-2490 from Springer Abstract: Abstract Several classical results of G. Szegö concern positive measures on the complex unit circle with log-integrable Radon–Nikodym derivatives with respect to Lebesgue measure. For example, given such a positive and absolutely continuous measure, μ $$\mu $$ , a theorem of Grenander and Szegö provides a formula for this log-integrable Radon–Nikodym derivative in terms of the sequence of orthogonal polynomials obtained by applying Gram–Schmidt orthogonalization to the analytic monomials in L 2 ( μ ) $$L^2 (\mu )$$ . We provide a new, functional analytic proof and extensions of this result to arbitrary pairs of positive measures, and even more generally to the Simon–Lebesgue decomposition of positive quadratic forms in Hilbert space. We further characterize the positive, absolutely continuous measures, μ $$\mu $$ , on the circle with log-integrable derivative, δ $$\delta $$ , as those positive measures for which H 2 ( μ ) = ℂ [ ζ ] − ∥ ⋅ ∥ L 2 ( μ ) $$H^2 (\mu ) = \mathbb {C} [ \zeta ] ^{- \| \cdot \| _{L^2 (\mu )}}$$ is a reproducing kernel Hilbert space of analytic functions in the disk, equal to the operator-range space of multiplication by g −1 $$g^{-1}$$ , where g is the unique outer function so that | g | 2 = δ $$|g| ^2 = \delta $$ . Date: 2026 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-16356-1_111 Ordering information: This item can be ordered from DOI: 10.1007/978-3-032-16356-1_111 Access Statistics for this chapter More chapters in Springer Books from Springer |
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