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Register a new userA spectral Szego theorem on the real line
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We characterize even measures $mu=wdx+mu_s$ on the real line with finite entropy integral $int_{R} frac{log w(t)}{1+t^2}dt>-infty$ in terms of $2times 2$ Hamiltonian generated by $mu$ in the sense of inverse spectral theory. As a corollary, we obtain criterion for spectral measure of Krein string to have converging logarithmic integral.
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Let $mathcal{H}_0=V, mathcal{H}_1=B+V$ and $mathcal{H}_2=mathcal{L}+V$ be the operators on the Heisenberg group $mathbb{H}^n$, where $V$ is the operator of multiplication growing like $|g|^kappa, 0<kappa<1$, $B$ is a bounded linear operator and $mathcal{L}$ is the sublaplacian on $mathbb{H}^n$. In this paper we prove Szego limit theorem for the operators $mathcal{H}_0, mathcal{H}_1$ and $mathcal{H}_2$ on $L^2(mathbb{H}^n).$
In this article, we extend a result of L. Loomis and W. Rudin, regarding boundary behavior of positive harmonic functions on the upper half space $R_+^{n+1}$. We show that similar results remain valid for more general approximate identities. We apply this result to prove a result regarding boundary behavior of nonnegative eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space $mathbb H^n$. We shall also prove a generalization of a result regarding large time behavior of solution of the heat equation proved in cite{Re}. We use this result to prove a result regarding asymptotic behavior of certain eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space $mathbb H^n$.
We present a characterization of sets for which Cartwrights theorem holds true. The connection is discussed between these sets and sampling sets for entire functions of exponential type.
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this paper, we use the expansion method to prove a quantitative version of this characterization.
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