We study dual integral equations associated with Hankel transforms, that is, dual integral equations of Titchmarsh's type. We reformulate these equations giving a better description in terms of continuous operators on $L^p$ spaces, and we solve them in these spaces. The solution is given both as an operator described in terms of integrals and as a series $∑_{n=0}^{∞} c_n J_{μ+2n+1}$ which converges in the $L^p$-norm and almost everywhere, where $J_ν$ denotes the Bessel function of order ν. Finally, we study the uniqueness of the solution.
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We consider the sample covariance matrices of large data matrices which have i.i.d. complex matrix entries and which are non-square in the sense that the difference between the number of rows and the number of columns tends to infinity. We show that the second-order correlation function of the characteristic polynomial of the sample covariance matrix is asymptotically given by the sine kernel in the bulk of the spectrum and by the Airy kernel at the edge of the spectrum. Similar results are given for real sample covariance matrices.
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This paper is concerned with the action of a special formally real Jordan algebra U on an Euclidean space E, with the decomposition of E under this action and with an application of this decomposition to the study of Bessel functions on the self-adjoint homogeneous cone associated to U.
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Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in $ℂ^n$ of codimension 1, $vol_{2n-2}(D^{n-1}) ≤ vol_{2n-2}(H ∩ D^{n}) ≤ 2vol_{2n-2}(D^{n-1})$. The lower bound is attained if and only if H is orthogonal to the versor $e_{j}$ of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector $e_{j} + σe_{k}$ for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify $ℂ^n$ with $ℝ^{2n}$; by $vol_{k}(·)$ we denote the usual k-dimensional volume in $ℝ^{2n}$. The result is a complex counterpart of Ball's [B1] result for cube slicing.
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