We give a new poised bivariate Hermite scheme and a formula for the interpolation polynomial. We show that the Hermite interpolation polynomial is the limit of bivariate Lagrange interpolation polynomials at Bos configurations on circles.
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Let F be the Cartesian product of N closed sets in ℂ. We prove that there exists a function g which is continuous on F and holomorphic on the interior of F such that $Γ_g(F):={(z, g(z)): z ∈ F}$ is complete pluripolar in $ℂ^{N+1}$. Using this result, we show that if D is an analytic polyhedron then there exists a bounded holomorphic function g such that $Γ_g(D)$ is complete pluripolar in $ℂ^{N+1}$. These results are high-dimensional analogs of the previous ones due to Edlund [Complete pluripolar curves and graphs, Ann. Polon. Math. 84 (2004), 75-86] and Levenberg, Martin and Poletsky [Analytic disks and pluripolar sets, Indiana Univ. Math. J. 41 (1992), 515-532].
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We give an elementary proof of the product formula for the multivariate transfinite diameter using multivariate Leja sequences and an identity on vandermondians.
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