34 research outputs found

    Graphs with multiple sheeted pluripolar hulls

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    In this paper we study the pluripolar hulls of analytic sets. In particular, we show that hulls of graphs of analytic functions can be multiple sheeted and sheets can be separated by a set of zero dimension.Comment: 12 page

    Pade interpolation by F-polynomials and transfinite diameter

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    We define FF-polynomials as linear combinations of dilations by some frequencies of an entire function FF. In this paper we use Pade interpolation of holomorphic functions in the unit disk by FF-polynomials to obtain explicitly approximating FF-polynomials with sharp estimates on their coefficients. We show that when frequencies lie in a compact set K⊂CK\subset\mathbb C then optimal choices for the frequencies of interpolating polynomials are similar to Fekete points. Moreover, the minimal norms of the interpolating operators form a sequence whose rate of growth is determined by the transfinite diameter of KK. In case of the Laplace transforms of measures on KK, we show that the coefficients of interpolating polynomials stay bounded provided that the frequencies are Fekete points. Finally, we give a sufficient condition for measures on the unit circle which ensures that the sums of the absolute values of the coefficients of interpolating polynomials stay bounded.Comment: 16 page

    Transcendence measures and algebraic growth of entire functions

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    In this paper we obtain estimates for certain transcendence measures of an entire function ff. Using these estimates, we prove Bernstein, doubling and Markov inequalities for a polynomial P(z,w)P(z,w) in C2{\Bbb C}^2 along the graph of ff. These inequalities provide, in turn, estimates for the number of zeros of the function P(z,f(z))P(z,f(z)) in the disk of radius rr, in terms of the degree of PP and of rr. Our estimates hold for arbitrary entire functions ff of finite order, and for a subsequence {nj}\{n_j\} of degrees of polynomials. But for special classes of functions, including the Riemann ζ\zeta-function, they hold for all degrees and are asymptotically best possible. From this theory we derive lower estimates for a certain algebraic measure of a set of values f(E)f(E), in terms of the size of the set EE.Comment: 40 page
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