34 research outputs found
Graphs with multiple sheeted pluripolar hulls
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
We define -polynomials as linear combinations of dilations by some
frequencies of an entire function . In this paper we use Pade interpolation
of holomorphic functions in the unit disk by -polynomials to obtain
explicitly approximating -polynomials with sharp estimates on their
coefficients. We show that when frequencies lie in a compact set
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 .
In case of the Laplace transforms of measures on , 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
In this paper we obtain estimates for certain transcendence measures of an
entire function . Using these estimates, we prove Bernstein, doubling and
Markov inequalities for a polynomial in along the graph
of . These inequalities provide, in turn, estimates for the number of zeros
of the function in the disk of radius , in terms of the degree
of and of .
Our estimates hold for arbitrary entire functions of finite order, and
for a subsequence of degrees of polynomials. But for special classes
of functions, including the Riemann -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 , in terms
of the size of the set .Comment: 40 page