28 research outputs found
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