Abstract:
The study of entire functions is of central importance in complex function theory. We
consider the ring of entire functions either on subfields of C or on some subfields of
Cp . By using a technique based on admissible filters we study the ideal structure of
the ring of entire functions. Then we prove the B ́zout property for the ring of entire
e
functions over Cp independent of Mittag-Leffler theorem.
An important problem in complex function theory is to find an entire function from
its values on a given sequence. By means of so-called Newton entire functions we solve
a series of interpolation problems. Then we obtain a general result which implies the
results of P ́lya and Gel’fond on the entire functions which are polynomials. We
o
prove a similar result for the entire functions f such that f (D) ⊂ D, where D is a
particular bounded set. As an application we replace the use of power series for the
initial value problems for ODE’s with Newton series for boundary value problems.
