The generalization of Borel's theorem
by
Nadir V. Ibadov
The Republic of Azerbaijan, Ganja State University

In this paper Borel's theorem (see [1], page 34, theorem 1.5.4.) is generalized for that case in which for any task of the constants C_\alpha, which are depend on every possible sets of \alpha = (\alpha₁, \alpha₂, ... , \alpha_n) non-negative integers there is a function f, which is infinitely differentiable in all space and which has got C_\alpha, by its Taylor factors in any point (or, otherwise, that reflection from C^\infty(Rⁿ) into a ring of formal power serieses is surjective).

Let K = [-a, a] subset R, where R is the set of material numbers, C- is a complex plane, H(C)- is the space of entire functions above the field C.

Let's designate as

PKn = f(z):f(z) in H(C)

(1)

the normalized space of functions f(z) in H(C) with the norm

||f||n = sup z in C   |f(z)| expaIm|z|+nln(|z|+1) < \infty.

(2)

Let's consider the sets of entire functions

PK = f(z):f(z) in H(C), |f(z)| <= Cf1expaIm|z|+Cf2ln(|z|+1).

(3)

The set P_K coincides with the unification of the spaces P_Kⁿ, i.¥.

PK = \cup n=1\infty PKn.

In this unification we can define the topology of inductive limit, i.¥.

PK = lim n --> \infty  ind PKn.

Let's introduce the system of open sets \Omega_m. Let at the beginning of \Omega_m - there is an interval in R which includes K .

The unification of the space C^\infty(\Omega_m) coincides with C^\infty(K), i.e.

C\infty(K) = \cup m=1\infty C\infty(\Omegam),

where C^\infty(\Omega_m) - is the space of infinitely differentiable functions on \Omega_m and C^\infty(K) - is the space of infinitely differentiable functions on K.

In the unification let's consider the topology of inductive limit

C\infty(K) = lim n --> \infty  ind C\infty(\Omegam).

Let's introduce as [C^\infty(K)]^* the space which is conjugated to C^\infty(K) where strong topology is introduced.

Let s = {(a₀, a₁, a₂, ... )} - is the totality of every possible number succesions.

The function of function F generates some operator M_F functioning from C^\infty(K) into the spaces s, by the rule if f(x) in C^\infty(K), then

MF[f] = {(F, f(k)(x))}k=0\infty in s.

Let's consider the conjugated reflection M_F^* to the reflection M_F which are functioning by the rules:

MF*:s* --> [C\infty(K)]*,

where s^* - is the space which is conjugated to the space s.

Theorem 1 The direct image of operator [^(M_F^*)] is closed-loop i.¥.

Im ^ MF*   = Im ^ MF*     .

The problem is also solved in the case when D - is limited area in a plane C.

References

1. Narasimchan R. The analysis on real and complex multiplicities., Mir., Moscow 1971, page.232.

Date received: May 29, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahs-22.