Linear functional in space of entire functions
We consider the the linear topology space of entire functions of a proximate order and normal type less of the given type with respect to the proximate order. We obtain the form of continuous linear functional on this space. Let ρ(r) be a proximate order. A single valued function f(z) of the complex variable z is said to belong to the space if f(z) has the order less than ρ(r)or equal ρ(r) but in this second case type less than σ. A sequence of functions from space converges in the sense of space if (i) it converges uniformly on compacts, (ii) there exists such that, where does not depend on. The space is the linear topology space with sequence topology. We introduce the function defined to be the unique solution of the equation. Set. For a function, we associate the function. It is regular in the close circle, r > 1. Conversely, for any analytical in the disk function F(z), the function f(z) from corresponds. Fact mapping function f(z) of to the function F(z) as indicated above will be celebrating a record. In the present paper we prove. 1) In order to be a sequence of functions from to converge in the sense of necessary and sufficient that there exists r>1 such that the sequence converges uniformly inside the disk, and are regular in this disk. 2) Continuous linear functional l on the space has the form, where the quantities satisfy. We remark, the case considered A. F Leont'ev. The case of the space of entire functions of type less then or equal σ with respect to the proximate order ρ(r) was consider by first author.