Some properties of principal submodules in the module of entire functions of exponential type and polynomial growth on the real axis

Abstract

In the work we consider a topological module of entire functions (;), which is the isomorphic image of Fourier-Laplace transform of Schwarz space formed by distributions with compact supports in a finite or infinite segment (;) ⊂ R. We study the conditions ensuring that the principal submodule of module (;) can be uniquely recovered by the zeroes of a generating function.

On $2$-generateness of weakly localizable submodules in the module of entire functions of exponential type and polynomial growth on the real axis

“…the relation | | = (̃︀ ( )), | | → +∞, holds true. By estimate (2.23) it yields for the function 2, that | 2, ( )|︀ ( ) → 0, | | → +∞.Arguing as in the proof of Lemma 3 in the work[9], we obtain that there exists a sequence of polynomials { } converging to the function 2, in the weighted norm ‖ • ‖ = sup∈R | • | ( ) , where ( ) = (1 + | |) 2̃︀ ( ).We let ( ) = ln ( ), 2 + 2 d is the Poisson integral of a function , = + i . By condition (2.13) it is easy to obtain that the function belongs to the class of slowly varying canonical weights introduced in the monograph[19, S1.3].…”

“…It can be also checked straightforwardly by employing the definition of stability and the description of the topology in ( ; ). On the other hand, an example constructed in [8] as well as Theorem 3 of work [9] show that not all principle submodules in the submodule ( ; ) are weakly localizable. Thus, the statement that each stable finite generated submodule in ( ; ) is weakly localizable is wrong.…”

Spectral synthesis for the differentiation operator in the Schwartz space

“…Clearly, the relations (3.9) hold for an invertible function ϕ. [6,Theorem 1] shows that they may also be valid when the function ϕ is not invertible. Let ϕ ∈ P(a; b) be such a function that the submodule J (ϕ) contains at least one function Φ = ωϕ, where ω is not a polynomial.…”

“…hold, where c 1 = c 2 0 π 2 . We are going to apply the following lemma (see [6,Lemma 1]) to get a proper estimate for the function ϕ on the real axis.…”

“…. } is not complete in the spaces E(I), C(I), L p (I), 1 ≤ p < ∞ (see [5] [6,Theorem 3]. In the other hand, as our example in Section 3 below shows, D-invariant subspace W with discrete spectrum σ(W ) = −i Λ satisfyng the relation 2ρ Λ = |I W |, may also be of the form (1.3).…”