On Shifts of the Sequence of Integers Generating Functions that are Invertible in the Sense of Ehrenpreis
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ON SHIFTS OF THE SEQUENCE OF INTEGERS GENERATING FUNCTIONS THAT ARE INVERTIBLE IN THE SENSE OF EHRENPREIS N. F. Abuzyarova∗
UDC 517.538.2+517.984.26+517.547
We consider the Schwartz algebra P which consists of all entire functions of exponential type having the polynomial growth along the real axis. Given an unbounded function l : [0; +∞) → R, we study under which conditions the perturbed sequence {k + l(|k|)}, k = ±1, ±2, . . . , forms the zero set of the function that is invertible in the sense of Ehrenpreis. Bibliography: 14 titles.
Introduction Let P denote the set of all entire functions ϕ such that ∃Cϕ > 0 : |ϕ(z)| ≤ Cϕ (1 + |z|)Cϕ exp(Cϕ |Im z|), ∀z ∈ C. Further, let [a1 ; b1 ] ⊂ [a2 ; b2 ] ⊂ . . . be a sequence of segments exhausting R, and Pk denote the Banach space consisting of all entire functions ϕ with a finite norm ϕ k = sup z∈C
|ϕ(z)| , (1 + |z|)k exp(bk y + − ak y − )
y±
= max{0, ±y}, z = x + iy. where Pk . Equipped with the topology of the inductive limit of the Clearly, we have P = k
sequence {Pk }, P becomes a topological algebra over the ring of polynomials C[z] which is called Schwartz algebra. Also, we use the following conventional notations: E = C ∞ (R), E = (C ∞ (R)) , D = C0∞ (R), D = (C0∞ (R)) . Considered as a linear topological space, P is isomorphic to the Schwartz space E [1, Theorem 7.3.1]. The isomorphism is implemented by the Fourier-Lapalce transform F : S → ϕ, ϕ(z) = S(e−itz ). A function ϕ ∈ P is called slowly decreasing if there exists a > 0 such that ∀x ∈ R ∃x ∈ R : |x − x | ≤ aln (2 + |x|), |ϕ(x )| ≥ (a + |x |)−a . L. Ehrenpreis [2] introduced and considered this notion in connection with the invertibility of distribution S ∈ E in the spaces E and D . Recall that the invertibility of S in each of these spaces means that S ∗ E = E, (0.1) S ∗ D = D ,
(0.2)
respectively. It turns out that each of these relations holds if and only if the function ϕ = F(S) is slowly decreasing ([2, Theorems 1, 2.2, Proposition 2.7]). It is also obtained in [2] and [3] that if ϕ = F(S) is slowly decreasing, then each solution f ∈ E of the homogeneous convolution equation S∗f =0 ∗
Bashkir State University, 450076 Ufa, Russia, e-mail: [email protected].
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 480, 2019, pp. 5–25. Original article submitted February 11, 2019. 1072-3374/20/2512-0161 ©2020 Springer Science+Business Media, LLC 161
can be represented as a series (with grouping) of exponential monomials which are elementary solutions of this equation. Also, the series converges, after some grouping of its terms, in the space E. A function ϕ ∈ P is called invertible in the sense of Ehrenpreis if the principal ideal Iϕ generated by it in P is closed: Φ ∈ P,
Φ/ϕ ∈ H(C) =⇒ Φ/ϕ ∈ P.
Theorem A. (The analytical criterion of the invertibility in the sense of Ehrenpreis [2, Theorem 2.6].) A function ϕ ∈ P is invertible in the sense of Ehrenpreis if and only if it is slowly decreasing. Theorem A and the above-cited results from [2] guarantee that ϕ ∈ P is invertible i
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