Abstract
In this paper, we study an integral representation of one class of entire functions. Conditions for the existence of this representation in terms of certain solutions of some differential equations are found. We obtain asymptotic estimates of entire functions from the considered class of functions. We also give examples of entire functions from this class.
Highlights
IntroductionLet Lp(X) be the space of all measurable functions f : X → C on a measurable set X ⊆ R with the norm f p Lp (X )
Let Lp(X) be the space of all measurable functions f : X → C on a measurable set X ⊆ R with the norm f p Lp (X ) :=|f (x)|p dx, 1 ≤ p < +∞
An entire function G is said to be of exponential type σ ∈ [0; +∞) ( [3, p. 4], [14, p. 12]) if for any ε > 0, there exists a constant c(ε) such that
Summary
Let Lp(X) be the space of all measurable functions f : X → C on a measurable set X ⊆ R with the norm f p Lp (X ). Denote by P Wσ2 the set of all entire functions of exponential type σ whose narrowing on R belongs to the space L2(R), and by P Wσ2,+ denote the class of even entire functions from P Wσ2. 4) G is an even entire function, and G(z) := G(z) − z w−1G (w) dw belongs to the space P W12,+; 5) G is an even entire function of exponential type σ ≤ 1, w−2(G(w) −. G1(z√) + G1(−z), where G1 is an entire function satisfying |G1(z)| ≤ c1(1 + |z|)/ 1 + Im z, c1 > 0, Im z ≥ 0 Such a class E arises in the investigation of some boundary-value problems ( [11], [12]), whose singularity lies in the fact that the set of their canonical eigenfunctions can be overflowed. We obtain an analog of Theorem A for this class of functions. Close assertions can be found in [4], [6]
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