Integral Representation of One Class of Entire Functions

. 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


Introduction
Let L p (X) be the space of all measurable functions f : X → C on a measurable set X ⊆ R with the norm f p L p (X) := X |f (x)| p dx, 1 ≤ p < +∞.
Denote by E the class of even entire functions of exponential type σ ≤ 1 The following conditions are equivalent: 3) on (0; +∞), the equation f (z) − zf (z) = G(z) has a solution f = F belonging to P W 2 1,+ ; 4) G is an even entire function, and G(z) := G(z) − z z 0 w −1 G (w) dw belongs to the space P W 2 1,+ ; 5) G is an even entire function of exponential type σ ≤ 1, w −2 (G(w) − G(0)) belongs to L 1 (R), z +∞ z w −2 G(w) dw belongs to L 2 (0; +∞), and G(z) = The aim of the present paper is to give a description of the class E of entire functions Q which can be presented in the form Q(z) = 1 0 (−z 2 t 2 cos(tz)+3tz sin(tz)+3 cos(tz))g(t) dt, g ∈ L 2 (0; 1). (1) 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. This result is contained in Theorems 1-3. Close assertions can be found in [4], [6].

Main results
Our principal results are the following statements.
Theorem 1 An entire function Q has the representation (1) if and only if the differential equation has a solution f = G belonging to E.
Proof. Necessity. Let Q have the representation (1) and Therefore, the necessity has been proved.
Sufficiency. If f = G is the indicated solution to the differential equation (2), then according to Theorem A, we have Hence, Theorem 1 is proved.
be a Bessel function of the first kind of index ν ∈ R, where Γ is the gamma function. Since (see [13, p. 55 we obtain that the function Q ∈ E can be represented in the form Example 1 Let α := 4(−π + 2)/π 3 . The function belongs to E with Indeed, the function belongs to E (see [7, Example 2, p. 13]) and is a solution to the equation Therefore, according to Theorems A and 1, the function Q admits representation (1).
Theorem 2 An entire function Q has the representation (1) if and only if the differential equation has a solution f = F belonging to P W 2 1,+ . In this case, the function z −1 (z −1 Q (z)) also belongs to the space P W 2 1,+ and g can be found by one of the following formulas Proof. Necessity. Let the function Q can be presented in the form (1) and F (z) = 1 0 cos(tz)g(t) dt, g ∈ L 2 (0; 1).

Example 2
The function Q(z) = 2z 4 cos z does not belong to E. In fact, for this function Q the differential equation (3) has a solution F (z) = C 1 z + C 2 z 3 − 2z 2 cos z + 2z sin z. But there are no constants C 1 and C 2 for which the function F belongs to P W 2 1,+ . Indeed, F is an even entire function only if C 1 = C 2 = 0, and in this case, the function G(z) = −2z 2 cos z + 2z sin z does not belong to W 2 1,+ since G / ∈ L 2 (R). Hence, the equation (3) with Q(z) = 2z 4 cos z has no solution belonging to P W 2 1,+ . Thus, according to Theorem 2, the function Q cannot be represented in the form (1).
Remark 2 Theorems 1 and 2 give a description of functions Q, for which differential equations (2) and (3) have solutions in the corresponding spaces. Similar problems are considered in many investigations (see, for example, [2]).
Theorem 3 Let an entire function Q ∈ E be defined by the formula (1). Then and Q is an even entire function of exponential type σ ≤ 1. Moreover, w −4 Q(w) belongs to L 1 (1; +∞), and Q(z) = Q 1 (z) + Q 1 (−z) where Q 1 is an entire function satisfying Proof. Indeed, let Q ∈ E and Q(z) Due to the Paley-Wiener theorem, the functions F 1 (z)/z 2 , F 2 (z)/z and F 3 (z) belong to the space P W 2 1 , and A simple estimate with the use of the Schwarz inequality shows that Hence, Q is an even entire function of exponential type σ ≤ 1. Further, and from the Schwarz inequality we obtain Therefore, w −4 Q(w) belongs to L 1 (1; +∞). Furthermore, Q(z) = Q 1 (z) + Q 2 (z), where Here Q 2 (z) = Q 1 (−z), and using Schwarz inequality, we get for z = x + iy and y ≥ 0, where c 4 > 0. Theorem 3 is proved.

Conclusions
The paper is devoted to the study of an integral representation of the class E of entire functions. The conditions for this representation in terms of the existence of certain solutions of some differential equations are found (see Theorems 1 and 2). Asymptotic estimates of entire functions Q ∈ E are obtained (see Theorem 3). The corresponding examples are given. Those results can be used for the investigation of completeness and minimality of the trigonometric system {−t 2 ρ 2 k cos(tρ k )+3tρ k sin(tρ k )+3 cos(tρ k ) : k ∈ N} in the space L 2 (0; 1), where (ρ k ) k∈N is a sequence of distinct nonzero complex numbers.

Acknowledgments
The author is very grateful to the anonymous referees for their valuable remarks and suggestions toward the improvement of the paper.