LINEAR DIFFERENTIAL EQUATIONS WITH COEFFICIENTS IN FOCK TYPE SPACE

In this paper we deal with complex differential equa- tions of the form f (k) + ak−1(z)f (k−1) + · · · + a1(z)f 0 + a0(z)f = 0 with the coefficients in Fock type space. The relation betweenthe solutions and coefficients in Fock type space is obtained.


Introduction
Motivated by the work in [6], [7] and [8], we will study complex differential equations of the form where the coefficients are entire functions.
In [8], equations of the form (1) with coefficients in weighted Bergman or Hardy spaces are studied.The direct problem is proved, that is, if the coefficients a j (z), j = 0, ..., k − 1 of (1) belong to the weighted Bergman space, then all solutions are of finite order of growth and belong to weighted Bergman space.The inverse problem is also investigated, that is, if all solutions are of finite order of growth, then the coefficient is proved to belong to weighted Bergman space.
The Bargmann-Fock space (see [1], [2]) is the Hilbert space of entire functions equipped with the inner product < f, g >= 1 π C f (z)g(z)e −z•z dx dy, normed by f = √ < f, f >.This space has been studied by many authors and it is rooted from mathematical problems of relativistic physics (see [12]) or from quantum optics (see [10]).In physics the Bargmann-Fock space contains the canonical coherent states, so it is the main tool for studying the bosonic coherent state theory of radiation field (see [11]).The Bargmann-Fock space has also been proved invaluable in the theory of the wavelets.In fact, the Bargmann transform is a unitary map from L 2 (R) onto the Bargmann-Fock space which transforms the family of evaluation functionals at a point into canonical coherent states which are nothing but the Gabor wavelets.
The Fock-type space F α (see [3]) is the Hilbert space of entire functions equipped with the inner product In this paper, we will consider the growth relation between the coefficients and the solutions of (1).We are particularly interested in the Fock-type spaces F α and F e α case: (i)Find the conditions imposed on the coefficients a j (z), j = 0, ..., k − 1 of (1) which make all of the solutions belong to the Fock-type space F e α .
(ii)Suppose that all solutions of (1) belong to the Fock-type space F e α , find out whether all of the coefficients a j (z), j = 0, ..., k − 1 belong to the Fock-type space F α .Hereafter, problems (i) and (ii) will be referred to as the direct problem and the inverse problem, respectively.
Throughout this paper, A will denote positive constants, it may be different at each occurrence.

direct problem
In this section, sufficient conditions for all of the solutions of (1) belong to F e α will be obtained.We need the following result on growth estimate for solutions of (1) in [6].
Lemma 2.1.Let f be a solution of (1) in the disk {z ∈ C : |z| < r}, where 0 < r ≤ ∞, let n c ∈ {1, ..., k} be the number of nonzero coefficients where A is some positive constant depends on the values of the derivatives of f and the values of a j (z θ ) at z θ .
The main result of this section is as follows.
Proof.Since a j (z) ∈ F α for j = 0, 1, ..., k − 1, we have If f (z) is a solution of (1), from Lemma 2.1, we have where A is some positive constant depends on the values of the derivatives of f and the values of a j (z θ ) at z θ and n c is defined in Lemma 2.1, combination with (3) yields By (2), we have Thus proving that all of solutions of (1) belong to F e α .
Remark 2.1.Although we are unable to show the sharpness of the constant 1 in (4), we remark it is necessary.Actually, if (4) does not hold, we may suppose Taking α(r) ∼ log r η 0 where 0 < η 0 < 1 2 , for example, here the symbol ∼ denotes that α(r) and log r η 0 have the same growth as r tends to infinity, then By the proof of Theorem 2.1, we know that f / ∈ F e α holds in this case.

inverse problem
To study the inverse problem, we need some background knowledge and some lemmas.
We present the following elementary result on inequality in [4] for later use.We also need the growth estimates of meromorphic functions in [5].Lemma 3.2.Let f(z) be a transcendental meromorphic function, furthermore, let β > 1 be a positive constant.Then there exist a set E ⊂ [0, 2π) that has linear measure zero, a constant A > 0 that depends only on β, and a constant r 0 = r 0 (θ) > 1 such that where r > r 0 and θ ∈ [0, 2π) \ E .EJQTDE, 2011 No. 48, p. 4 Recall that the order reduction procedure is as follows(see [8]): if {f 1 , ..., f k } is a solution base of (1) in |z| < r, then the first order reduction of (1) results in where for j = 0, ..., k − 2 and the meromorphic functions for j = 0, ..., k − 1 are linearly independent solutions of (5) in |z| < r.
We have the following relations between the solutions and its reductions.
We also need the following result on reduction.
Our result on the inverse problem is as follows.