MATEMATIQKI VESNIK 48 (1996), 77–82

UDK 517.518.3

originalni nauqni rad research paper

A BASIS PROPERTY OF FAMILIES OF THE MITTAG-LEFFLER FUNCTIONS

Milutin Dostanić

Abstract. In this paper we study basis properties of the Mittag-Leffler functions {y(x, λ_{k})} where

y(x, λ) = |x|^{ω/2}E_{1}(ixλ; 1 + ω/2), ω ∈ (−1, 1),

E_{1}(z; α) = ∑_{k=0}^{∞} \frac{z^{k}}{Γ(α + k)} and \{λ_{k}\} is a sequence of complex numbers. In the case ω = 0 this system is reduced to the exponential system.

1. Introduction

In [3] necessary and sufficient conditions for the system {x^{α−1}E_{1}(iλ_{n}x; α)} to form an unconditional basis in L^{2}(0, 1) are obtained. The same conditions are obtained in [9] as a consequence of a more general result. These conditions are expressed in terms of the Muckenhoupt condition which should be satisfied by an entire function with zeros {λ_{k}}. Conditions that the system {e^{iλ_{n}x}} is a Riesz basis of L^{2}(−π, π) can also be expressed in terms of the Muckenhoupt condition [8].

These conditions are not always easy to check. In [5] some sufficient conditions that the exponential system is a Riesz basis in L^{2}(−π, π) are given. In this paper we give some sufficient conditions that the system of the Mittag-Leffler functions is a Riesz basis in L^{2}(−π, π).

2. Preliminaries

Given f, g: X → R_{+} we write f(x) ≍ g(x) (x ∈ X) if there exist constants C_{1}, C_{2} > 0 such that C_{1} ⩽ f(x)/g(x) ⩽ C_{2} (x ∈ X).

Definition. [1] An entire function F of exponential type is said to be of sine type if

1) the zeros of F lie in { z ∈ C | |ℑz| ⩽ h } for some h > 0.

2) there is y_{0} ∈ R such that |F (x + iy_{0})| ≍ 1 (x ∈ R) holds.

AMS Subject Classification: 30B60

Keywords and phrases: Basis property; Riesz basis; Mittag-Leffler functions

Supported by Ministry of Science and technology RS, grant number 04M01

If f is an entire function of exponential type, then by h_{f}(θ) we denote the indicator function

h_{f}(θ) = \overline{lim}_{r→∞} \frac{ln |f(re^{iθ})|}{r}.

By W_{π}(ω) we denote the class of entire functions of exponential type σ ⩽ π such that ∫_{−∞}^{∞} |x|^{ω}|f(x)|^{2} dx < ∞ for some ω ∈ (−1, 1). A characterization of this class is given by the following theorem:

Theorem 1. [4]

W_{π}(ω) = \{ f | f(λ) = ∫_{−π}^{π} E_{1} (iλt; 1 + \frac{ω}{2} ) |t|^{ω/2}φ(t) dt and φ ∈ L^{2}(−π, π) \} .

The function φ is almost everywhere determined by the formula

\frac{1}{2π} \frac{d}{dt} ∫_{−∞}^{∞} \frac{e^{−itv} − 1}{−iv} f(v) (|v|ei\frac{π}{2} sgn v)^{\frac{ω}{2}} dv = \{ \begin{matrix}φ(t); & t ∈ (−π, π), \\ 0; & t ∉ (−π, π). \end{matrix}

The next theorem gives some sufficient conditions for the exponential system to be a Riesz basis in L^{2}(−π, π).

Theorem 2. [7], [8] Let S be an entire function of sine type with zeros {λ_{n}}_{n∈Z} such that h_{s}(±π/2) = π and inf_{m≠n} |λ_{n} − λ_{m}| > 0. Then the system of functions {e^{iλ_{n}x}}_{n=−∞}^{∞} is a Riesz basis in L^{2}(−π, π).

3. Main result

Let the sequence {λ_{n}}_{−∞}^{∞} satisfies the conditions of Theorem 2 and φ(x) = |x|^{ω/2}E_{1}(ix; 1 + ω/2).

Theorem 3. If ω ∈ (−1, 1), then the system of functions {φ(xλ_{n})}_{−∞}^{∞} is a Riesz basis in L^{2}(−π, π).

First we prove the following lemma.

Lemma. If f ∈ W_{π}(ω) (ω ∈ (−1, 1)), {λ_{n}}_{−∞}^{∞} is the sequence of zeros of a sine type function S which satisfies the conditions of Theorem 2, then

∑_{n=−∞}^{∞} |λ_{n}|^{ω}|f(λ_{n})|^{2} ≍ ∫_{R} |x|^{ω}|f(x)|^{2} dx.  (1)

Proof. Let L_{2}^{0}(R) be the set of functions in L^{2}(R) such that their Fourier transformations vanish a.e. on R \ [−π, π]. Since \frac{S(λ)}{S′(λ_{n})(λ−λ_{n})} is an entire function of exponential type and S is a sine type function, then

\frac{S(λ)}{S′(λ_{n})(λ − λ_{n})} ∈ W_{π}(ω), ω ∈ (−1, 1).

From the Theorem of B. Y. Levin and V. D. Golovin [5] it follows that the system of functions {e^{iλ_{n}x}}_{−∞}^{∞} is a Riesz basis in L^{2}(−π, π).

Since \frac{S(λ)}{S′(λ_{n})(λ−λ_{n})} ∈ L^{2}(R) and it is an entire function of exponential type, from the Paley-Wiener theorem it follows that there exists a function ψ_{n} ∈ L^{2}(−π, π) such that

\frac{1}{2π} ∫_{−π}^{π} e^{iλx}\overline{ψ_{n}(x)} dx = \frac{S(λ)}{S′(λ_{n})(λ − λ_{n})}.  (2)

From (2) it follows that the system of functions {(2π)^{−1}ψ_{n}}_{−∞}^{∞} is biorthogonal to the system {e^{iλ_{n}x}}_{n=−∞}^{∞}. Since the system {e^{iλ_{n}x}}_{−∞}^{∞} is a Riesz basis in L^{2}(−π, π), we conclude that the system {ψ_{n}}_{−∞}^{∞} is also a Riesz basis in L^{2}(−π, π). Hence, the system of functions {ψ_{n}(−x)}_{−∞}^{∞} is also a Riesz basis in L^{2}(−π, π).

Now we shall prove that the system \{\frac{S(λ)}{S′(λ_{n})(λ−λ_{n})}\}_{−∞}^{∞} is a Riesz basis in L_{2}^{0}(R).

It is enough to show ([2]) that this system is complete in L_{2}^{0}(R) and that for arbitrary constants C_{ν} there holds

∥∑_{ν} C_{ν} \frac{S(λ)}{S′(λ_{n})(λ − λ_{n})}∥ ≍ ∑_{ν} |C_{ν}|^{2}.  (3)

But this follows directly from the Parseval relation, (2) and the fact that the system {ψ_{n}(x)} is a Riesz basis in L^{2}(−π, π).

Let f ∈ W_{π}(ω). From Theorem 1 it follows that f(v) (|v|ei\frac{π}{2} sgn v)^{ω/2} ∈

L_{2}^{0}(R),

f(x) (|x|ei\frac{π}{2} sgn x)^{\frac{ω}{2}} = ∑_{ν} d_{ν} \frac{S(x)}{S′(λ_{ν})(x − λ_{ν})}  (4)

and

∑_{ν} |d_{ν}|^{2} ≍ ∫_{−∞}^{∞} |x|^{ω}|f(x)|^{2} dx  (5)

(because the system \{\frac{S(x)}{S′(λ_{n})(x−λ_{n})}\}_{−∞}^{∞} is a Riesz basis in L_{2}^{0}(R)).

Since S is a sine type function, then (see [7]) |S′(λ_{n})| ⩾ ε > 0 for each n ∈ Z. Let δ = inf_{m≠n} |λ_{n} − λ_{m}| and D = ⋃_{ν=−∞}^{∞}{ λ | |λ − λ_{ν}| < δ/3 }. The series ∑_{−∞}^{∞} |λ − λ_{n}|^{−2} converges uniformly on compact subsets of C \ D (because S(λ_{ν}) = 0 and the function S is of exponential type).

From the maximum modulus principle we get that the series ∑_{−∞}^{∞} \frac{S^{2}(x)}{S′^{2}(λ_{ν})(x−λ_{ν})^{2}} converges uniformly on compact subsets of C. Then from (5) it follows that the series ∑_{ν} d_{ν} \frac{S(x)}{S′(λ_{ν})(x−λ_{ν})} converges uniformly on compact subsets of C and represents an entire function.

The convergence in (4) is convergence in the subspace L_{2}^{0}(R). Since the func-

tion f(x) (|x|ei\frac{π}{2} sgn x)^{ω/2} is continuous for x > 0 and the series ∑_{ν} d_{ν} \frac{S(x)}{S′(λ_{ν})(x−λ_{ν})}

converges uniformly on compact subsets, then from (4) it follows

f(x) (|x|ei\frac{π}{2} sgn x)^{\frac{ω}{2}} = ∑_{ν} d_{ν} \frac{S(x)}{S′(λ_{ν})(x − λ_{ν})}, x > 0.  (6)

From (6), by the uniqueness theorem, we get

e^{\frac{iπω}{4}} f(λ)λ^{\frac{ω}{2}} = ∑_{ν} d_{ν} \frac{S(λ)}{S′(λ_{ν})(λ − λ_{ν})},

(here λ^{\frac{ω}{2}} = e\frac{ω}{2} ln λ, ln λ = ln |λ| + i arg λ, −π ⩽ arg λ < π). The previous relation gives

|d_{ν}| = |f(λ_{ν})| |λ_{ν}|^{ω/2}.  (7)

From (5) and (7) it follows (1).

Proof of Theorem 3. It is well known [2] that a minimal system {e_{k}}_{−∞}^{∞} is a Riesz basis in a Hilbert space H if and only if ∑_{k} |(f, e_{k})|^{2} ≍ ∥f∥^{2} for each f ∈ H (with (·, ·) we denote the scalar product in H). Now we shall prove that the system of functions {φ(xλ_{n})} is minimal in L^{2}(−π, π).

Really, the functions \frac{S(λ)}{|λ_{n}|^{ω/2}S′(λ_{n})(λ−λ_{n})} are in W_{π}(ω) and by Theorem 1 there exist functions φ_{n} ∈ L^{2}(−π, π) such that

∫_{−π}^{π} E_{1} (iλt; 1 + \frac{ω}{2} ) |t|^{\frac{ω}{2}} φ_{n}(t) dt = \frac{S(λ)}{|λ_{n}|^{ω/2}S′(λ_{n})(λ − λ_{n})}.  (8)

From (8) it follows (φ(xλ_{m}), φ_{n})_{L^{2}(−π,π)} = δ_{nm} which proves the minimality of {φ(xλ_{n})}.

Let e_{n}(x) = |λ_{n}x|^{ω/2}E_{1}(iλ_{n}x; 1 + ω/2) and h ∈ L^{2}(−π, π). To prove Theorem 3 it is enough to show that

∑_{n} |(h, e_{n})|^{2} ≍ ∥h∥^{2}.  (9)

Since ∑_{n} |(h, e_{n})|^{2} = ∑_{n} |(e_{n}, h)|^{2} = ∑_{n} |λ_{n}|^{ω}|f(λ_{n})|^{2} where

f(λ) = ∫_{−π}^{π} E_{1} (iλt; 1 + \frac{ω}{2} ) |t|^{\frac{ω}{2}} \overline{h(t)} dt (∈ W_{π}(ω)),

the relation (9) follows from the Lemma.

Notice that the basis property of {φ(xλ_{n})} (in L^{2}(−π, π)) does not follow from the basis property of {E_{1}(iλ_{n}x; 1+ω/2)} in L^{2}(0, π) because E_{1}(z_{1} +z_{2}; α) ≠ E_{1}(z_{1}; α) · E_{1}(z_{2}; α).

Theorem 4. If {λ_{n}}_{−∞}^{∞} is a complex sequence such that sup_{n∈Z} |λ_{n} − n| = q < 1/8 and −1 < ω < 1 − 8q, then the system {φ(xλ_{n})}_{−∞}^{∞} is a Riesz basis in L^{2}(−π, π).

Proof. Let G(λ) = (λ − λ_{0}) ∏_{n=1}^{∞} (1 − \frac{λ}{λ_{−n}} ) (1 − \frac{λ}{λ_{n}} ). If we prove

a) \frac{G(λ)}{G′(λ_{n})(λ − λ_{n})} ∈ W_{π}(ω),

b) |G′(λ_{n})| ⩾ C(1 + |λ_{n}|)^{−ε} (C > 0 and does not depend on n and ε < 1/2),

the assertion of the Theorem will follow directly by the method used in the proof of Theorem 3.

Since q < 1/8 and −1 < ω < 1−8q we have |G(x)| ⩽ C_{1}(1+|x|)^{4q} (C_{1} does not depend on x, see [6]) and hence the function \frac{G(λ)}{G′(λ_{n})(λ−λ_{n})} satisfies the condition a). The function G satisfies the condition b) and in that case ε = 4q (< 1/2). In the proof we used the fact (see [6]) that if arg λ = θ, ℜλ ⩾ 0, |λ| ⩾ 1/2 and N is a natural number defined by N − \frac{1}{2} ⩽ |λ| sec θ < N + \frac{1}{2}, then

|G(λ)| ⩾ C_{2} \frac{|λ_{N} − λ|e^{π|ℑλ|}}{(1 + |λ − N|)(1 + |λ|)^{4q}}  (10)

where the constant C_{2} does not depend on λ and N. Let D_{n} = { λ | |λ − λ_{n}| < δ < 1 − 8q \}. Since for λ ∈ ∂D_{n} the inequality N − \frac{1}{2} ⩽ |λ| sec θ < N + \frac{1}{2} holds for N = n, applying (10) we obtain

|\frac{G(λ)}{λ − λ_{n}} | ⩾ C′_{2} \frac{e^{π|ℑλ|}}{(1 + |λ|)^{4q}}  (11)

where C′_{2} does not depend on λ ∈ ∂D_{n} and n.

From (11) by the minimum modulus principle we get |G′(λ_{n})| ⩾ C(1+|λ_{n}|)^{−4q} where the constant C does not depend on n. Similarly we prove that the last inequality holds for n = −1, −2, …

REFERENCES

[1] S. A. Avdonin, I. Joo, Riesz bases of exponentials and sine type functions, Acta Math. Hung. 51, 1–2 (1988), 3–14

[2] I. C. Gohberg, M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Moscow 1965.

[3] G. M. Gubreev, A basis property of families of the Mittag-Leffler functions, Dzhrbashyan’s transform and Muckenhoupt condition, Funktsional. Anal. Prilozhen., 20, 4 (1987), 71–72

[4] M. M. Dzhrbashyan, Integral Transformation and Representation of Functions in Complex Domain, Moscow 1966.

[5] V. E. Kaznelson, Bases of exponential functions, Funktsional. Anal. Prilozhen., 5, 1 (1971), 37–47

[6] N. Levinson, Gap and density theorems, Colloquium publications 26, 1940

[7] B. Ya. Levin, Entire Functions, Lectures, Moscow 1971.

82 M. Dostanić

[8] S. V. Hruščev, Perturbation theorem for exponential system and Muckenhoupt condition, Soviet Math. Dokl. 274, 1 (1979), 44–48

[9] S. V. Hruščev, Unconditional bases in L^{2}(0, a), Proc. Amer. Math. Soc. 99, 4 (1987) (received 04.12.1995.) Faculty of Mathematics, Studentski trg 16, 11000 Beograd, Yugoslavia