Georgian Mathematical Journal Volume 10 (2003), Number 1, 157–164

LOCAL GROWTH OF WEIERSTRASS σ-FUNCTION AND WHITTAKER-TYPE DERIVATIVE SAMPLING

TIBOR K. POGÁNY

Dedicated to N. N. Leonenko on the occasion of his 50th birthday

Abstract. Two explicit guard functions K_{j} = K_{j}(δ_{z}), j = 1, 2, are obtained, which depend on the distance δ_{z} between z and the nearest point of the integer lattice in the complex plane, such that δ_{z}K_{1}(δ_{z}) ≤ |σ(z)|e^{−π|z|^{2}/2} ≤ δ_{z}K_{2}(δ_{z}), z ∈ ℂ, where σ(z) stands for the Weierstraß σ-function. This result is used to improve the circular truncation error upper bound in the qth order Whittaker-type derivative sampling for the Leont’ev functions space [2, \frac{πq}{2} ), q ≥ 1.

2000 Mathematics Subject Classification: 30D15, 30E10, 94A20; Secondary: 30A10, 41A05, 94A12. Key words and phrases: Circular truncation error, derivative sampling, entire function spaces [ρ, ψ], [ρ, ψ), sampling truncation error upper bound, Weierstraß σ-function, Whittaker-type derivative sampling reconstruction.

1. Introduction

The Weierstraß σ-function described as an infinite product is given by

σ(z) = z\begin{matrix}∏ & ′ \\ (m,n)∈ℤ^{2} \end{matrix} (1 − \frac{z}{m + ni}) exp \{\frac{z}{m + ni} + \frac{z^{2}}{2(m + ni)^{2}} \} ,

where the dashed product means that the factor with m = n = 0 is omitted. The local growth estimation of σ(z), interesting for different purposes, has quite a long history. The first result of this kind known by the author is given in [3], where ln M_{σ}(r) ∼ πr^{2}/2, r → ∞, is proved (M_{σ}(r) stands for the maximum modulus of σ on the circle |z| = r), see also [7, Chapter 4, §1, Problem 49], where this result is quoted from the book [3]. After that Hayman proved that there exist absolute constants K_{1}, K_{2}, K_{1} < K_{2}, for which

K_{1} ≤ \frac{|σ(z)|}{dist(z, ℤ^{2})} e^{−π|z|^{2}/2} ≤ K_{2}, z ∈ ℂ.  (1)

In his article [1] no specific comments were given on the nature of K_{j}’s. (Here we have to point out that Hayman’s proof is not correct; he wrongly deduced that the type of σ(z) is π/4 instead of the true value π/2, [1].) Seip confirms (1) on the lattice Λ_{α} = \sqrt{\frac{π}{α}}ℤ^{2}, α > 0, but without any closer specification of K_{j}’s, [8] (these estimates of σ(z) were needed as convergence tools).

In this short note we obtain the values for these uniform constants (K_{1} ≈ 0.266 and K_{2} = 1) as a corollary of our main result. Our principal result is to

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

establish two positive guard functions K_{j} = K_{j}(δ_{z}), j = 1, 2, depending just on δ_{z} = dist(z, ℤ^{2}) such that (1) holds below; the proposed numerical bounds (see the Proof of the Corollary 1), are the minimal and maximal values of these guard functions respectively (therefore K_{j} cannot be improved in our setting).

In the third section of the article we consider the truncation error upper bound appearing in the Whittaker-type derivative sampling restoration formula, i.e. the sampling restoration formula which involves not just the sampled values of the function f, but the sampled values of f^{(j)}, j = \overline{0, q − 1}, i.e., of the first q derivatives of the functions sampled at points of the lattice ℤ^{2}. This type of sampling reconstruction holds for the Leont’ev type functions space [2, \frac{πq}{2}) ^{1}, [4]. Our second main goal is to improve the truncation error upper bound in [6], Theorem 1, with the aid of inequalities (2).

2. Local Growth Behaviour of σ(z)-Function

Along with the notation already introduced the following symbols will also be used: ℕ, ℤ, ℂ denote the sets of natural, integer and complex numbers, respectively; the symbol Cl{A} denotes the closure of the set A, t^{⋆} is the complex conjugate of t ∈ ℂ, the circle Γ_{r} = {ζ| |ζ| = r} contains no point of ℤ^{2}.

Theorem 1. For all z ∈ ℂ we have

δ_{z}K_{1}(δ_{z}) ≤ |σ(z)| exp \{−\frac{π}{2} |z|^{2}\} ≤ δ_{z}K_{2}(δ_{z}),  (2)

where

K_{1}(δ_{z}) = (1 − \frac{π^{4}δ_{z}^{4}}{90} ) (1 − A_{22}δ_{z}^{4})^{2} exp \{−\frac{π}{2} δ_{z}^{2}\} ,  (3)
K_{2}(δ_{z}) = exp \{(\frac{π^{4}}{90} + A_{22}) δ_{z}^{4} − \frac{π}{2} δ_{z}^{2}\} ,  (4)

and A_{22} := ∑_{m,n∈ℕ}(m^{2} + n^{2})^{−2}.

Proof. Let P_{\frac{1}{2}} = (−\frac{1}{2}, \frac{1}{2})^{2} be the period cell of the Weierstraß σ-function. First we remark that for all z ∈ Cl\{P_{\frac{1}{2}} \},

δ_{z}(1 − A_{22}δ_{z}^{4})^{2} (1 − \frac{π^{4}δ_{z}^{4}}{90} ) ≤ |σ(z)| ≤ δ_{z} exp \{(\frac{π^{4}}{90} + A_{22}) δ_{z}^{4}\} . (5)

Indeed, if z = x + iy ∈ Cl\{P_{\frac{1}{2}} \}, then we conclude that δ_{z} = dist(z, ℤ^{2}) = dist(z, 0) = |z|. Put

a_{m,n} := (1 − \frac{z}{m + ni}) exp \{\frac{z}{m + ni} + \frac{z^{2}}{2(m + ni)^{2}} \} .

1The infrequently used notation [ρ, ψ), [ρ, ψ] stands for the functions space consisting of all entire functions of order less then or equal to ρ, and [ρ, ψ) denotes the case when the function of order ρ has a type less then ψ, while [ρ, ψ] is used in the case when the function of order ρ possesses a type less then or equal to ψ, [4].

LOCAL GROWTH OF WEIERSTRASS σ-FUNCTION 159

Consequently, we directly get

|σ(z)|^{2} = |z|^{2} ∏_{n∈ℕ}|a_{n,0}a_{−n,0}a_{0,n}a_{0,−n}|^{2} ∏_{m,n∈ℕ}|a_{m,n}a_{m,−n}a_{−m,n}a_{−m,−n}|^{2}

= |z|^{2} ∏_{n∈ℕ} |(1 − \frac{z^{2}}{n^{2}} ) (1 + \frac{z^{2}}{n^{2}} )|^{2}

× ∏_{m,n∈ℕ} |(1 − (\frac{z}{m + ni})^{2}) (1 − (\frac{z}{m − ni})^{2})

× exp \{z^{2} (\frac{1}{(m + ni)^{2}} + \frac{1}{(m − ni)^{2}} )\}|^{2}  (5a)

≤ |z|^{2} ∏_{n∈ℕ}(1 + \frac{|z|^{4}}{n^{4}} )^{2}  (5b)

× ∏_{m,n∈ℕ}(1 − 2\frac{(x^{2} − y^{2})(m^{2} − n^{2}) + 4mnxy}{(m^{2} + n^{2})^{2}} + \frac{|z|^{4}}{(m^{2} + n^{2})^{2}} ) (5c)

× (1 − 2\frac{(x^{2} − y^{2})(m^{2} − n^{2}) − 4mnxy}{(m^{2} + n^{2})^{2}} + \frac{|z|^{4}}{(m^{2} + n^{2})^{2}} )  (5d)

× exp \{4\frac{(x^{2} − y^{2})(m^{2} − n^{2})}{(m^{2} + n^{2})^{2}}\} .

Now, we use 1+t ≤ e^{t} to majorize the factors in displays (5b, c, d). So, straightforward calculation results in

|σ(z)|^{2} ≤ |z|^{2} exp \{2 (\frac{π^{4}}{90} + ∑_{m,n∈ℕ} \frac{1}{(m^{2} + n^{2})^{2}} ) |z|^{4}\} ,  (6)

which is the asserted upper bound in (5).

The lower bound derivation will be realized in a somewhat different way. At first an auxiliary inequality is established. Namely, let a ∈ ℂ, |a| < 1. Then there holds

|(1 − a)e^{a}| ≥ 1 − |a|^{2}.  (7)

Indeed, after fixing a = |a|e^{iϕ}, |a|, we get by direct calculation

h(ϕ) := |(1 − a)e^{a}| = e^{|a| cos ϕ}\sqrt{1 − 2|a| cos ϕ + |a|^{2}}

≥ min_{ϕ∈[0,2π)} h(ϕ) = h(0) = (1 − |a|)e^{|a|}.

Finally, by e^{t} ≥ 1 + t we deduce estimate (7).

To continue, assume λ_{n} ∈ (0, 1), n ∈ ℕ. Then if ∑_{n∈ℕ} λ_{n} converges, there holds

∏_{n∈ℕ}(1 − λ_{n}) ≥ 1 − ∑_{n∈ℕ}λ_{n}.  (8)

This result is a generalization of the inequality due to Weierstraß. (In fact, (8) is a straightforward consequence of the inequality concerning the finite product/sum case, cf. (1) in [5, 3.2.37., p. 207]).

Now we are ready to establish the lower bound for |σ(z)|, z ∈ Cl\{P_{\frac{1}{2}} \}. We have

|σ(z)| = |z| |∏_{n∈ℕ}(1 − \frac{z^{4}}{n^{4}} )|

× | ∏_{m,n∈ℕ}(1 − \frac{z^{2}}{(m + ni)^{2}} ) e^{\frac{z^{2}}{(m+ni)^{2}}} (1 − \frac{z^{2}}{(m − ni)^{2}} ) e^{\frac{z^{2}}{(m−ni)^{2}}} | (9)

≥ |z| ∏_{n∈ℕ}(1 − \frac{|z|^{4}}{n^{4}} ) ∏_{m,n∈ℕ}(1 − \frac{|z|^{4}}{(m^{2} + n^{2})^{2}} )^{2}  (10)

≥ |z| (1 − |z|^{4} ∑_{n∈ℕ} \frac{1}{n^{4}} ) (1 − |z|^{4} ∑_{m,n∈ℕ} \frac{1}{(m^{2} + n^{2})^{2}} )^{2}

= |z| (1 − \frac{π^{4}}{90}δ_{z}^{4}) (1 − A_{22}δ_{z}^{4})^{2} .  (11)

Indeed, since for a = z^{2}(m±in)^{−2} we have |a| ≤ \frac{1}{4} for all z ∈ Cl\{P_{\frac{1}{2}} \}, m, n ∈ ℕ, this allows us to use estimate (7) in the double-indexed product in (9), and then, taking into account that the factors λ_{n}^{(1)} = |z|^{4}n^{−4}, λ_{n}^{(2)} = |z|^{4}(m^{2} + n^{2})^{−2} belong to the interval (0, 1) in display (10), with the help of the Weierstraß-type inequality (8) we easily deduce (11).

Finally, combining estimates (6) and (11), we finish the derivation of (5). Let τ_{k} : ℂ ↦ Cl\{P_{\frac{1}{2}} \} be the translation which defines a unique k = (k_{u}, k_{v}) ∈

ℤ^{2} and a unique w = (u, v) ∈ Cl\{P_{\frac{1}{2}} \} such that

z = k + w = k_{u} + ik_{v} + u + iv.

The quasi-periodicity property of the Weierstraß σ-function implies that

σ(z) = (−1)^{k_{u}+k_{v}+k_{u}k_{v}}σ(w) exp{πwk^{⋆} + π|k|^{2}/2}.

Since dist(z, ℤ^{2}) = dist(w, 0) = δ_{z}, we get

|σ(z)| = |σ(w)|| exp{πwk^{⋆} + π|k|^{2}/2}|

≤δ_{z} exp \{(\frac{π^{4}}{90} +A_{22}) δ_{z}^{4}+ \frac{π}{2} (|k|^{2} + 2k_{u}u + 2k_{v}v + |w|^{2})\} exp \{−\frac{π}{2} |w|^{2}\}

= δ_{z} exp \{(\frac{π^{4}}{90} + A_{22}) δ_{z}^{4} − \frac{π}{2} δ_{z}^{2}\} exp \{\frac{π}{2} |z|^{2}\} .

This finishes the proof of the upper bound assertion in (2). Finally it remains to derive the lower bound in (2). For this we can argue in a similar way so that by the left-hand estimate in (5) we deduce

|σ(z)| = |σ(w)|| exp{πwk^{⋆} + π|k|^{2}/2}|

≥ δ_{z} (1 − \frac{π^{4}δ_{z}^{4}}{90} ) (1 − A_{22}δ_{z}^{4})^{2} exp \{−\frac{π}{2} δ_{z}^{2}\} exp \{\frac{π}{2} |z|^{2}\} ,

which is the asserted lower bound. □

Remark 1. The Mathematica 4.0 gives A_{22} =∑_{m,n∈ℕ}(m^{2}+n^{2})^{−2} ≈ 0.42437977. So we can rearrange (3) and (4) with the approximate value of A_{22}.

Corollary 1.1. The following estimates hold:

δ_{z} (1 − \frac{π^{4}}{360}) (1 − \frac{A_{22}}{4} )^{2} exp \{−\frac{π}{4} \} ≤ |σ(z)| exp \{−\frac{π}{2} |z|^{2}\} ≤ δ_{z}, z ∈ ℂ.

Proof. We find min K_{1}(δ_{z}) and max K_{2}(δ_{z}) by (3), (4) as 0 ≤ δ_{z} ≤ 1/\sqrt{2}. So

K_{1} ⇐ min_{[0,1/\sqrt{2}]} K_{1}(δ_{z})=K_{1}(1/\sqrt{2}) = (1 − \frac{π^{4}}{360}) (1 − \frac{A_{22}}{4} )^{2} e^{− \frac{π}{4}} ≈ 0, 26574548,
K_{2} ⇐ max_{[0,1/\sqrt{2}]} K_{2}(δ_{z}) = K_{2}(0) = 1

are the uniform Hayman’s constants. □

Remark 2. The sharpness of inequalities (2) is an open question. Namely, starting with (9) and (5a), by the Euler’s infinite product representation of sin πδ_{z} and sinh πδ_{z} we get the lower and upper guard functions for |σ(z)| as follows:

K̃_{1}(δ_{z}) = (\frac{sin πδ_{z}}{πδ_{z}} )^{2} (1 − A_{22}δ_{z}^{4})^{2} exp \{−\frac{π}{4} δ_{z}^{2}\} ,

K̃_{2}(δ_{z}) = (\frac{sinh πδ_{z}}{πδ_{z}} )^{2} exp \{A_{22}δ_{z}^{4} − \frac{π}{4} δ_{z}^{2}\} .

Since K̃_{1}(δ_{z}) possesses lower minimum than K_{1}(δ_{z}), and K̃_{2}(δ_{z}) has larger maximum than K_{2}(δ_{z}) we propose the use of (3) and (4) respectively.

3. Whittaker Type Derivative Sampling

In this part of the article we consider the influence of bounds (2) on the convergence rate in the Whittaker-type q-th order derivative, uniformly spaced sampling restoration formula for the Leont’ev function space [2, \frac{πq}{2}).

For f ∈ [2, \frac{πq}{2}ϑ], ϑ ∈ [0, 1) we have

f(z) = σ^{q}(z) ∑_{(m,n)∈ℤ^{2}} ∑_{j=0}^{q−1} ∑_{k=0}^{q−1−j} \frac{f^{(q−1−j−k)}(m + ni)R_{mnj}^{q}}{j!(q − 1 − j − k)!(z − m − ni)^{k+1}}  (12)

uniformly on the compact subsets of ℂ; as usual, f^{(0)}(·) ≡ f(·). Here

R_{mnj}^{q} = lim_{w→m+ni} \frac{d^{j}}{dw^{j}} (\frac{w − m − ni}{σ(w)})^{q};

cf. [6], Theorem 1 and Corollary 1 for certain additional details, e.g. truncation error analysis, etc. (In fact, formula (12) for q = 1 belongs to Whittaker; the case q > 1 appears in [2] as well.)

Let us introduce a subset of ℤ^{2}:

N(r) := {(m, n)| |m + ni| < r}.

Under the Whittaker-type derivative sampling series (12) truncated to N(r) we mean the interpolation formula

I_{N}(z; f; σ; q; r) = ∑_{(m,n)∈N(r)} ∑_{j=0}^{q−1} ∑_{k=0}^{q−1−j} \frac{σ^{q}(z)f^{(q−1−j−k)}(m + ni)R_{mnj}^{q}}{j!(q − 1 − j − k)!(z − m − ni))^{k+1}} ; (13)

the so-called circular truncation error is

ϵ_{N}(f; z; q; r) := f(z) − I_{N}(z; f; σ; q; r).  (14)

Now, we apply estimates (2) to the results in [6], Theorem 1, Corollary 1, where we cannot avoid truncation error bound estimates such that depend on the Hayman constants K_{1}, K_{2}. To prove the principal result in this section, we will need the following evaluation.

Lemma 1. Consider a circle Γ_{r} = {ζ| |ζ| = r} such that contains no point of ℤ^{2}. Then there holds

δ_{ζ} ≥ dist(Γ_{r}, ℤ^{2}) ≥ \frac{1 − |1 − 2(r^{2} − [r^{2}])|}{4 r + \sqrt{2}} := H(r),  (15)

where [x] is the largest integer less than or equal to x.

Proof. Let A(𝔞_{1}, 𝔞_{2}) be the closest point in ℤ^{2} to Γ_{r} and denote by Δ_{A} the distance between A and the nearest point on Γ_{r}. Clearly, \sqrt{𝔞_{1}^{2} + 𝔞_{2}^{2}} = r ± Δ_{A}, choosing the sign according to where A lies, outside or inside the circle Γ_{r}. Thus we can write \sqrt{𝔞_{1}^{2} + 𝔞_{2}^{2}} ≤ r + Δ_{A} ≤ r + 1/\sqrt{2}. Since 𝔞_{1}^{2} + 𝔞_{2}^{2} is the positive integer nearest to r^{2}, we have

𝔞_{1}^{2} + 𝔞_{2}^{2} = [r^{2}] if A lies inside of Γ_{r} (= [r^{2}] + 1 if A lies outside of Γ_{r}) ,

which gives, that

Δ_{A} = dist(Γ_{r}, ℤ^{2}) = |r − \sqrt{𝔞_{1}^{2} + 𝔞_{2}^{2}}| = \frac{|r^{2} − (𝔞_{1}^{2} + 𝔞_{2}^{2})|}{r + \sqrt{𝔞_{1}^{2} + 𝔞_{2}^{2}}}

≥ \frac{min\{r^{2} − [r^{2}], [r^{2}] + 1 − r^{2}\}}{2r + 1/\sqrt{2}}.

According to the definition of a modulus this implies (15). □

Theorem 2. For all f ∈ [2, \frac{πq}{2}ϑ], ϑ ∈ [0, 1), and for all

N ≥ [\frac{|z|}{\sqrt{2(1 − ϑ)}} + \frac{1}{2}] + 1, z ∈ ℂ,

we have

|ϵ_{N}(f; z; q; \sqrt{2}(N + 1/2))| ≤ \frac{A_{f}(2N + 1)(4N + 3)^{q}e^{−2πq(1−ϑ)N}}{K_{1}^{q}(2(1 − \sqrt{1 − ϑ})N + 1 + \sqrt{1 − ϑ})} , (16)

where A_{f} is the absolute constant which characterizes the [2, \frac{πq}{2}ϑ]-function f, i.e. it is given by |f(z)| ≤ A_{f} exp \{\frac{πqϑ}{2} |z|^{2}\} , z ∈ ℂ. Moreover

lim_{N→∞} I_{N}(z; f; σ; q; r) = f(z)

uniformly in z ∈ ℂ.

Proof. First we repeat the procedure for deriving the truncation error upper bound in [6], Theorem 1 with the integration path Γ_{r} = {ζ| |ζ| = r} chosen according to the definition of the index set N(r) and assume z ∈ int(Γ_{r}). We point out that |f(z)| ≤ A_{f} exp \{\frac{πq}{2}ϑr^{2}\} on the circle Γ_{r} with A_{f} > 0. Then using (2), the numerical values of the Hayman constants K_{1}, K_{2} and (15) in Lemma 1 for estimating |σ(·)| in this result in [6], we get

|ϵ_{N}(f; z; q; r)| ≤ \frac{|σ(z)|^{q}}{2π} ∮_{Γ_{r}} \frac{|f(ζ)||dζ|}{|σ(ζ)|^{q}|ζ − z|}

≤ (\frac{δ_{z}K_{2}(δ_{z})}{min_{ζ∈Γ_{r}} δ_{ζ}K_{1}(δ_{ζ})})^{q} \frac{A_{f} r}{r − |z|} exp \{\frac{πq}{2} (|z|^{2} − (1 − ϑ)r^{2})\}

≤ (\frac{δ_{z}K_{2}}{H(r)K_{1}} )^{q} \frac{A_{f} r}{r − |z|} exp \{\frac{πq}{2} (|z|^{2} − (1 − ϑ)r^{2})\}

≤ \frac{A_{f} r(4r + \sqrt{2})^{q} exp \{\frac{πq}{2}(|z|^{2} − (1 − ϑ)r^{2})\}}{(r − |z|) (\sqrt{2}K_{1}(1 − |1 − 2(r^{2} − ⌊r^{2}⌋)|)^{q}}.  (17)

It is not difficult to see that the circle Γ_{r}, r = \sqrt{2}(N + 1/2), does not contain any integer point from ℤ^{2}, being r^{2} ∉ ℕ. So Γ_{\sqrt{2}(N+1/2)} is a suitable integration contour for which (17) holds. Then, substituting r = \sqrt{2}(N + 1/2) into the bound (17), we deduce

|ϵ_{N}(f; z; q; \sqrt{2}(N + 1/2))|

≤ \frac{A_{f}(N + 1/2)(4(N + 1/2) + 1)^{q}}{K_{1}^{q}((N + 1/2 − \sqrt{1 − ϑ}(N − 1/2))} e^{πq(1−ϑ)[(N−1/2)^{2}−(N+1/2)^{2}]}

= \frac{A_{f}(2N + 1)(4N + 3)^{q} exp\{−2πq(1 − ϑ)N\}}{K_{1}^{q}(2(1 − \sqrt{1 − ϑ})N + 1 + \sqrt{1 − ϑ})},  (18)

which is the asserted upper bound (16).

The uniform convergence in f(z) ≈ I_{N}(z; f; σ; q) follows from the truncation error upper bound (16) as N → ∞. Indeed, since the right-hand term in (16) does not depend on z and vanishes with the growth of N, the assertion follows. □

Remark 3. By fixing the values of z we get the mathematical model

\frac{A_{f}(2N + 1)(4N + 3)^{q}e^{−2πq(1−ϑ)N}}{K_{1}^{q}(2(1 − \sqrt{1 − ϑ})N + 1 + \sqrt{1 − ϑ})} < ϵ  (19)

for a pre-assigned approximation error level ϵ > 0 from (16). Then inequality (19) gives an optimal value of N in finding the minimal size of the approximation sum (13). Moreover, the convergence rate in (12) is

|ϵ_{N}(f; z)| = O(N^{q}e^{−2πq(1−ϑ)N})

under the assumptions of Theorem 2.

Acknowledgement

It is the author’s pleasure to thank Professor Emeritus J. R. Higgins for the suggestions he made and for many helpful discussions. Many thanks are also due to the unknown referee for detailed reading and making numerous valuable comments.

References

1. W. K. Hayman, On the local growth of power series: a survay of the Wiman–Valiron method. Canad. Math. Bull. 17(3)(1974), 317–358.

2. J. R. Higgins, Sampling theorem and the contour integral method. Appl. Anal. 41(1991), 155–171.

3. A. Hurwitz and R. Courant, Allgemeine Funktionentheorie und elliptische Funktionen. Geometrische Funktionentheorie. J. Springer, Berlin, 1922.

4. F. A. Leont’ev, Generalized series of exponentials. (Russian) Nauka, Moscow, 1981.

5. D. S. Mitrinović, Analytic inequalities. (Serbian) Grad̄evinska knjiga, Beograd, 1970.

6. T. K. Pogány, Derivative uniform sampling via Weierstraß σ(z). Truncation error analysis in [2, \frac{πq}{2s^{2}} ). Georgian Math. J. 8(2001), No. 1, 129–134.

7. G. Pólya and G. Szegő, Aufgaben und Lehrsätze aus der Analysis. Band II: Funktionentheorie. Nullstellen. Polynome. Determinanten. Zahlentheorie. Dritte berichtigte Auflage. Die Grundlehren der Mathematischen Wissenschaften, Band 20, Springer-Verlag, Berlin-New York, 1964 (Russian translation: Nauka, Moscow, 1978).

8. K. Seip and R. Wallstén, Density theorems for sampling and interpolation in the Bargmann–Fock space II. J. Reine Angew. Math. 429(1992), 107–113.

(Received 2.07.2001; revised 21.09.2002)

Author’s address:

Faculty of Maritime Studies University of Rijeka 51000 Rijeka, Studentska 2 Croatia E-mail: poganj@brod.pfri.hr