Georgian Mathematical Journal Volume 8 (2001), Number 1, 129-134

DERIVATIVE UNIFORM SAMPLING VIA WEIERSTRASS σ(z). TRUNCATION ERROR ANALYSIS IN [2, \frac{πq}{2s^{2}} )

TIBOR K. POGÁNY

Abstract. In the entire functions space [2, \frac{πq}{2s^{2}} ) consisting of at most second order functions such that their type is less than πq/(2s^{2}) it is valid the qorder derivative sampling series reconstruction procedure, reading at the von Neumann lattice {s(m + ni)| (m, n) ∈ ℤ^{2}} via the Weierstrass σ(·) as the sampling function, s > 0. The uniform convergence of the sampling sums to the initial function is proved by the circular truncation error upper bound, especially derived for this reconstruction procedure. Finally, the explicit second and third order sampling formulæ are given.

2000 Mathematics Subject Classification: Primary: 41A05, 94A20; Secondary: 30D15, 94A12. Key words and phrases: Derivative sampling, entire functions spaces [ρ, σ], [ρ, σ), Weierstrass sigma-function, plane sampling reconstruction, sampling circular truncation error upper bound.

1. Introduction and Preparation

The well-known sampling representation of some function f(z), z ∈ ℂ, means that f is completely reconstructed from the values f(sn) obtained by “sampling” f at the points of the one-dimensional lattice sℤ, s > 0. The reconstruction formula (often called by the names of Whittaker, Kotel’nikov and Shannon) is

f(z) = \frac{sin πz/s}{π/s} ∑_{n∈ℤ}f(sn)\frac{(−1)^{n}}{z − sn}.  (1)

One of the most interesting extensions of (1) was done by J. M. Whittaker. Namely, for f ∈ [2, π/2)^{1} for the two-dimensional so-called von Neumann lattice {s(m + ni)| (m, n) ∈ ℤ^{2}} one has

1The functions space [ρ, σ) is introduced as the space of all entire functions of the order less than ρ; when it is equal to ρ, it possesses type less than σ, i.e.,

lim sup_{r→∞} \frac{ln M_{f}(r)}{r^{ρ}} < σ

(cf. [1, p. 18]). Here M_{f}(r) := sup_{|z|=r} |f(z)| denotes the maximum modulus. Similarly, [ρ, σ] denotes the functions spaces of order at most ρ, and the type less than or equal to σ.

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f(z) = σ(z) ∑_{(m,n)∈ℤ^{2}}(−1)^{m+n+mn} \frac{f(m + ni)}{z − m − ni}e^{− \frac{π}{2} (m^{2}+n^{2})},  (2)

uniformly in all z ∈ ℂ, and

σ(z) = z ∏_{m,n∈ℤ}′ (1 − \frac{z}{ζ_{m,n}} ) exp (\frac{z}{ζ_{mn}} + \frac{z^{2}}{2ζ_{mn}^{2}} ) ,

where the primed product means that the term ζ_{m,n} = 0 is omitted, see [2, pp. 72–73]. (Throughout we use the notation ζ_{mn} ≡ m + ni, (m, n) ∈ ℤ^{2}.)

J. R. Higgins extended (2) in a derivative sampling manner, i.e., when the reconstruction procedure involves not only the sampled values of the initial function f, but also the sampled values of first q − 1 derivatives f^{(j)}(z), j = \overline{1, q − 1} of f. When f belongs to the Paley–Wiener type functions space (we deal with the exponential type functions), then the role of the sine function is crucial. But in that case the Weierstrass σ(z) replaces the sine in (1). We give this result now. If f ∈ [2, \frac{πq}{2s^{2}} ) for some positive real s we have

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

× \frac{R_{mnj}^{q}}{(z − s(m + ni))^{k+1}} ,  (3)

with

R_{mnj}^{q} = s^{q−j} lim_{w→ζ_{mn}} \frac{d^{j}}{dw^{j}} (\frac{w − m − ni}{σ(w)})^{q} ,

uniformly on all compact z-sets from ℂ [3, Ch. 5], [4]; here we follow the usual convention f^{(j)} = \frac{d^{j}f}{dz^{j}} := f for j = 0.

Our main goal in this paper is the following. Let us introduce

N_{δ}(r) := \{(m, n)| s|m + ni| < r; s(N + δ)\sqrt{2} < r < s(N + \sqrt{δ})\sqrt{2}; δ ∈ (0, 1)\}.

The truncated to N_{δ}(r) variant of the sampling series (3) is

I_{N}(z; f; σ; q)
= σ^{q}(z/s) ∑_{(m,n)∈N_{δ}(r)} ∑_{j=0}^{q−1} ∑_{k=0}^{q−1−j} \frac{f^{(q−1−j−k)}(s(m + ni))R_{mnj}^{q}}{j!(q − 1 − j − k)!(z − s(m + ni))^{k+1}} , (4)

and the so-called circular truncation error is

ε_{N}(f; z) := f(z) − I_{N}(z; f; σ; q).  (5)

To derive a uniform upper bound for |ε_{N}(f; z)|, when f ∈ [2, \frac{πq}{2s^{2}} ) we use the contour integration method following the approach by Whittaker and Higgins.

DERIVATIVE UNIFORM SAMPLING VIA WEIERSTRASS σ(z) 131

2. Derivation of an Upper Bound of the Truncation Error

The main tool in deriving upper bounds of the truncation error consists in using the contour integral method in complex integration. Most authors used positively oriented rectangles as an integration path like C_{M,N,R} which surrounds the set \{z| − M − \frac{δ}{2} < ℜ\{z\} < N + \frac{δ}{2}; |ℑ\{z\}| < R\} for some positive integers M, N, R (cf. [3, Ch. 5], [4, p. 215]). In the case of circular truncation there is no need to take C_{M,N,R} and therefore we choose a more suitable circular path Γ_{r}, the circle of radius r with center at the origin. Another mathematical tool is the use of the previous results in evaluating the growth of the Weierstrass σ-function. Namely, we have

|σ(z)| ≍ de^{\frac{π}{2} |z|^{2}}, z ∈ ℂ,  (6)

where d = dist(z, ℤ) is the distance from z to the nearest point in the von Neumann lattice ℤ^{2} (see [5, pp. 346–347]).

Theorem 1. Assume f ∈ [2, \frac{πq}{2s^{2}} θ_{f}], where the parameter θ_{f} ∈ [0, 1) depends of the function f. Then for all N ≥ (s\sqrt{2})^{−1}| max(ℜ\{z\}, ℑ\{z\})| and δ ∈ (0, 1) we have

|ε_{N}(f; z)| ≤ \frac{|σ(z/s)|^{q}(N + \sqrt{δ})}{(C_{1}d)^{q} (N + δ − \frac{|z|}{s\sqrt{2}})}e^{−πq(1−θ_{f})N^{2}},  (7)

where d = dist(z, sℤ^{2}) and the absolute constant C_{1} is defined by (6).

Proof. Let us estimate the truncation error of the nonharmonic Fourier series expansion of the function f(z) with respect the complex variable z. By the calculus of residues we find that

ε_{N}(f; z) = f(z) − I_{N}(z; f; σ; q) = \frac{σ^{q}(z/s)}{2πi} ∮_{Γ_{r}} \frac{f(ζ)dζ}{σ^{q}(ζ/s)(ζ − z)}.

The positively oriented circle Γ_{r} forms the integration contour containing and bounding the square

−N − δ < ℜ \{\frac{z}{s} \} , ℑ \{\frac{z}{s} \} < N + δ, δ ∈ (0, 1)

and the set N_{δ}(r) so, that the radius r satisfies s(N +δ)\sqrt{2} < r < s(N +\sqrt{δ})\sqrt{2}. Now it is clear that

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

≤ \frac{r|σ(z/s)|^{q}}{(C_{1}d)^{q}} e^{−[πq/(2s^{2})]r^{2}} sup_{ζ∈Γ_{r}} |f(ζ)| sup_{ζ∈Γ_{r}} |ζ − z|^{−1}

≤ \frac{r|σ(z/s)|^{q}}{(C_{1}d)^{q}|r − |z||}exp \{−\frac{πq(1 − θ_{f})}{2s^{2}}r^{2}\}

≤ \frac{|σ(z/s)|^{q}(N + \sqrt{δ})}{(C_{1}d)^{q} (N + δ − \frac{|z|}{s\sqrt{2}})}e^{−πq(1−θ_{f})(N+δ)^{2}},  (8)

where in (8) it is enough to use tha maximum modulus principle to take the suprema on the countour Γ_{r}, bearing in mind the Hayman estimate (7). After that straightforward transformations lead to the estimate (7).

Corrolary 1. Let the situation be the same as in the previous theorem. Then the qth order derivative plane sampling reconstruction formula (3) holds true for all f ∈ [2, \frac{πq}{2s^{2}} ), where, in (3) the convergence is uniform on compact subsets of ℂ. The uniform convergence rate of ε_{N}(f; z) defined by (4) and (5) is O (e^{−πq(1−θ_{f})N^{2}}) .

Proof. Taking z fixed in the truncation error upper bound (7), we revisit the Higgins extension (3) of the Whittaker result (2) in the uniform manner as N → ∞. On the other hand, according to the assumptions of Theorem 1, estimate (7) becomes

sup_{z∈ℂ}|ε_{N}(f; z)| ≤ C_{2} \frac{|σ(z/s)|^{q}(N + \sqrt{δ})}{N + δ − \frac{|z|}{s\sqrt{2}}}e^{−πq(1−θ_{f})N^{2}} = O (e^{−C_{3}N^{2}}) ,

for N large enough and C_{2}, C_{3} ∈ ℝ_{+}. This means an exponential convergence rate of the symmetric partial sums sequence I_{N}(f; z; σ; q) of the non-harmonic Fourier series type derivative plane sampling reconstruction formula (3) as N → ∞.

3. Second and Third Order Derivative Sampling

In this section we give the exact derivative plane sampling reconstruction formulae for q = 3, 4 (the case q = 2 was considered already by Higgins under s = \sqrt{2}, see [3, Problem 9.4, p. 101]. Another reason, which restricts ourselves to considereing these formulæ, is that the first four derivatives of the Weierstrass σ(z) do not contain the invariants g_{2}, g_{3} of the Weierstrass elliptic function ℘(·). Indeed, since σ(z) is entire, we have the following straigthforward extension:

σ(z) = z − \frac{g_{2}}{240}z^{5} − \frac{g_{3}}{840}z^{7} + O(z^{9}),  (9)

where

g_{2} = 60 ∑_{(m,n)∈ℤ^{2}}′ \frac{1}{(m + ni)^{4}} ; g_{3} = 140 ∑_{(m,n)∈ℤ^{2}}′ \frac{1}{(m + ni)^{6}} .

Consequently, having in mind the quasi-periodicity property

σ(z + m + ni) = (−1)^{m+n+mn}σ(z)e^{π(m−ni)z+π(m^{2}+n^{2})/2},

we obtain by extension (9) the following results.

Lemma 1.

σ^{(p)}(m + ni) = (−1)^{m+n+mn}p[π(m − ni)]^{p−1}e^{−(π/2)(m^{2}+n^{2})}, p = \overline{1, 4}, (10)
R_{mnj}^{q} = s^{q−j}[−qπ(m − ni)]^{j}(−1)^{q(m+n+mn)}e^{−(πq/2)(m^{2}+n^{2})}, j = \overline{0, 3}. (11)

For a detailed derivation of these relations consult [6, §4].

Now we are ready to formulate our results on the derivative plane sampling reading on the von Neumann lattice s(ℤ)^{2} for q = 3, 4.

Corrolary 2. Let f ∈ [2, \frac{3π}{2s^{2}} ). Then

f(z) = sσ^{3} (\frac{z}{s} ) ∑_{(m,n)∈ℤ^{2}}(−1)^{m+n+mn}e^{−3π|ζ_{mn}|^{2}/2}

× \{[\frac{s^{2}}{(z − sζ_{mn})^{2}} − \frac{3πsζ_{mn}^{∗}}{z − sζ_{mn}} + \frac{(3πζ_{mn}^{∗})^{2}}{2} ] \frac{f(sζ_{mn})}{z − sζ_{mn}}

+ [\frac{s}{z − sζ_{mn}} − 3πζ_{mn}^{∗}] \frac{sf′(sζ_{mn})}{z − sζ_{mn}} + \frac{s^{2}f′′(sζ_{mn})}{2(z − sζ_{mn})}\} ,  (12)

where the convergence is uniform in z, on all compact subsets of ℂ; z^{∗} denotes the complex conjugate of z.

Proof. Using formulæ (10), (11), the rearrangement of (3) for q = 3 gives us the asserted display (12).

Corrolary 3. Let f ∈ [2, \frac{2π}{s^{2}} ). Then

f(z) = ∑_{(m,n)∈ℤ^{2}} \frac{sσ^{4}(z/s)}{6(z − sζ_{mn})} \{[\frac{6s^{3}}{(z − sζ_{mn})^{3}} − \frac{24πs^{2}ζ_{mn}^{∗}}{(z − sζ_{mn})^{2}} + \frac{3s(4πζ_{mn}^{∗})^{2}}{z − sζ_{mn}}

− (4πζ_{mn}^{∗})^{3}] f(sζ_{mn}) + [\frac{6s^{2}}{(z − sζ_{mn})^{2}} − \frac{24πsζ_{mn}^{∗}}{z − sζ_{mn}} + 48(πζ_{mn}^{∗})^{2}] sf′(sζ_{mn})

+ [\frac{3s}{z − sζ_{mn}} − 12πζ_{mn}^{∗}] s^{2}f′′(sζ_{mn}) + s^{3}f′′′(sζ_{mn})\} e^{−2π|ζ_{mn}|^{2}},

where the convergence is uniform in z, on all compact subsets of ℂ.

4. Final Remarks

The presented method and the truncation error upper bound results have their counterparts in the derivative plane sampling reconstruction of the socalled Piranashvili α-processes defined on the probability space (Ω, F, P), and constituting the space

L^{α}(Ω) := {ξ(t)| ∥ξ∥_{α} < ∞},

such that is with the quasi norm ∥ · ∥_{α} := (E|ξ(t)|^{α})^{1/α} endoved, where α ∈ [0, 2] and E is the expectation operator. The spectral representation of such a stochastic process ξ is given by

ξ(t) = ∫_{Λ} f(t, λ)dZ_{ξ}(λ),

where Λ is a linear Borel set, Z_{ξ} is the spectral measure of the process ξ and f(z, λ) is in the entire functions space [1, ∞), i.e., the kernel function f(z, λ) is an exponentially bounded finite-type function, see [5] for appropriate definitions and stochastic sampling results. Nevertheless, our reconstruction procedure is applicable to such f, so we get a very powerful tool for the derivative plane sampling reconstruction of the Piranashvili α-processes reading at the von Neumann lattice sℤ^{2}. However it has to be pointed out that the stochastic process results of [6] are not scaled (s ≡ 1), since the Weierstrass σ ∈ [2, ∞). So there is no need to introduce some scaling parameter s > 0 because the convergence is controlled by the behaviour of σ(·) in sampling formulæ similar to (3).

Acknowledgement

The author wishes to take this opportunity to thank the anonymous referee for his suggestions which led the manuscript its final form.

References

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

2. J. M. Whittaker, Interpolatory function theory. Cambridge University Press, Cambridge, 1935.

3. J. R. Higgins, Sampling theory in Fourier and signal analysis-foundations. Oxford Science Publications, Clarendon Press, Oxford, 1996.

4. T. Pogány, Some irregular-derivative sampling theorems. Proc. Functional Analysis IV, D. Butković, et al. (eds.), Various Publication Series 43, 205–219, Matematisk Institut, Aarhus, 1994.

5. W. K. Hayman, On the local growth of power series: a surway of the Wiman–Valiron method. Canad. Math. Bull. 17(1974), No. 3, 317–358.

6. T. Pogány, Derivative Kotel’nikov–Shannon formulæ for L^{α}(Ω)-stochastic processes sampled at lattice points in the complex plane. (Submitted to Sampling Theory in Signal and Image Processing.)

(Received 09.10.2000)

Author’s address:

University of Rijeka

Department of Maritime Studies

51000 Rijeka, Studentska 2

Croatia

E-mail: poganj@brod.pfri.hr