In simplistic terms, a signal might be discrete, such as letters or digits sequences, or it might be analogue, i.e. continuous functions, reading such as a temperature or a pressure. It is a remarkable fact that under additional assumptions, analogue signals and discrete signals are equivalent: an analogue signal f can be recovered exactly from its samples {f(xj)}j=−ωω , i.e. from a digital signal. This is the essence of the sampling theorems. These theorems are fundamental, in particular, in information theory and communication, particularly since the advent of modern digital computers.

An analogue signal is, in the classical sense, a function such that it is continuous with respect to a real variable (for example, time) and has finite energy, i.e. it is a square-integrable on R . The following classical Shannon sampling theorem (see, e.g. Higgins, 1996; p. 51) states that if f∈L2(R) is bandlimited to [−σ,σ] , σ>0 0$]]>, i.e. if the Fourier transform of f is supported on [−σ,σ] , then (1.1) f(x)= ∑k=−∞∞f(πkσ)sin(σx−πk)σx−πk and this series converge in L2(R) -norm and uniformly on R . This theorem was the starting point for many further developments in sampling theory. Series (1.1) is called a cardinal series of f. The sequence {πk/σ}k=−∞∞ is called a sampling sequence and the corresponding values f(πk/σ) are called sample values with Nyquist sampling rate of σ/π samples per unit time. The first problem is to find conditions under which f can be reconstructed completely by (1.1). In most applications, the assumption that f is bandlimited, or equivalently that the spectrum of f is supported to certain bounded area in R , is completely justified. More precisely, if a signal f is bandlimited to [−σ,σ] , 0<σ<∞ , or in other words, the quantity σ is the maximal frequency in the spectrum of f, then (1.1) perfectly recovers f by samples, taken every π/σ seconds.

There are several ways by which (1.1) may be generalized (see, e.g. Zayed and Schmeisser, 2014). For example, in 1955, Fogel (Jagerman and Fogel, 1956) was motivated by a problem in aircraft instrument communications (a pilot in the case of pointer-on-scale displays may estimate the pointer position and also the rate information concerning the acceleration of the pointer, corresponding to first- and second-time derivatives) and studied a sampling expansion which involved sample values of a function and its derivatives.

Next, one possible application of sampling theory is in sensor networks. In this case, a large number of sensors is used to monitor some quantity, e.g. temperature. This quantity varies continuously in space and hence, can be viewed as a signal in multidimensional space.

In this paper, we concern with the sampling representation for bandlimited signals f of several variables such that f has finite p-energy for certain 1⩽p<∞ (see Nashed and Sun, 2010 and Nguyen and Unser, 2017).

Note that the reconstruction (1.1) is called stable, if it converges uniformly on R . Such a stability is important not only from the theoretical point of view but, in particular, for applications. If (1.1) is stable, then it is possible to bound the approximation error, which is made by using in (1.1) only finitely many samples. Finally, we note that our sampling series have such a stable reconstruction properties. Namely, they converge absolutely and uniformly on the whole Rn .

Let Zn , Rn and Cn be the usual n-dimensional integer lattice, real Euclidean space and complex Euclidean space, respectively. Next, Lp(Rn) , 1⩽p⩽∞ , denotes the Lebesgue space of complex-values measurable functions on Rn with finite p-energy, i.e. such that |f|p is integrable on Rn . For f∈L1(Rn) , we define the Fourier transform of f as usual by fˆ(x)=∫Rne−i⟨x,t⟩f(t)dt, x∈Rn , where ⟨x,t⟩=∑k=1nxktk is the scalar product on Rn . If f∈Lp(Rn) and 1<p⩽∞ , then we understand fˆ in a distributional sense of tempered distributions S′(Rn) . We recall that the Schwartz space of test functions S(Rn) consists of the complex-valued infinitely differentiable functions φ on Rn satisfying supx∈Rn;|α|⩽m((1+‖x‖)k|Dαφ(x)|)<∞ for all nonnegative integers m and k, also for any nonnegative multi-index α, where Dα is the partial derivative of order α and ‖x‖ denotes the standard Euclidean norm on Rn . The dual space S′(Rn) of S(Rn) is called the space of tempered distributions.

Given a closed subset Ω of Rn , a function ω:Rn→C is called bandlimited to Ω if ωˆ vanishes outside Ω. For σ=(σ1,…,σn)∈Rn with σm>0 0$]]>, m=1,…,n , we define the spaces of bandlimited signals by BQσnp={f∈Lp(Rn):suppfˆ⊂Qσn}, where Qσn={x∈Rn:|xk|⩽σk,k=1,…,n}. Hence, any f∈BQσnp is bandlimited to hyperrectangle or n-orthotope Qσn , i.e. fˆ vanishes outside Qσn . If we equip BQσnp with the norm ‖f‖p=(∫Rn|f(x)|pdx)1/pfor1⩽p<∞,or‖f‖∞=ess suppx∈Rn|f(x)|forp=∞, then BQσnp is a Banach space and it is called the Bernstein space. By the Paley–Wiener–Schwartz theorem (see Hörmander, 1990; p. 181), any f∈BQσnp , 1⩽p⩽∞ , is infinitely differentiable on Rn and has an (unique) extension onto the complex space Cn to an entire function.

Below we shall use the technique of complex variable functions on Cn . In particular, we shall need certain facts in entire functions. Therefore, we use below only function theory terms, i.e. “bandlimited function” term.

Next, to simplify the notation, we shall write Bσp in the case of functions of one variable Bσp instead of BQσ1p .

If we define the sinc -function to be sinc(z)=sin(πz)πz,z≠0,1,z=0, z∈C , then (1.1) can be rewritten as (2.1) f(x)=∑k∈Zf(πkσ)sinc(σπx−k). This expression holds for each f∈Bσp , 1⩽p<∞ . Next, (2.1) converges absolutely and uniformly on compact subsets of C . The Nyquist sampling rate of σ/π samples per unit time in (2.1) is exact, i.e. f∈Bσp cannot be recovered from its samples taken at a lower rate. This means that, for any ε>1 1$]]>, there exist two fj∈Bσp , j=1,2 such that f1≢f2 , but f1 coincides with f2 on every point of the sampling sequence {(επk/σ}k∈Z .

On the other hand, if we know sample values of f∈Bσp only the sampling sequence {2πk/σ}k with the sampling rate σ/(2π) , then reconstruction of f is also possible if we know, in addition, its derivative values {f′(2πk/σ)}k . More precisely, if f∈Bσp , 1⩽p<∞ , then (2.2) f(z)=∑k∈Z(f(2πkσ)+f′(2πkσ)(z−2πkσ))sinc2(σ2πz−k) (see Jagerman and Fogel, 1956; p. 145 or Butzer et al., 2011; p. 442).

The following n-dimensional sampling theorem is a standard extension of (2.1) in BQσnp f(z)=∑u∈Zn(f(πσu) ∏m=1nsinc(σmπzm−um)), z∈Cn (Gosselin, 1977; p. 172) (see also Jerri, 2017 for references). Here and below we are taking (τab)=(τa1b1,…,τanbn) for any τ∈C and all a,b∈Cn such that b1≠0,…,bn≠0 . These series converge absolutely and uniformly on compact subsets of Cn .

The aim of this paper is to prove a multidimensional version of (2.2). In the case of two variables, in Fang and Li (2006) (see also a tutorial review Jerri, 2017; p. 40) a version of (2.2) involving only the following sample values sequences was given (2.3) {f(2πuσ)}u∈Z2,{∂f∂x1(2πuσ)}u∈Z2and{∂f∂x2(2πuσ)}u∈Z2. Namely, in Fang and Li (2006) the following representation was given: (2.4) f(z)=∑u∈Z2[f(2πuσ)+(z1−2πu1σ1)·∂f∂x1(2πuσ)+(z2−2πu2σ2)·∂f∂x2(2πuσ)]×sinc2(σ12πz1−u1)sinc2(σ22πz2−u2) for all f∈BQσ2p , 1⩽p<∞ . We say that such a sampling theorem fails in general. Indeed, let us take any χ∈S(R2) , χ≢0 , such that suppχ⊂{x∈R2:|xk|⩽σk/2,k=1,2} and define Λ(x)=sin(σ12x1)sin(σ22x2)·χˆ(x), x∈R2 . Since S(R2) is invariant under the Fourier transform, it follows that χˆ∈S(R2) , and consequently Λ is in BQσ2p for all 1⩽p⩽∞ . On the other hand, if f=Λ , then all sequences in (2.3) are null sequences. Hence, (2.4) generates null function, but not Λ. Even more, this example with our function Λ shows that the same is still true if we add to (2.4) an arbitrary number of sample values sequences {∂jΛ∂xkj(2πuσ)}u∈Z2, k=1,2 , j=2,3,…  . Therefore, any multidimensional version of (2.2) must necessarily contain also mixed partial derivatives of f.

Our basic idea based on observation that any sampling series are also an interpolation formula. For example, the m-th coefficient in (2.1) equals to the value of the sum, i.e. the value of f at the point with the number m, i.e. at πm/σ .

We now state the main result of this paper. Theorem 1.

If f∈BQσ2p , 1⩽p<∞ , then (2.5) f(z)=∑u∈Z2[f(2πuσ)+(z1−2πu1σ1)·∂f∂x1(2πuσ)+(z2−2πu2σ2)·∂f∂x2(2πuσ)+(z1−2πu1σ1)(z2−2πu2σ2)·∂2f∂x1∂x2(2πuσ)]×sinc2(σ12πz1−u1)sinc2(σ22πz2−u2). The series (2.5) converge absolutely and uniformly on R2 and on compact subsets of C2 .

Set Qπ2={x∈R2:|x1|,|x2|⩽π} . We may assume without loss of generality below that σ1=σ2=π , since the operator A(f)(z)=f(πσ1z1,πσ2z2) is an isometric isomorphism between BQσ2p and BQπ2p .

Let 1⩽p<q⩽∞ . Then there exists 0<M(p;q)<∞ such that (3.1) ‖f‖BQπ2q⩽M(p;q)‖f‖BQπ2p for all f∈BQπ2q (see Triebel, 1983; pp. 21–22). Therefore, (3.2) BQπ21⊂BQπ2p⊂BQπ2q⊂BQπ2∞. If f∈BQπ2p with 1⩽p<∞ , then (3.3) limx∈Rn;x→∞f(x)=0 (see Nikol’skii, 1975; p. 118). If f∈BQπ2∞ , then (see, e.g. Nikol’skii, 1975; p. 117; Jagerman and Fogel, 1956; p. 181 or Hörmander, 1990; p. 181). (3.4) |f(z)|⩽supx∈R2|f(x)|eπ(|y1|+|y2|) for all z=x+iy , x,y∈R2 . Note that (3.2) shows that (3.4) also holds true for any f∈BQπ2p , 1⩽p<∞ .

In the sequel, we shall use several times the following function in BQπ/22∞ : sπ(z)=sin(πz12)sin(πz22). Let us start by proving a simple auxiliary statement for functions of one variable. For completeness, we also give its proof.

Set BQσ2={f∈C0(R2):suppfˆ⊂Qσ2}, where C0(R2) is the usual space of continuous functions on R2 that vanish at infinity. BQσ2 is a Banach space with respect to the sup -norm inherited from C0(R2) . Next, (3.3) implies that (3.30) BQσ2p⊂BQσ2 for all 1⩽p<∞ .

Let us look again at the proof of Theorem 2. Then we can see that together with (3.20) and (3.21), we used, in addition, only the following two properties of fj , j=1,2 : a) fjˆ is supported in Qπ2 ; b) fj satisfies (3.3). Hence, the following corollary is true:

Next, for each 1<r<∞ and any w>0 0$]]>, the following estimate is true (see Splettstösser, 1982; p. 811): (3.31) ∑k∈Z|sinc(wt−k)|r⩽rr−1 for all t∈R . In particular, this implies that the series in (3.31) converge on the whole R and its sum is a bounded function on R . Proof of Theorem 1.

To begin, we recall the following Nikol’skii’s inequality (Nikol’skii, 1975; p. 123): for any 1⩽p<∞ and each 0<ϱ<∞ , there exists a finite positive number c(σ;p;ϱ) such that (∑u∈Z2|f(ϱu)|p)1/p⩽c(σ;p;ϱ)‖f‖BQσ2p for all f∈BQσ2p . A similar inequality holds also for all partial derivatives of f, since by Bernstein’s inequality (Nikol’skii, 1975; p. 116), any partial derivative of f also is an element of BQσ2p .

Suppose again that σ1=σ2=π and let l2p be the usual space of sequences {cu∈C:∑u∈Z2|cu|p<∞} . First, we claim that the series (2.5) converge absolutely and uniformly on R2 . Indeed, if z=x∈R2 , then (2.5) can be divided into four series of the following type (3.32) Iν;∞(x)=∑m∈Z2[αm(x12−m1)ν1(x22−m2)ν2×sinc2(x12−m1)sinc2(x22−m2)] with certain α∈l2p and some ν=(ν1,ν2) , where ν1,ν2∈{0,1} . Therefore, it suffices to show that for any ε>0 0$]]> there is a positive integer N such that (3.33) |Iν;N(x)|=|∑m∈Z2|m1|,|m2|⩾N[αm(x12−m1)ν1(x22−m2)ν2sinc2(x12−m1)×sinc2(x22−m2)]|<ε for all x∈R2 . Take any p1⩾p such that 1<p1<∞ . Then Hölder’s inequality implies that (3.34) |Iν;N(x)|⩽(∑m∈Z2|m1|,|m2|⩾N|αm|p1)1/p1(∑m∈Z2|m1|,|m2|⩾N|x12−m1|q1ν1|x22−m2|q1ν2×|sinc(x12−m1)|2q1|sinc(x22−m2)|2q1)1/q1 with 1<q1<∞ such that 1/p1+1/q1=1 . Since α∈l2p⊂l2p1 , it follows that, for any ε1>0 0$]]>, there exists a positive integer N1 such that (3.35) (∑m∈Z2|m1|,|m2|⩾N1|αm|p1)1/p1<ε1. On the other hand, it is clear that |xj2−mj|q1νj|sinc(xj2−mj))|2q1⩽|sinc(xj2−mj))|(2−νj)q1⩽1 for j=1,2 , mj∈Z and all xj∈R . Hence, by (3.31), we see that the second series in (3.34) converge and (3.36) (∑m∈Z2|m1|,|m2|⩾N|x12−m1|q1ν1|x22−m2|q1ν2|sinc(x12−m1)|2q1|sinc(x22−m2)|2q1)1/q1⩽(∑m1∈Z,|m1|⩾N|sinc(x12−m1)|(2−ν1)q1)1/q1)⩽((2−ν1)q1(2−ν1)q1−1)1/q1 for all x∈R2 and each N⩾0 . Combining this with (3.33), (3.34) and (3.35), we get that (3.32) converges absolutely and uniformly on R2 .