Cardinal interpolation with periodic polysplines on strips

CALCOLO 44, 203 – 217 (2007) DOI: 10.1007/s10092-007-0137-9

A. Bejancu · O.I. Kounchev · H. Render

Cardinal interpolation with periodic polysplines on strips

Received: September 2006 / Accepted: April 2007 – © Springer-Verlag 2007

Abstract We obtain a multivariate extension of a classical result of Schoenberg on cardinal spline interpolation. Specifically, we prove the existence of a unique function in C 2 p−2 Rn+1 , polyharmonic of order p on each strip ( j, j + 1) × Rn , j ∈ Z, and periodic in its last n variables, whose restriction to the parallel hyperplanes {j }×Rn , j ∈ Z, coincides with a prescribed sequence of n-variate periodic data functions satisfying a growth condition in | j |. The constructive proof is based on separation of variables and on Micchelli’s theory of univariate cardinal L-splines. Keywords: cardinal L-splines, polyharmonic functions, multivariable interpolation Mathematics Subject Classification (2000): 41A05, 41A15, 41A63 A. Bejancu (B) Department of Mathematics, Kuwait University, PO Box 5969, Safat 13060, Kuwait. E-mail: [email protected] O.I. Kounchev Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev St. 8, 1113 Sofia, Bulgaria. E-mail: [email protected] H. Render Departamento de Matemáticas y Computación, Universidad de La Rioja, Edificio Vives, Luis de Ulloa, s/n. 26004 Logroño, Spain. E-mail: [email protected]

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1 Introduction Cardinal spline interpolation represents a classical model in approximation theory (Schoenberg [12]) with implications for wavelet analysis (Chui [4]). For an integer p ≥ 1, a cardinal spline of degree 2 p−1 is a function s ∈ C 2 p−2 (R) whose restriction to each ( j, j + 1), j ∈ Z, is a polynomial of degree interval 2 p−1. For a sequence y j j ∈Z of real numbers satisfying, for constants C > 0, γ ≥ 0, the power growth condition y j ≤ C (1 + | j |γ ) , j ∈ Z, a central result of Schoenberg’s theory [12] asserts the existence of a unique cardinal spline s of degree 2 p − 1 interpolating the data sequence on the cardinal grid Z and growing no faster than the data sequence, i.e., j ∈ Z, s ( j) = yj, (1) |s (t)| ≤ D (1 + |t|γ ) , t ∈ R, where D depends on C, γ , and p. This result was generalized by Micchelli [9] to cardinal L-splines, i.e., functions in C N−2 (R) whose restrictions to each interval ( j, j + 1), j ∈ Z, are in the null-space of a differential operator L of order N with constant coefficients. An important research topic over the last two decades has been the multivariate extension of cardinal interpolation to box-splines and radial basis functions on the discrete lattice Zn (see [2, 3]). The present paper obtains a new multivariable extension of Schoenberg’s result in which the cardinal grid Z is replaced by the set Z × Rn of hyperplanes in Rn+1 and the role of the univariate operator L is played by the iterated Laplacian in n + 1 variables. In order to state the main result, Theorem 1.1, we introduce the following notation and terminology. Let N0 be the set of non-negative integers. For any α α multi-index α ∈ Nd0 , we write |α| := α1 + . . . + αd and D α := ∂∂x α11 · · · ∂∂x αdd . 1 d 2 2 d Also, |x |2 := x 1 + . . . + x d for x ∈ R . Recall that a function ϕ : U → C defined on an open subset U of the Euclidean space Rd is polyharmonic of order p on U if ϕ ∈ C 2 p (U ) and p ϕ (x ) = 0 for all x ∈ U , where p is the 2 2 pth iterate of the Laplace operator = ∂∂x 2 + . . . + ∂∂x 2 . 1

d

Definition 1.1 A function S : Rn+1 → C is called a cardinal polyspline of order p on strips in Rn+1 if S ∈ C 2 p−2 Rn+1 and S is polyharmonic of order p on each strip j := (t, y) ∈ R × Rn : t ∈ ( j, j + 1) , j ∈ Z. If, in addition, S is 2π -periodic in each of its last n variables, then S is called a cardinal periodic polyspline of order p on strips in Rn+1 .


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The general concept of a multivariate polyspline whose domain pieces are separated by arbitrary hypersurfaces rather than hyperplanes was introduced and studied by Kounchev ([5, 6]). Let Q = [−π, π ] and, for any non-negative integer k, denote by W k (Q n ) the space of all continuous functions f : Rn → C which are 2π -periodic in 1 −iξ,y f (y) d y, each variable and whose Fourier coefficients f (ξ ) = (2π) n Qn e n ξ ∈ Z , satisfy f k := f (ξ ) (1 + |ξ |2 )k < ∞. ξ∈Zn

Here, and in the sequel, the brackets , denote the usual dot product in Rn and i is the imaginary root of −1 situated in the upper semiplane. Note that C ∞ (Q n ) ⊂ W k (Q n ) ⊂ C k (Q n ). Theorem 1.1 Let γ ≥ 0 and p ∈ {1, 2, . . .} be fixed. Assume that a sequence of data functions f j ∈ W 2 p−2 (Q n ), j ∈ Z, satisfies the growth condition fj ≤ C1 (1 + | j |γ ) , j ∈ Z, (2) 2 p−2 for some constant C1 > 0. Then there exists a unique cardinal periodic polyspline S of order p on strips in Rn+1 , satisfying the interpolation conditions (3) S ( j, y) = f j (y) , j ∈ Z, y ∈ Q n , as well as the following growth property for any α ∈ N0n+1 with |α| ≤ 2 p − 2: |D α S (t, y)| ≤ C2 (1 + |t|γ ) ,

t ∈ R, y ∈ Q n ,

(4)

where the constant C2 > 0 depends on C1 , γ , and p. For data functions prescribed on concentric spheres, a similar result on cardinal interpolation with polysplines was obtained in [7]. As in that case, the present Lagrange scheme construction of the interpolating polyspline S (Sect. 2) is reduced by separation of variables to the analysis of a family of univariate cardinal L-splines. However, our analysis of cardinal L-splines (Sect. 3) employs a Fourier transform representation due to Micchelli [9] instead of B-spline representations as in [7]. This enables direct proofs of the decay properties of fundamental cardinal L-splines (Sect. 4), avoiding the intermediate estimates of Euler functions and B-splines that were needed in [7]. For p = 2 and non-periodic data, the existence part of Theorem 1.1 was established in [1] using explicit computations. The present approach for arbitrary p works by replacing these computations with the key inequality of Lemma 3.1.

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2 Lagrange scheme and separation of variables The concept of a ‘fundamental spline’ — the unique bounded cardinal spline L of degree 2 p − 1 such that L (0) = 1 and L ( j ) = 0 for all integers j = 0 — plays a key role in Schoenberg’s theory [12]. Indeed, the cardinal spline s satisfying (1) is constructed as a Lagrange series ∞

s (t) =

y j L (t − j ) ,

t ∈ R,

j =−∞

absolutely and uniformly convergent on compact sets due to the exponential decay of |L (t)| as |t| → ∞. The analogous concept in the context of cardinal periodic polysplines on strips is introduced as follows. Definition 2.1 Let f ∈ W 2 p−2 (Q n ). A function L f : Rn+1 → C is called a fundamental interpolation polyspline of order p for f if L f is a cardinal periodic polyspline of order p on strips in Rn+1 such that L f is bounded on Rn+1 and satisfies f (y) , if j = 0, ∀y ∈ Rn . (5) L f ( j, y) = 0, if j ∈ Z \ {0} , The existence of fundamental interpolation polysplines of order p with suitable decay properties is established in the next theorem, whose proof is postponed to Sect. 4. Theorem 2.1 For each f ∈ W 2 p−2 (Q n ) there exists a fundamental interpolation polyspline L f of order p for f . In addition, there exist positive constants C3 and β, independent of f , such that, for any multi-index α ∈ N0n+1 with |α| ≤ 2 p − 2, and any f ∈ W 2 p−2 (Q n ), we have α D L f (t, y) ≤ C3 e−β|t | f |α| ∀t ∈ R, ∀y ∈ Q n . (6) This theorem implies the existence part of our main result. Proof of existence in Theorem 1.1 For each j ∈ Z, by Theorem 2.1 there exists a fundamental interpolation polyspline L f j of order p satisfying (5) with f replaced by f j ∈ W 2 p−2 (Q n ). Define S by the Lagrange-type series S (t, y) =

∞ j =−∞

L f j (t − j, y) ,

t ∈ R, y ∈ Rn .

(7)


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By formal differentiation for any multi-index α ∈ N0n+1 with |α| ≤ 2 p − 2, and by the decay estimates (6), the inequality f r1 ≤ f r2 for 0 ≤ r1 ≤ r2 , and the hypothesis (2), we obtain ∞ α D L f (t − j, y) |D S (t, y)| ≤ j α

j =−∞

≤ C3

∞

e−β|t − j | f j 2 p−2

j =−∞

≤ C3

∞

e−β|t − j |C1 (1 + | j |γ )

j =−∞

≤ C2 (1 + |t|γ ) , the last inequality being proved in [11, Sect. 5]. Therefore (7) is absolutely and uniformly convergent on compact sets, the interpolation conditions (3) are satisfied, and S is 2π -periodic in its last n variables. Further, the series can be differentiated term-by-term and its partial derivatives D α S are continuous and satisfy the growth property (4) for any multi-index α ∈ N0n+1 , |α| ≤ 2 p − 2. We also deduce that S is polyharmonic of order p on each strip j , j ∈ Z, since every term of the locally and uniformly convergent series (7) has this property (see Nicolesco [10, p. 23]). The existence part of the theorem is established.

For the proof of uniqueness in Theorem 1.1, we use the following separation of variables result. Lemma 2.1 Let S be a cardinal periodic polyspline of order p on strips in Rn+1 , and, for each t ∈ R, let

1 e−iξ,y S (t, y) d y, ξ ∈ Zn , (8) S (t, ξ ) = (2π )n Q n be the sequence of Fourier coefficients of S with respect to the last n variables. If Lξ denotes the ordinary differential operator 2 p d 2 |ξ | − , (9) Lξ = 2 dt 2 S (·, ξ ) that maps t to S (t, ξ ) is a univariate then, for each ξ ∈ Zn , the function S (t, ξ ) = 0 for t ∈ ( j, j + 1), j ∈ Z, and S (·, ξ ) ∈ cardinal Lξ -spline, i.e., Lξ 2 p−2 (R). C

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Proof The continuity property S (·, ξ ) ∈ C 2 p−2 (R) follows from (8) and S ∈ C 2 p−2 Rn+1 . Let j ∈ Z. By hypothesis, S (t, y) is polyharmonic of order p S (t, ξ ) is also of class C ∞ as and hence of class C ∞ in the strip j . Therefore n a function of t ∈ ( j, j + 1), for each ξ ∈ Z , and standard arguments imply the validity of the Fourier series representation (10) S (t, ξ ) eiξ,y , S (t, y) = ξ∈Zn

absolutely and uniformly convergent on compact subsets of j . By termwise differentiation, valid here, we obtain p n 2 2 ∂ ∂ + S (t, y) 0 = p S (t, y) = ∂t 2 ∂ yk2 k=1 p d2 2 = − |ξ |2 S (t, ξ ) eiξ,y dt 2 n ξ∈Z = Lξ S (t, ξ ) eiξ,y , (t, y) ∈ j , ξ∈Zn

S (t, ξ ) = 0 for ξ ∈ Zn , t ∈ ( j, j + 1). which implies Lξ

Proof of uniqueness in Theorem 1.1 Assume that S1 and S2 are two cardinal periodic polysplines of order p on strips in Rn+1 such that the interpolation conditions (3) and the growth property (4) are satisfied for S1 and S2 in place S (t, ξ ), ξ ∈ Zn , be the sequence of S. Let S := S1 − S2 and, for each t ∈ R, let of Fourier coefficients defined by (8). By Lemma 2.1 and the assumptions on S (·, ξ ) is a univariate cardinal Lξ -spline for each ξ ∈ Zn , satisfying S1 and S2, S ( j, ξ ) = 0 for all j ∈ Z, ξ ∈ Zn , as well as

1 |S (t, y)| d y ≤ 2C2 (1 + |t|γ ) , t ∈ R, ξ ∈ Zn , S (t, ξ ) ≤ (2π )n Q n where C2 is the constant from (4). Since the cardinal interpolation problem with cardinal L-splines of polynomial growth has a unique solution (Micchelli [9]), it follows that S (t, ξ ) = 0 for t ∈ R, ξ ∈ Zn . Then (10) implies that

S ≡ 0, i.e., S1 ≡ S2. Remark 2.1 The above uniqueness proof relies on the compactness of the integration domain Q n in (8) and does not work in the non-periodic case in which Q n is replaced by Rn . By contrast, the existence proof of Theorem 1.1 carries over to the non-periodic case with only minor changes, as illustrated for biharmonic polysplines ( p = 2) in [1, Theorem 11].


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The remaining two sections are devoted to the proof of Theorem 2.1. Arguing conversely to Lemma 2.1, in Sect. 4 we construct the required fundamental interpolation polyspline as a Fourier series whose coefficients are obtained from the family of fundamental cardinal L-splines introduced below. 3 A family of cardinal L-splines This auxiliary section studies the Fourier transform representation of the fundamental cardinal L-spline associated to the operator L = Lξ of (9), for the wider range ξ ∈ Rn of the parameter ξ . First, let N ∈ {1, 2, . . .}, := (λ1 , ..., λ N ) ∈ R N and consider the linear differential operator of order N with constant coefficients L = q dtd , where q is the polynomial N z − λj . q (z) = j =1

A function s : R → C is a ‘cardinal L-spline’ if s ∈ C N−2 (R) and, for every j ∈ Z, s ∈ C ∞ ( j, j + 1) and Ls (t) = 0 for t ∈ ( j, j + 1). The following is a special case of a more general result due to Micchelli [9]. Proposition 3.1 Suppose that the set {λ1, ..., λ N } is symmetric about the origin and N is even. Then there exists a unique cardinal L-spline L , bounded on R, such that L (0) = 1 and L ( j ) = 0 ∀ j ∈ Z \ {0}. In addition, |L (t)| ≤ Ae−B|t | , ∀t ∈ R, for some positive constants A and B. Of particular importance to our analysis is Micchelli’s representation [9, p. 224] of the ‘fundamental’ function L as an absolutely convergent inverse Fourier transform:

∞ 1 ω (u) du, t ∈ R, eiut L (t) = 2π −∞ q (iu) where

ω (u) =

k∈Z

1 q (i (u + 2π k))

−1 ,

u ∈ R.

Returning to our parameter operator (9), we specialize the above result to the case N = 2 p, λ1 = . . . = λp = |ξ |2 , λp+1 = . . . = λ2 p = − |ξ |2 , where ξ ∈ Rn . Thus = ξ := λ1 , ..., λ2 p and each piece of a cardinal ±t |ξ| 2 ±t |ξ| 2 p−1 ±t |ξ| 2 is spanned by the set e , te , . . . , t e if ξ = 0, Lξ -spline 2 p−1 if ξ = 0. The latter case corresponds to Schoenberg’s or 1, t, . . . , t polynomial cardinal splines. Let qξ (z) := z2 + |ξ |22 ,

z ∈ C, ξ ∈ Rn ,

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p so that qξ (iz) = (−1) p qξ (z) , and define −1 1 , ω p (z, ξ ) := (−1) p ωξ (z) = p qξ (z + 2π k) k∈Z

z ∈ C, ξ ∈ Rn .

Hence, for each ξ ∈ Rn , the absolutely convergent integral

∞ 1 ω p (u, ξ ) p du, t ∈ R, eiut L ξ (t) := L ξ (t) = 2π −∞ qξ (u)

(11)

defines the fundamental cardinal Lξ -spline satisfying (12) L ξ (0) = 1 and L ξ ( j ) = 0 for j ∈ Z\ {0} , as well as L ξ (t) ≤ Aξ e−Bξ |t |, ∀t ∈ R, for positive constants Aξ and Bξ . Note that ω p is continuous in (z, ξ ), and ω p (z + 2π, ξ ) = ω p (z, ξ ). The additional properties of ω p stated in the following lemma are used in the next section to prove the remarkable result that Aξ and Bξ actually do not depend on the parameter ξ (see Theorem 4.1). Lemma 3.1 There exists ε = ε ( p) > 0 such that, for all z = u + iη with |u| ≤ π , u = 0, |η| ≤ ε, and for all ξ ∈ Rn , 1 3 ≥ . (13) ω p (z, ξ ) qξ (z + 2π k) p

k∈Z

− p are analytic functions of z in the comAlso, ω p (z, ξ ) and ω p (z, ξ ) qξ (z) mon strip {z ∈ C : |Im z| < ε}, for any ξ ∈ Rn . Proof Note that if |u| ≤ π and u = 0, then qξ (z) = 0. To prove (13) for positive ε, it is thus sufficient to establish the inequality q (z) p qξ (z) p ξ (14) 3 ≥1+ qξ (z + 2π k) p ω p (z, ξ ) k∈Z\{0}

for |u| ≤ π , |η| ≤ ε, and ξ ∈ Rn . Letting G p,ξ (z) :=

k∈Z\{0}

1 , qξ (z + 2π k) p

we have the identity p p qξ (z) = 1 + qξ (z) G p,ξ (z) ω p (z, ξ )

(15)


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for |u| ≤ π , η ∈ R, and ξ ∈ Rn . Since qξ (u + 2π k + iη) = (u + 2π k)2 − η2 + |ξ |22 + 2iη (u + 2π k) , for k ∈ Z \ {0}, |u| ≤ π , and |η| ≤

π 2

(16)

we obtain

2 2 qξ (z + 2π k) ≥ Re qξ (z + 2π k) ≥ π 2 (2 |k| − 1)2 − π ≥ 3π , 4 4

(17)

which implies that 1 G p,ξ (z) ≤ qξ (z + 2π k) p k∈Z\{0}

≤

1 1 p =: M < ∞. 2 p π (2 |k| − 1)2 − 14 k∈Z\{0}

(18)

Consequently, G p,ξ (z) is well-defined and analytic for z = u + iη satisfying |u| < π , |η| < π2 . Let δ ∈ (0, 1) and ε > 0 be such that

3 δ 2

p ≤

1 2M

and

4π ε π ≤ tan . δ 8p

(19)

To prove (14) for this choice of ε, let |u| ≤ π , |η| ≤ ε, and consider two cases. 1. First, assume that u 2 + |ξ |22 ≤ δ. Letting k = 0 in (16) and noting that (19) implies 4π ε ≤ δ, we obtain qξ (z) ≤ u 2 + |ξ |2 − η2 + 2 |uη| ≤ max δ, ε 2 + 2π ε ≤ 3 δ. (20) 2 2 Then (18)–(20) yield 1 + qξ (z) p G p,ξ (z) ≥ 1 − qξ (z) p G p,ξ (z) ≥ 1 − qξ (z) p M ≥ 1 , 2 as well as qξ (z) p

1 1 1 p ≤ ≤ . qξ (z + 2π k) qξ (z + 2π k) p 2M 2 k∈Z\{0} k∈Z\{0}

The last two displays and (15) show that (14) holds for u 2 + |ξ |22 ≤ δ.

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2. Assume now that u 2 + |ξ |22 ≥ δ. In this case we prove below that arg qξ (z + 2π k) ≤ π ∀k ∈ Z, 8p

(21)

where arg w ∈ [−π, π ) denotes the principal argument of w ∈ C. From (qξ (z)) p π ≤ for any k ∈ Z \ {0}, hence this we deduce arg q (z+2πk) (ξ )p 4 p qξ (z) p qξ (z) 1 p , k ∈ Z \ {0} . Re p ≥ √ qξ (z + 2π k) 2 qξ (z + 2π k) Since by the inequality |w| ≥ Re w we have 1 + qξ (z) p G p,ξ (z) ≥ 1 + Re k∈Z\{0}

p qξ (z) qξ (z + 2π k)

p ,

the last two displays and (15) show that (14) also holds for u 2 + |ξ |22 ≥ δ. To prove (21), let k = 0 first and note that 1 Re qξ (u + iη) = u 2 + |ξ |22 − η2 ≥ δ − ε 2 > δ. 2 Since Im qξ (u + iη) = 2 |uη| ≤ 2π ε whenever |u| ≤ π and |η| ≤ ε, we obtain Im qξ (u + iη) 4π ε ≤ , 0< Re qξ (u + iη) δ ≤ 8πp . For k ∈ Z \ {0}, we use so (19) implies arg qξ (u + iη) ≤ arctan 4πε δ Im qξ (u + 2π k + iη) = 2η (u + 2π k) together with inequality (17) to obtain Im qξ (u + 2π k + iη) 2ε (π + 2π |k|) 8ε ≤ . ≤ 0< 2 1 Re qξ (u + 2π k + iη) π π 2 (2 |k| − 1) − 4 ≤ 4πε , from (19) it follows that (21) also holds for k ∈ Z \ {0}. This Since 8ε π δ completes the proof of inequality (13). For the second statement of the lemma, we adapt Madych and Nelson’s arguments [8, Lemma 1], used in the context of polyharmonic radial basis functions. For any interval I ⊂ R, p we write Iε = I × (−ε, ε). Relations (14) and (15) imply that 1 + qξ (z) G p,ξ (z) = 0 for all z ∈ (−π, π )ε . Since qξ (z) and G p,ξ (z) are analytic for z ∈ (−π, π )ε , from (15) it follows that − p are analytic functions of z in (−π, π )ε for ω p (z, ξ ) and ω p (z, ξ ) qξ (z) any ξ ∈ Rn . In addition, by its periodicity and continuity properties, ω p (z, ξ ) extends as an analytic function of z in Rε . It remains π only to observe that π qξ (z) is analytic and does not vanish for z ∈ Rε \ − 2 , 2 ε , which implies − p also extends as an analytic function of z in Rε for any that ω p (z, ξ ) qξ (z)

ξ ∈ Rn . The proof is complete.


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4 Decay estimates and proof of Theorem 2.1 The following two lemmas are used to estimate in terms of ξ the exponential decay of the fundamental function L ξ and of its derivatives. Lemma 4.1 There exists√C4 > 0 such that, for any m ∈ {0, 1, ..., 2 p − 2}, any η ∈ R with |η| ≤ 1/ 2, and any ξ ∈ Rn with |ξ |2 ≥ 1, we have

∞ |u + iη|m C4 p du ≤ 2 p−m−1 . |ξ | −∞ qξ (u + iη) 2

Proof The relations qξ (u + iη) ≥ Re qξ (u + iη) = u 2 − η2 + |ξ |22 and |ξ |22 ≥ 1 ≥ 2η2 imply qξ (u + iη) ≥ u 2 + 12 |ξ |22 . Since |u + iη| < |u| + 1, the integral is bounded above by

∞ m ∞ (|u| + 1)m uk m du = 2 p du. p 1 2 k u 2 + 12 |ξ |22 −∞ u 2 + 2 |ξ | 2 0 k=0 The conclusion follows by changing the variable of integration and using the

inequality |ξ |22 ≥ 1. Lemma 4.2 Let ε be the value provided by the statement of Lemma 3.1. There exists C5 > 0 such that ω p (z, ξ ) ≤ C5 |ξ |2 p−1 2 for all z = u + iη ∈ C and ξ ∈ Rn with u ∈ R, |η| ≤ ε, and |ξ |2 ≥ 4π . Proof By Lemma 3.1, inequality (13) holds for all ξ ∈ Rn and z = u + iη with |u| ≤ π , u = 0, |η| ≤ ε. To find a lower bound estimate of the right-hand side of (13), note that k = 0, |u| ≤ π , and |η| ≤ ε < π imply that |z + 2π k| ≤ |z| + 2π |k| < 2π + 2π |k| ≤ 4π |k| , and hence qξ (z + 2π k) = (z + 2π k)2 + |ξ |22 < 16π 2k 2 + |ξ |22 . Consequently, ∞ 1 3 >2 p , ω p (z, ξ ) 16π 2k 2 + |ξ |2 k=1

2

whenever |u| ≤ π , u = 0, and |η| ≤ ε. Since the infinite sum of the last display is not less than

∞ 1 p dx , 2 2 16π x + |ξ |22 1

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a change of variable shows that, for |ξ |2 ≥ 4π , |u| ≤ π , u = 0, and |η| ≤ ε,

∞ 1 1 3 > p dx . 2 p−1 2 ω p (z, ξ ) x +1 2π |ξ |2 1 By the continuity and 2π -periodicity of ω p (·, ξ ), this implies the inequality of the statement for all u ∈ R, |η| ≤ ε, and |ξ |2 ≥ 4π .

Theorem 4.1 There exist positive constants C6 and β such that, for all ξ ∈ Rn and m ∈ {0, 1, .., 2 p − 2}, m d −β|t | m ∀t ∈ R. (22) dt m L ξ (t) ≤ C6 1 + |ξ |2 e Proof Consider the absolutely convergent representation (11) of L ξ as an inverse Fourier transform. Differentiating under the integral and taking account p |u| → ∞, and of the boundedness of ω p (·, ξ ) of the magnitude of qξ (u) as m for each fixed ξ , it follows that dtd m L ξ exists for any m ∈ {0, 1, .., 2 p − 2}, and

∞ 1 ω p (u, ξ ) dm L ξ (t) = (iu)m eiut p du, t ∈ R. m dt 2π −∞ qξ (u) − p Let ε be the value for which, by Lemma 3.1, ω p (z, ξ ) and ω p (z, ξ ) qξ (z) are analytic in the common strip {z ∈ C : |Im z| < ε}, and choose β ∈ (0, ε). Let m ∈ {0, 1, .., 2 p − 2} and assume t > 0 (the case t < 0 has a similar treatment). For any R > 0, we apply Cauchy’s theorem to the integral

ω p (z, ξ ) p dz, (iz) m ei zt qξ (z) where is the oriented boundary of the rectangle [−R, R] × [0, β]. The ‘side’ parts of this integral on each of the vertical segments {±R+ iτ : 0 ≤ τ ≤ β} p tend to zero as R → ∞, due to the magnitude of max qξ (±R + iτ ) : 0 ≤ τ ≤ β}, and to the boundedness of ω p (z, ξ ) for z ∈ R × [0, β] for each fixed ξ . Hence, letting R → ∞, we obtain

∞ 1 ω p (u + iβ, ξ ) dm p du, L (t) = (i (u + iβ))m ei(u+iβ)t ξ m dt 2π −∞ qξ (u + iβ) which implies that m

e−βt ∞ |u + iβ|m ω p (u + iβ, ξ ) d du. dt m L ξ (t) ≤ 2π qξ (u + iβ) p −∞

(23)


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√ For |ξ |2 ≥ 4π, since the proof of Lemma 3.1 ensures ε < 1/ 2, Lemmas 4.1 and 4.2 imply that m C4 C5 −βt m d L (t) (24) dt m ξ ≤ 2π e |ξ |2 . For |ξ |2 ≤ 4π , we show that there exists C7 > 0, independent of ξ , such that

∞ |u + iβ|m ω p (u + iβ, ξ ) du ≤ C7 , m ∈ {0, 1, .., 2 p − 2} . (25) qξ (u + iβ) p −∞

To this aim, we write the integral as |u|>β + |u|≤β and find an upper bound for each of the subintegrals. Firstly, since ω p (u + iβ, ξ ) is continuous in (u, ξ ) and 2π -periodic in u, there exists C8 > 0 such that ω p (u + iβ, ξ ) ≤ C8 for u ∈ R and |ξ |2 ≤ 4π. Using the identity qξ (z) = z2 + |ξ |22 = |iz − |ξ |2 | |iz + |ξ |2 |, we also have qξ (u + iβ) = u 2 + (β + |ξ |2 )2 u 2 + (β − |ξ |2 )2 ≥ u 2 . Hence there exists C9 > 0 such that, for |ξ |2 ≤ 4π and m ∈ {0, 1, .., 2 p − 2},

|u|>β

|u + iβ|m ω p (u + iβ, ξ ) |u|m p du ≤ C8 2m du ≤ C9 . 2p qξ (u + iβ) |u|>β u

(26)

Secondly, the proof of Lemma 3.1 shows that the right-hand side of (15) does |η| ≤ ε. not vanish for any ξ ∈ Rn and any z = u + iη with |u| ≤ π, − p is In particular, for η := β, the expression ω p (u + iβ, ξ ) qξ (u + iβ) a continuous function of (u, ξ ), hence bounded above in absolute value by a constant independent of ξ for |u| ≤ β and |ξ |2 ≤ 4π . Thus there exists C10 > 0 such that

|u + iβ|m ω p (u + iβ, ξ ) du ≤ C10 , (27) qξ (u + iβ) p |u|≤β for all ξ ∈ Rn with |ξ |2 ≤ 4π . Relations (26) and (27) imply (25). From (23), (24) and (25) we obtain the estimate (22) for all ξ ∈ Rn .

Proof of Theorem 2.1 Define L f : Rn+1 → R as a Fourier series with respect to the last n coordinates: (28) f (ξ ) L ξ (t) eiξ,y , t ∈ R, y ∈ Rn . L f (t, y) := ξ∈Zn

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The series is absolutely convergent since f ∈ W 2 p−2 (Q n ) ⊂ W 0 (Q n ) and the estimate (22) imply that L f (t, y) ≤ f (ξ ) L ξ (t) ≤ C11 e−β|t | f 0 ξ∈Zn

for some constant C11 . This also proves (6) for α = 0 and shows that L f ∈ C Rn+1 . Moreover, differentiating term-by-term and using (22), we deduce that the partial derivatives D α L f are continuous and satisfy (6) for any multiindex α ∈ N0n+1 with |α| ≤ 2 p − 2. On the other hand, each term of the expansion (28) is polyharmonic of order p for (t, y) ∈ j , j ∈ Z, since

p

L ξ (t) e

iξ,y

p n ∂2 ∂2 = L ξ (t) eiξ,y + 2 2 ∂t ∂ yk k=1 p n d2 = − ξk2 L ξ (t) eiξ,y 2 dt k=1 2 p d iξ,y 2 − |ξ |2 L ξ (t) = 0, t ∈ ( j, j + 1) . =e dt 2

The last equality holds because L ξ is a cardinal Lξ -spline. Since the series (28) is absolutely and uniformly convergent on compact sets, Nicolesco’s result [10, p. 23] implies that L f is polyharmonic of order p on each strip j , j ∈ Z. Therefore L f is a cardinal periodic polyspline of order p on strips in Rn+1 . To finish the proof, it only remains to observe that the interpolation property (5) of L f is a consequence of the representation (28) and the interpolation

conditions (12) for L ξ .

Remark 4.1 If the data function f is not periodic, the Fourier series that defines L f has to be replaced by a Fourier integral over ξ ∈ Rn . Since Lemma 3.1 and Theorem 4.1 hold for ξ ∈ Rn , the above proof can be adapted to obtain a non-periodic version of Theorem 2 extending the biharmonic case p = 2 treated in [1, Theorem 7]. Acknowledgement. This work was supported by Nuffield Foundation research grant NAL/00619/G held by the first author while at the School of Mathematics, University of Leeds, UK. The second- and third-named authors have been partially sponsored by the Institutes partnership project with the Humboldt Foundation, Germany.


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