PPOISSON SUMMATION FORMULA AND BOX SPLINES
MICH`ELE VERGNE
Contents
1. Introduction 1
Part 1. The results
52. Piecewise analytic functions 53. Box splines with parameters 103.1. Inversion formula for the box spline. The unimodular case 103.4. Inversion of the Box spline with parameters 123.8. Translated Box spline 164. Rational functions on U and functions on T Part 2. Proofs
Part 3. Applications
Introduction
Let V be a finite dimensional real vector space, of dimension d , equippedwith a lattice Λ. Let U be the dual vector space of V , and Γ ⊂ U the duallattice of Λ. For v ∈ V , denote by ∂ v the differentiation in the direction v .We denote by δ λ the Dirac measure at the point λ ∈ V .Let Φ := [ α , α , . . . , α N ] be a list of elements in Λ. The box spline B (Φ)is the measure on V such that, for a continuous function F on V , (cid:104) B (Φ) , F (cid:105) = (cid:90) · · · (cid:90) F ( N (cid:88) k =1 t k α k ) dt · · · dt N . The support of B (Φ) is the zonotope Z (Φ) := { (cid:80) Nk =1 t k α k ; 0 ≤ t k ≤ } .If Φ generates V , then the measure B (Φ) is given by integration against apiecewise polynomial function, that we denote by b (Φ). a r X i v : . [ m a t h . C O ] M a r MICH`ELE VERGNE
The Fourier transform ˆ B (Φ)( x ) is the analytic function of x ∈ U :ˆ B (Φ)( x ) = (cid:89) α ∈ Φ e i (cid:104) α,x (cid:105) − i (cid:104) α, x (cid:105) . Remark that the inverse of the Box spline is related to the generating func-tion for Todd classes. We thus denote it by
T odd (Φ , x ) = (cid:89) α ∈ Φ i (cid:104) α, x (cid:105) e i (cid:104) α,x (cid:105) − . It is only defined for x small enough.Denote by C (Λ) the space of complex valued functions on Λ. If m ∈ C (Λ),let b ( m ) = (cid:88) λ ∈ Λ m ( λ ) δ λ ∗ b (Φ)the convolution of the discrete measure (cid:80) λ ∈ Λ m ( λ ) δ λ with b (Φ). Thus b ( m )is a locally polynomial measure on V .Recall that the list Φ is called unimodular if any basis of V contained inΦ is a basis of the lattice Λ. For simplicity, we restrict to the unimodularcase in this introduction. In this case, Dahmen-Micchelli [2] proved thatthe convolution m (cid:55)→ b ( m ) is injective. Let us recall Dahmen-Micchelliformula for the inverse map. By Fourier transform, convolution becomesthe multiplication by ˆ B (Φ), and it is thus tempting to use Fourier transformto invert the convolution. Indeed we obtain (in case where m is compactlysupported)(1) (cid:88) λ m ( λ ) e i (cid:104) λ,x (cid:105) = (cid:90) V e i (cid:104) v,x (cid:105) T odd (Φ , x ) b ( m )( v ) dv for x small.Replace T odd (Φ , x ) by its Taylor series and consider the Todd operator T odd (Φ , ∂ ) = (cid:89) α ∈ Φ ∂ α − e − ∂ α = 1 + 12 (cid:88) α ∈ Φ ∂ α + · · · , an infinite series of constant coefficients differential operators on V . Thus,for x small, after integration by parts, we still have(2) (cid:88) λ ∈ Λ m ( λ ) e i (cid:104) λ,x (cid:105) = (cid:90) V ( T odd (Φ , ∂ ) b ( m ))( v ) e i (cid:104) v,x (cid:105) dv. Miraculously, this equation still holds if we replace the integral on V bethe summation in Λ, in an appropriate limit sense. Indeed we have theidentity for all x ∈ U ,(3) (cid:88) λ ∈ Λ m ( λ ) e i (cid:104) λ,x (cid:105) = lim t> ,t → (cid:88) λ ∈ Λ ( T odd (Φ , ∂ ) b ( m ))( λ + t(cid:15) ) e i (cid:104) λ,x (cid:105) . Here (cid:15) is a generic vector in the cone generated by the α k , and λ + t(cid:15) isa regular point in V (the notion of generic and regular vectors is defined OISSON SUMMATION AND BOX SPLINE 3 in the article). As b ( m ) is a piecewise polynomial function, only a finitenumber of terms in the series T odd (Φ , ∂ ) b ( m ) do not vanish at the regularpoints λ + t(cid:15) , so that the formula is well defined. Thus Dahmen-Micchellideconvolution formula is:(4) m ( λ ) = lim t> ,t → ( T odd (Φ , ∂ ) b ( m ))( λ + t(cid:15) ) . In this article, we prove a slightly more general formula.Let y = [ y , y , . . . , y N ] be a list of N complex numbers. The box spline B (Φ , y ), with parameter y , is the measure on V such that, for a continuousfunction F on V , (cid:104) B (Φ , y ) , F (cid:105) = (cid:90) · · · (cid:90) e i (cid:80) Nk =1 t k y k F ( N (cid:88) k =1 t k α k ) dt · · · dt N . The Fourier transform ˆ B (Φ , y )( x ) is the functionˆ B (Φ , y )( x ) = N (cid:89) k =1 e i ( (cid:104) α k ,x (cid:105) + y k ) − i ( (cid:104) α k , x (cid:105) + y k ) . If m ∈ C (Λ), let b ( y , m ) = (cid:88) λ ∈ Λ m ( λ ) δ λ ∗ B (Φ , y ) . We also consider centered Box splines, defined using convolution of inter-vals { tα k , − ≤ t ≤ } , and more generally translation of the Box splineby a parameter r in the zonotope. We prove a deconvolution formula sim-ilar to (4) for the translated box spline with parameters. We show thatthe deconvolution formula allows us to recover m from b ( y , m ) by an uni-form formula on all points of Λ ∩ ∆, where ∆ is a generic translation of thezonotope. In particular, an interesting case is when y = 0. If the function b (0 , m ) is polynomial on a domain Ω, we obtain that m is polynomial on theenlarged domain Ω − Z (Φ). Interesting examples of this phenomenon occurin the case of the Kostant partition function. Indeed in this case the convo-lution of the partition function with the Box spline is simply a convolutionof Heaviside functions, with domains of polynomiality given by the so calledbig chambers. More generally, these examples occur in Hamiltonian geome-try, where b (0 , m ) is the Duistermaat-Heckman measure and is polynomialon each connected component of the set of regular values of the momentmap. For example, we have used the deconvolution formula in [4] to studyqualitative properties of some branching rules, for reductive noncompact Liegroups, even in the absence of explicit character formulae.Let us comment on the technique used in this article.The problem of inverting the convolution with the Box spline is equiva-lent to the problem of describing the function number of integral points inpolytopes in terms of volumes, and we could have applied results of [5], [1]. MICH`ELE VERGNE
We also gave a proof of the deconvolution formula in [3] for y = 0, based ona detailed study of the Dahmen-Micchelli spaces of functions on Λ.A method by Poisson formula was used in an unpublished article withMichel Brion to obtain formulae for partition functions, as an alternatemethod to the cone decomposition method of [1]. Here we use a mixedmethod between [5], [1]. We use a crucial lemma of [5], and we follow severalof the steps of the unpublished article with Brion. So there is no new ideain this article. However our inversion formula is slightly more general, andwe believe we have clarified some of the delicate points. In particular wedescribe more precisely the regions of quasi polynomial behavior of m interms of the regions of polynomiality of b ( m ). Furthermore, it is stated ina quite natural way ( we state the inversion formula in such a way that itis impossible to make signs mistakes, for example), and we believe that ourPoisson method is straightforward. However, as it should be, in all thesemethods, the limiting procedures are delicate. Indeed the deconvolutionformula itself is delicate.Let us sketch the proof of Formula (3) in the case where m is the deltafunction at 0 of the lattice Λ, and Φ is unimodular. We thus need to provethe identity 1 = lim t> ,t → (cid:32)(cid:88) λ ∈ Λ ( T odd [ L ] (Φ , ∂ ) b (Φ))( λ + t(cid:15) ) e i (cid:104) λ,x (cid:105) (cid:33) where T odd [ L ] (Φ , ∂ ) is the series T odd (Φ , ∂ ) truncated at some sufficientlylarge order L , the higher terms giving a zero contribution on the regularelement λ + t(cid:15) of V .We compute the term in parenthesis using Poisson formula.Define F ([ q ] , x ) = (cid:80) ∞ a =0 q a F a ( x ) to be the Taylor series of T odd (Φ , qx ) ˆ B (Φ)( x ) = N (cid:89) k =1 q ( e i (cid:104) α k ,x (cid:105) − e iq (cid:104) α k ,x (cid:105) − q = 0. The Fourier transform of T odd [ L ] (Φ , ∂ ) b (Φ) is the truncated series F ( x ) = (cid:80) La =0 F a ( x ), thus formally (cid:88) λ ∈ Λ T odd [ L ] (Φ , ∂ ) b (Φ)( λ + t(cid:15) ) e i (cid:104) λ,x (cid:105) = e − i (cid:104) t(cid:15),x (cid:105) (cid:88) γ ∈ Γ F ( x − πγ ) e i (cid:104) t(cid:15), πγ (cid:105) . We compute the full formal series (cid:88) γ ∈ Γ F ([ q ] , x − πγ ) e i (cid:104) t(cid:15), πγ (cid:105) , this is well defined when (cid:15) is generic and t small, coefficients of q a obtainedby summations over Γ are 0 on the regular element t(cid:15) when a is sufficientlylarge, and the limit when t → q = 1 is 1 provided (cid:15) is in the conegenerated by the α k . This gives us the wanted result. In fact, the main OISSON SUMMATION AND BOX SPLINE 5 result of this article (Theorem 4.5) is a generalisation of the equality of L -functions of t ∈ R / Z : e i { t } x = (cid:88) n ∈ Z e ix − x − in e iπnt . Here { t } = t − [ t ] ∈ [0 ,
1] is the fractional part of t . As shown on thisexample, limits from right or left of the summation over the lattice are notthe same. It is equal to 1 only if t tends to 0 from the right.We need furthermore to introduce parameters y . In fact, the use of genericparameters y simplify the proofs.Equation (1) is very reminiscent of the “delocalized” equivariant indexformula for an equivariant elliptic operator. If G is a torus, we have employedthe deconvolution formula for box splines in order to obtain multiplicitiesformula for the index of a G -equivariant elliptic (or transversally elliptic)operator in terms of spline functions on the lattice Λ of characters of G ([3]).This slightly more general deconvolution formula proved here is similarlyneeded for the proofs of the results announced in [6].We thank Michel Duflo for several comments on this manuscript. Part The results
In this part, we state precisely the theorems proven in this paper. Thuswe start by definitions and notations.2.
Piecewise analytic functions
Let V be a finite dimensional real vector space equipped with a latticeΛ. Denote by C (Λ) the space of complex valued functions on Λ. For λ ∈ Λ,we denote by δ Λ λ the δ function on Λ such that δ Λ λ ( ν ) = 0, except for ν = λ where δ Λ λ ( λ ) = 1.If h is a distribution on V , we denote by (cid:82) V h ( v ) f ( v ) its value on atest function f . If v ∈ V , we denote by δ Vv the δ distribution on V : (cid:82) V δ Vv ( v ) f ( v ) = f ( v ).Choosing the Lebesgue measure dv determined by Λ, we identify a gen-eralized function f ( v ) on V to the distribution f ( v ) dv .Consider Φ := [ α , α , . . . , α N ] be a list of elements in Λ. The box spline B (Φ) is the measure on V such that, for a continuous function F on V , (cid:104) B (Φ) , F (cid:105) = (cid:90) · · · (cid:90) F ( N (cid:88) k =1 t k α k ) dt · · · dt N . Let Z (Φ) = { z = (cid:80) Nk =1 t k α k ; 0 ≤ t k ≤ } be the Minkowski sum of thesegments [0 , α i . The polytope Z (Φ), called the zonotope, is the supportof B (Φ). MICH`ELE VERGNE
Assume that Φ generates V . An hyperplane spanned by elements of Φwill be called a wall. A translate of a wall by an element of Λ will be calledan affine wall.A point (cid:15) ∈ V is called Φ-generic if (cid:15) does not lie on any wall. A point v ∈ V is called Φ-regular if v does not lie on any affine wall. A connectedcomponent τ of the set of Φ-generic elements is called a tope. Thus topes areopen cones in V . A connected component c of the set of Φ-regular elementsis called an alcove. We denote by V reg the set of Φ-regular elements, that isthe disjoint union of the alcoves. In the rest of this article, we often just saythat v is generic, regular, etc., the system Φ being implicitly understood.A piecewise polynomial function b is a function on V reg such that for eachalcove c , there exists a polynomial function b c on V satisfying b ( v ) = b c ( v )for v ∈ c . A piecewise analytic function is a function on V reg such that foreach alcove c , there exists an analytic function b c on V satisfying b ( v ) = b c ( v )for v ∈ c .We denote by P W the space of piecewise polynomial functions. We denoteby
P W ω the space of piecewise analytic functions. Definition 2.1. If f ∈ P W ω , and c is an alcove, we denote by f c theanalytic function on V coinciding with f on c .The lattice Λ acts on P W, P W ω by translation.If V = R with lattice Λ = Z , such a piecewise analytic function admitsleft and right limits at any point of V . Let us generalize the notion of leftor right limits to our multidimensional context.If v ∈ V , and (cid:15) is a generic vector, then v + t(cid:15) is in V reg if t > Definition 2.2.
Let v ∈ V , and f ∈ P W (or
P W ω ) . Let (cid:15) be a genericvector. We define (lim (cid:15) f )( v ) = lim t> ,t → f ( v + t(cid:15) ).Clearly (lim (cid:15) f )( v ) depends only of the tope τ where (cid:15) belongs and isdenoted by (lim τ f )( v ) in [3].Consider f ∈ P W (defined on V reg ) as a locally L -function on V , thus f ( v ) defines a generalized function on V . An element of P W , considered as ageneralized function on V , will be called a piecewise polynomial generalizedfunction . Multiplying by dv , we obtain the space of piecewise polynomialdistributions on V . We define similarly piecewise analytic generalized func-tions and piecewise analytic distributions on V .The box spline is an important example of piecewise polynomial distribu-tion.Indeed, if Φ spans the vector space V , B (Φ) is in the space P W of piece-wise polynomial distributions. We will write B (Φ) = b (Φ)( v ) dv , where b (Φ)( v ) is a locally polynomial function on V .If Ψ is a sublist of Φ still spanning V , then B (Ψ) and its translates byelements of Λ are again in P W . In fact alcoves for the system Ψ are larger
OISSON SUMMATION AND BOX SPLINE 7 than alcoves for the system Φ, and B (Ψ) is given by a polynomial functionon each alcove for Ψ. If Ψ does not span V , the distribution B (Ψ) vanisheson V reg .Let y = [ y , y , . . . , y N ] be a list of N complex numbers. The box spline B (Φ , y ), with parameter y , is the measure on V such that, for a continuousfunction F on V , (cid:104) B (Φ , y ) , F (cid:105) = (cid:90) · · · (cid:90) e i ( (cid:80) Nk =1 t k y k ) F ( N (cid:88) k =1 t k α k ) dt · · · dt N . Then, if Φ spans V , B (Φ , y ) = b (Φ , y )( v ) dv where b (Φ , y )( v ) is in thespace P W ω of piecewise analytic functions. In fact, it is a piecewise expo-nential polynomial function of v .When y = [0 , , . . . , B (Φ , y ) = B (Φ).We give examples in dimension 1. Example 2.3.
Let V = R ω . We identify V with R : t ∈ R is the element tω of V . • Φ = [ ω ], y = [ y ]. Then: b (Φ )( t ) = t <
01 if 0 < t <
10 if t > b (Φ , y )( t ) = t < e ity if 0 < t <
10 if t > . • Let Φ (cid:48) = [ ω, − ω ].Then b (Φ (cid:48) )( t ) = t < − t + 1 if − < t < t − < t <
10 if t >
MICH`ELE VERGNE b (Φ (cid:48) , y )( t ) = t < − e ity e i ( y y i ( y + y ) − e − ity i ( y + y ) if − < t < e i ( y y e − ity i ( y + y ) − e ity i ( y + y ) if 0 < t <
10 if t > • Φ = [ ω, ω ] , y = [ y , y ] . Then B (Φ)( t ) = t < t if 0 < t < if 1 < t < (3 − t )2 if 2 < t <
30 if t > B (Φ , y )( t ) = t < e ity i (2 y − y ) − e it y i (2 y − y ) if 0 < t < e it y ( e i ( y − y
22 ) − i (2 y − y ) if 1 < t < − e ity e i ( y − y i (2 y − y ) + e it y e i ( y − y
22 ) i (2 y − y ) if 2 < t <
30 if t > B (Φ , y )( t )seems to have poles in y , it is easy to verify that on each alcove B (Φ , y )( t )is an analytic function on t, y . Definition 2.4.
Assume Φ generates V . We denote by S the space ofgeneralized functions on V generated by the action of constant coefficientsdifferential operators and translations by elements of Λ. on the piecewisepolynomial b (Φ). OISSON SUMMATION AND BOX SPLINE 9
We denote by S y the space of generalized functions on V generated bythe action of constant coefficients differential operators and translations byelements of Λ. on the piecewise analytic function b (Φ , y )For example, N (cid:89) k =1 ( − ∂ α k + iy k ) b (Φ , y ) = N (cid:89) k =1 ( e iy k δ Vα k − δ V )is in the space S y . Here the product in the right hand side of this equationis the convolution product, thus the right hand side is a sum of δ -functionson V with coefficients depending on y .Elements of S y can be evaluated at any regular point v ∈ V reg . Thus, if (cid:15) is generic, we denote by Λ lim (cid:15) : S y → C (Λ)the map ( Λ lim (cid:15) f )( λ ) = (lim (cid:15) f )( λ ) . Let q be a formal variable. If E is a vector space and f ([ q ]) = (cid:80) ∞ a =0 q a f a is a formal series of elements of E , we write f ∈ E [[ q ]]. If f ( q, x ) is a smoothfunction of q , defined near q = 0, and depending of some parameters x , wedenote by f ([ q ] , x ) = (cid:80) ∞ a =0 q a f a ( x ) its Taylor series at q = 0, a formal seriesof functions of x . If the series f ([ q ]) is finite (or convergent), we write f ([1])or f ([ q ]) | q =1 for the sum (cid:80) ∞ a =0 f a .Introduce formal series m ([ q ]) = (cid:80) ∞ a =0 q a m a of generalized functions (orof distributions) on V . Then if f is a test function (cid:90) V m ([ q ])( v ) f ( v ) dv = ∞ (cid:88) a =0 q a (cid:90) V m a ( v ) f ( v ) dv is a formal power series in q . It may be evaluated at q = 1 on a testfunction f if the preceding series is finite (or convergent). Formal series ofdistributions occur naturally in the context of Euler-MacLaurin formula.If all the elements m a belong to S y , we write m ([ q ]) ∈ S y [[ q ]]. In thiscase, lim Λ (cid:15) m ([ q ]) = (cid:80) ∞ a =0 q a lim Λ (cid:15) m a is a formal power series of q with valuesin C (Λ). It can be evaluated at q = 1 if the corresponding series of elementsof C (Λ) is convergent at q = 1. This is for example the case if all, but afinite number, the generalized functions m a are supported on affine walls. Definition 2.5.
Let m ([ q ]) = (cid:80) ∞ a =0 q a m a be a series of generalized func-tions, with m a ∈ S y .Assume that the series (cid:80) ∞ a =0 q a lim Λ (cid:15) m a is convergent at q = 1. We denote Λ lim (cid:15) m ([ q ]) | q =10 MICH`ELE VERGNE the corresponding element (cid:80) ∞ a =0 lim Λ (cid:15) m a of C (Λ).We say that m ([ q ]) = (cid:80) ∞ a =0 q a m a is supported on affine walls if all theelements m a are supported on affine walls. In this case, m a = 0 on V reg forall a , and Λ lim (cid:15) m ([ q ]) | q =1 = 0 . Box splines with parameters
Inversion formula for the box spline. The unimodular case.
Let U be the dual vector space of V . If b is a function on V , we denote byˆ b ( x ) = (cid:82) V e i (cid:104) x,v (cid:105) b ( v ) dv its Fourier transform.The Fourier transform(5) ˆ B (Φ)( x ) = (cid:90) V e i (cid:104) x,v (cid:105) B (Φ)( v )of the box spline B (Φ) is the analytic function of U :ˆ B (Φ)( x ) = N (cid:89) k =1 e i (cid:104) α k ,x (cid:105) − i (cid:104) α k , x (cid:105) . Let us first explain the Dahmen-Micchelli inversion formula in the casewhere Φ (spanning V ) is unimodular, that is any basis of V consisting ofelements of Φ is a basis of the lattice Λ.Consider the function T odd (Φ)( x ) = N (cid:89) k =1 i (cid:104) α k , x (cid:105) e i (cid:104) α k ,x (cid:105) − , that is T odd (Φ)( x ) is the inverse of ˆ B (Φ)( x ). The function T odd (Φ)( x ) isdefined near x = 0. Then, when x is small, T odd (Φ)( x ) ˆ B (Φ)( x ) = 1 = T odd (Φ)( x ) (cid:90) V e i (cid:104) x,v (cid:105) b (Φ)( v ) dv. It is tempting to use the fact that Fourier transform exchange the multi-plication by i (cid:104) α, x (cid:105) on functions on U and the derivation − ∂ α on functionson V . We are not allowed to do this as T odd (Φ)( x ) is not a polynomial (andis not defined when x is large). Thus introduce a variable q , and considerthe Taylor series at q = 0 of T odd (Φ)( qx ). T odd (Φ)( qx ) = ∞ (cid:88) a =0 q a T a ( x ) = 1 − q N (cid:88) k =1 iα k ( x ) + · · · . We denote by
T odd (Φ)([ q ] , ∂ ) = N (cid:89) k =1 q∂ α k − e − [ q ] ∂ αk = 1 + q N (cid:88) k =1 ∂ α k + · · · OISSON SUMMATION AND BOX SPLINE 11 the corresponding series of differential operators with constant coefficients.Thus we obtain, for q sufficiently small, T odd (Φ)( qx ) ˆ B (Φ)( x ) = (cid:90) V e i (cid:104) x,v (cid:105) ( T odd (Φ)([ q ] , ∂ ) b (Φ))( v ) dv. In a certain sense, this equation still holds for q = 1, provided we replacethe integral over V by the sum over Λ. Indeed, T odd (Φ)([ q ] , ∂ ) b (Φ) | q =1 restricted to Λ in the sense explained below is equal to the Dirac function δ Λ0 on Λ, and this clearly satisfies1 = (cid:88) λ ∈ Λ ( T odd (Φ)([ q ] , ∂ ) b (Φ))( λ ) e i (cid:104) x,λ (cid:105) | q =1 = T odd (Φ)( qx ) ˆ B (Φ)( x ) | q =1 . Let us explain now precisely the results obtained in [3] (for which we willgive another proof in this article).Consider the series
T odd (Φ)([ q ] , ∂ ) b (Φ) of generalized functions on V .As b (Φ) is piecewise polynomial, this is a series of generalized functions (cid:80) ∞ a =0 q a m a , where all the m a are in S and all, but a finite number, gener-alized functions m a are supported on affine walls.Dahmen-Miccheli theorem is: Theorem 3.2. If (cid:15) is a generic vector belonging to the cone Cone (Φ) generated by the elements of Φ , then Λ lim (cid:15) T odd (Φ)([ q ] , ∂ ) b (Φ) | q =1 = δ Λ0 . Note that we do not assume that Φ generates a salient cone.Let us rephrase this theorem in order that it becomes easy to remember(and to prove and generalize).Consider the analytic function F ( q, x ) = T odd (Φ)( qx ) ˆ B (Φ)( x )that is F ( q, x ) = q N (cid:32) N (cid:89) k =1 e i (cid:104) α k ,x (cid:105) − e iq (cid:104) α k ,x (cid:105) − (cid:33) . For q = 1, this function is identically equal to 1. The power q N is suchthat F ( q, x ) has no pole at q = 0.The Taylor series F ([ q ] , x ) at q = 0 of F ( q, x ) is a series of analyticfunctions of x . We have: F ([ q ] , x ) = ˆ B (Φ)( x ) − q N (cid:88) k =1 iα k ( x )) ˆ B (Φ)( x ) + · · · . Theorem 3.3.
Assume that Φ is unimodular.Consider F ( q, x ) = q N (cid:32) N (cid:89) k =1 e i (cid:104) α k ,x (cid:105) − e iq (cid:104) α k ,x (cid:105) − (cid:33) . Denote by F ([ q ] , x ) the Taylor series at q = 0 of F ( q, x ) and write F ([ q ] , x ) = (cid:90) V e i (cid:104) x,v (cid:105) m ([ q ])( v ) dv where m ([ q ]) = (cid:80) ∞ a =0 q a m a is a series of generalized functions on V . • Then m ([ q ]) ∈ S [[ q ]] . • All, but a finite number, the generalized functions m a are supported onwalls. Thus, for any generic vector (cid:15) , the series lim Λ (cid:15) m ([ q ]) is a polynomialin q . • If (cid:15) is a generic vector belonging to the cone Cone (Φ) generated bythe elements of Φ , Λ lim (cid:15) m ([ q ]) | q =1 = δ Λ0 . Inversion of the Box spline with parameters.
To explain whathappens when Φ is not unimodular, we need more notations. We also intro-duce parameters. We assume that Φ spans V .Let Γ ⊂ U be the dual lattice of the lattice Λ. Thus (cid:104) γ, λ (cid:105) ∈ Z if γ ∈ Γ and λ ∈ Λ. We consider Λ as the group of characters of the torus T = U/ π Γ,and use the notation s λ for the value of λ ∈ Λ at s ∈ T . If S ∈ U is arepresentative of s ∈ U/ π Γ, then, by definition, s λ = e i (cid:104) S,λ (cid:105) .We denote by ˆ s the character λ (cid:55)→ s λ of Λ. If m is a function on Λ, thenˆ sm is a function on Λ: ˆ sm ( λ ) = s λ m ( λ ).For s ∈ T , let Φ( s ) = { α ∈ Φ; s α = 1 } . Definition 3.5.
Let V be the subset of T consisting of the elements s suchthat Φ( s ) still spans V .Thus V is a finite subset of T called the vertex set.Let y = [ y , y , . . . , y N ] be a list of N complex numbers. The box spline B (Φ , y ), with parameter y , is the measure on V such that, for a continuousfunction F on V , (cid:104) B (Φ , y ) , F (cid:105) = (cid:90) · · · (cid:90) e i ( (cid:80) Nk =1 t k y k ) F ( N (cid:88) k =1 t k α k ) dt · · · dt N . The Fourier transform ˆ B (Φ , y )( x ) is the functionˆ B (Φ , y )( x ) = N (cid:89) k =1 e i ( (cid:104) α k ,x (cid:105) + y k ) − i ( (cid:104) α k , x (cid:105) + y k ) . If s ∈ T , we define F s ( q, x, y ) = q | Φ( s ) | (cid:32) N (cid:89) k =1 e i ( (cid:104) α k ,x (cid:105) + iy k ) s α k − e iq ( (cid:104) α k ,x (cid:105) + y k ) s α k − (cid:33) . We denote by F s ([ q ] , x, y ) the Taylor series of F s ( q, x, y ) at q = 0, a seriesof analytic functions of x , depending of the parameter y . OISSON SUMMATION AND BOX SPLINE 13
Write F s ([ q ] , x, y ) = (cid:90) V e i (cid:104) x,v (cid:105) m s ([ q ] , y )( v ) dv where m s ([ q ] , y ) is a series of generalized functions on V . Theorem 3.6. • The series m s ([ q ] , y ) of generalized functions on V is in S y [[ q ]] . • If s is not in V , the generalized function m s ([ q ] , y ) is supported on walls.If y is sufficiently small, and (cid:15) is a generic vector, the series lim Λ (cid:15) m s ([ q ] , y ) is convergent at q = 1 . • Assume y is sufficiently small. If (cid:15) is a generic vector belonging tothe cone Cone (Φ) generated by the elements of Φ , then (6) (cid:88) s ∈V ˆ s − lim (cid:15) m s ([ q ] , y ) | q =1 = δ Λ0 . It is easy to see (see Formula (17)) that m s ([ q ] , y ) is obtained by anexplicit expression in terms of derivatives and translates of the Box spline B (Φ( s ) , y ). Thus, in particular, it is supported on affine walls if s is not in V . Equation (6) is clearly equivalent to the identity:(7) (cid:88) λ ∈ Λ ( (cid:88) s ∈V ˆ s − lim (cid:15) m s ([ q ] , y )( λ )) e i (cid:104) λ,x (cid:105) | q =1 = 1 . Example 3.7.
We verify Theorem 3.6 in the cases of Examples 2.3.We use the following Taylor series expansions at z = 0.(8) ze z − ∞ (cid:88) a =0 b ( a ) z a /a !where b ( a ) are the Bernoulli numbers. • Φ = [ ω ], y = [ y ]. Then V = { } .Then F ( q, x, y ) = q e i ( x + y ) − e iq ( x + y ) − e i ( x + y ) − i ( x + y ) iq ( x + y ) e iq ( x + y ) − . The Taylor series in q is F ([ q ] , x, y ) = e i ( x + y ) − i ( x + y ) ∞ (cid:88) a =0 q a b ( a ) ( i ( x + y )) a a ! . We write F ([ q ] , x, y ) = e i ( x + y ) − i ( x + y ) + (cid:80) ∞ a =1 f a ( x, y ) q a where f a ( x, y ) = ( e i ( x + y ) − b ( a ) ( i ( x + y )) a − a ! and take the Fourier transform: F ([ q ] , x, y ) = (cid:82) R e itx m ([ q ] , y )( t ) dt with m ([ q ] , y ) = (cid:80) ∞ a =0 q a m a ( y ) . As, for a ≥ ( i ( x + y )) a − a ! is a polynomial function of x , the Fourier trans-form m a ( y ) of f a ( x, y ) is supported on Z (more precisely on { , } ).Then m ([ q ] , y ) = B (Φ , y ) + ∞ (cid:88) a =1 q a m a ( y )and m a ( y ) for a ≥ R \ Z . We obtain that the restrictionof m ([ q ] , y ) to R \ Z is given by m ([ q ] , y )( t ) = t < e ity if 0 < t <
10 if t > . Thus we see that the right limit at all elements of Λ is 0, except at λ = 0,where it is 1. This prove that the right limit lim Λ (cid:15) m ([ q ] , y ) | q =1 is equal to δ Λ0 . Theorem 3.6 would not be true with left limits. • Φ (cid:48) = [ ω, − ω ], y = [ y , y ]. Again, V = { } . We have F ( q, x, y ) = q (cid:32) e i ( x + y ) − e iq ( x + y ) − (cid:33) (cid:32) e i ( − x + y ) − e iq ( − x + y ) − (cid:33) . Thus F ([ q ] , x, y ) is equal to (cid:32) e i ( x + y ) − i ( x + y ) (cid:33) (cid:32) e i ( − x + y ) − i ( − x + y ) (cid:33) (cid:32) ∞ (cid:88) a =0 q a b ( a )( i ( x + y )) a /a ! (cid:33) (cid:32) ∞ (cid:88) (cid:96) =0 q (cid:96) b ( (cid:96) )( i ( − x + y )) (cid:96) /(cid:96) ! (cid:33) . We write F ([ q ] , x, y ) = (cid:82) R e itx m ([ q ] , y )( t ) dt . By Fourier transform, i ( x + y ) acts by iy − ∂ t and this operator annihilates the function e ity while (cid:32) ∞ (cid:88) a =0 q a b ( a )( iy − ∂ t ) a /a ! (cid:33) · e − ity = q ( i ( y + y )) e i [ q ]( y + y ) − e − ity , the series being convergent at q = 1 for y small. Thus using Formulae for B (Φ (cid:48) , y ), we obtain that the restriction to R \ Z of the Fourier transform m ([ q ] , y ) of F ([ q ] , x, y ) is given by m ([ q ] , y )( t ) = t < − qe i ( y + y ) e ity e i [ q ]( y y − − q e − ity e i [ q ]( y y − if − < t < qe ( i ( y + y )) e − ity e i [ q ]( y y − − q e ity e i [ q ]( y y − if 0 < t <
10 if t > OISSON SUMMATION AND BOX SPLINE 15
We verify that m ([1] , y )( t ) is continuous and that its restriction to Λ isequal to δ Λ0 . As asserted by Theorem 3.6, we can take limits from the leftor right, as the cone generated by Φ (cid:48) is equal to R . • Φ = [ ω, ω ], y = [ y , y ].Then V = { s = 1 , s = − } .We have F ( q, x, y ) = q (cid:32) e i ( x + y ) − e iq ( x + y ) − (cid:33) (cid:32) e i (2 x + y ) − e iq (2 x + y ) − (cid:33) . We write F ([ q ] , x, y ) = (cid:82) R e itx m ([ q ] , y )( t ) dt .The restriction to R \ Z of m ([ q ] , y )( t ) is given by m ([ q ] , y )( t ) = t < q ( e ity (1 − e i [ q ]( y − y ) − e it y ( e i [ q ]( y − y
22 ) − ) if 0 < t < q e it y ( e i ( y − y
22 ) − e i [ q ]( y − y
22 ) − if 1 < t < − q e ity e i ( y − y (1 − e i [ q ]( y − y ) + q e it y e i ( y − y
22 ) ( e i [ q ] ( y − y ) − if 2 < t <
30 if t > s = −
1. Then F − ( q, x, y ) = q (cid:32) e i ( x + y ) + 1 e iq ( x + y ) + 1 (cid:33) (cid:32) e i (2 x + y ) − e iq (2 x + y ) − (cid:33) . We write F − ([ q ] , x, y ) = (cid:82) R e itx m − ([ q ] , y )( t ) dt .The restriction to R \ Z of m − ([ q ] , y ) is given by m − ([ q ] , y )( t ) = t < e it y (1+ e i [ q ]( y − y / ) if 0 < t < e it y e i ( y − y / e i [ q ]( y − y / if 1 < t < e ity / e i ( y − y / (1+ e i [ q ]( y − y / ) if 2 < t <
30 if t > We verify that, for (cid:15) > (cid:15) ( m ([1] , y ))( n ) + ( − n lim (cid:15) ( m − ([1] , y ))( n ) = n = 00 if n (cid:54) = 0This formula is not true for (cid:15) < Translated Box spline.
It is quite natural to introduce translatedBox splines.Assume that Φ spans V . Let y = [ y , y , . . . , y N ] be a sequence of complexnumbers, as before, and let r be a point of V . We choose r = [ r , r , . . . , r N ]a sequence of real numbers so that r = (cid:80) Nk =1 r k α k . We say that r is aΦ-representation of r . Let (cid:104) r , y (cid:105) = (cid:80) Nk =1 r k y k . Definition 3.9.
We define B r (Φ , y )( v ) = e − i (cid:104) r , y (cid:105) B (Φ , y )( v + r ) . Although B r (Φ , y ) depends of the representation of r as r = (cid:80) Nk =1 r k α k ,we do not include this in the notation.The support of B r (Φ , y ) is Z (Φ) − r . Thus 0 is in the support of B r (Φ , y )if and only if r belongs to the zonotope Z (Φ). If r ∈ Z (Φ), we denote by Cone ( r, Φ) the tangent cone at r to the zonotope Z (Φ).The Fourier transform ˆ B r (Φ , y )( x ) = e − i ( (cid:104) r,x (cid:105) + (cid:104) r , y (cid:105) ) ˆ B (Φ , y )( x ) is thusequal to e − i ( (cid:104) r,x (cid:105) + (cid:104) r , y (cid:105) ) N (cid:89) k =1 e i ( (cid:104) α k ,x (cid:105) + y k ) − i ( (cid:104) α k , x (cid:105) + y k ) = N (cid:89) k =1 e i (1 − r k )( (cid:104) α k ,x (cid:105) + y k ) − e − ir k ( (cid:104) α k ,x (cid:105) + y k ) i ( (cid:104) α k , x (cid:105) + y k ) . A natural point of translation is the center of the Box spline ρ = (cid:80) Nk =1 α k represented by r ρ = [ , . . . , ]. Then B ρ (Φ , y ) has Fourier transform N (cid:89) k =1 e i ( (cid:104) αk,x (cid:105) + yk )2 − e − i ( (cid:104) αk,x (cid:105) + yk )2 i ( (cid:104) α k , x (cid:105) + y k ) . However, we can consider any point r ∈ V .We write B r (Φ , y ) = b r (Φ , y )( v ) dv . We define V reg,r = V reg − r whichis the disjoint union of translated alcoves c − r , with boundaries the trans-lated walls. We construct similarly the space P W r of piecewise polynomialfunctions on V reg,r , the space P W ωr of piecewise analytic functions on V reg,r .Then b r (Φ , y ) ∈ P W ωr . We denote by S y r the space of derivatives by con-stant coefficients differential operators of b r (Φ , y ). Spaces P W r , P W ωr , S y r are isomorphic to the space P W, P W ω , S y by the translation( τ ( r ) f )( v ) = f ( v + r ) . OISSON SUMMATION AND BOX SPLINE 17 If (cid:15) is a generic vector, we can still take the limit on Λ (Note: we do nottranslate our lattice) of a function in S y r : Λ lim (cid:15) : S y r → C (Λ) . If s ∈ T , we define F s ( q, r , x, y ) = q | Φ( s ) | e i ( q − (cid:104) r,x (cid:105) + (cid:104) r , y (cid:105) ) (cid:32) N (cid:89) k =1 e i ( (cid:104) α k ,x (cid:105) + iy k ) s α k − e iq ( (cid:104) α k ,x (cid:105) + y k ) s α k − (cid:33) = q | Φ( s ) | (cid:32) N (cid:89) k =1 e − ir k ( (cid:104) α k ,x (cid:105) + y k ) ( e i ( (cid:104) α k ,x (cid:105) + iy k ) s α k − e − iqr k ( (cid:104) α k ,x (cid:105) + y k ) ( e iq ( (cid:104) α k ,x (cid:105) + y k ) s α k − (cid:33) . When q = 1, F s ( q, r, x, y ) = 1.We denote by F s ([ q ] , r, x, y ) the Taylor series of F s ( q, r, x, y ) at q = 0.Write F s ([ q ] , r, x, y ) = (cid:90) V e i (cid:104) x,v (cid:105) m s ([ q ] , r, y )( v ) dv where m s ([ q ] , r, y ) is a series of generalized functions on V . The followingtheorem is the main theorem of this article. Theorem 3.10. • The series m s ([ q ] , r , y ) of generalized functions on V isin S y r [[ q ]] . • If s is not in V , the generalized function m s ([ q ] , r , y ) is supported ontranslated walls.If y is sufficiently small, and (cid:15) is a generic vector, the series lim Λ (cid:15) m s ([ q ] , r , y ) is convergent at q = 1 . • Assume y is sufficiently small. If r belongs to the zonotope , andif (cid:15) is a generic vector belonging to the cone Cone ( r, Φ) tangent at r tothe zonotope Z (Φ) , (9) (cid:88) s ∈V ˆ s − lim (cid:15) m s ([ q ] , r , y ) | q =1 = δ Λ0 . Remark . • The function F s ([ q ] , r , x, y ) is equal to e i ([ q ] − (cid:104) r,x (cid:105) e i ([ q ] − (cid:104) r , y (cid:105) ) F s ([ q ] , x, y ) . When we evaluate at q = 1, we see that (cid:104) r , y (cid:105) plays no role. In particular,Theorem 3.10 does not depend of the way r is represented as (cid:80) Nk =1 r k α k . • The series of functions m s ([ q ] , r , y ) is obtained from the series of func-tions m s ([ q ] , y ) (defined in the preceding subsection, Subsection 3.4) by arather amusing operation. On each alcove c , the series of functions m s ([ q ] , y )is given by the restriction to c of a series of analytic functions m c s ([ q ] , y ) de-fined on all V . Then we see that m s ([ q ] , r , y ) on c − r is just equal to theseries e i ([ q ] − (cid:104) r , y (cid:105) m c s ([ q ] , y ), restricted to c − r . Indeed the Fourier transformof the operator e i ([ q ] − (cid:104) r,x (cid:105) acts by on f ∈ P W ωr by taking the Taylor series Figure 1. r = 0, r = , r = 1, r = 2of f at v + r , evaluated at v + r − qr . It is NOT the identity at q = 1, as f is not analytic on V !!.Let us consider the corresponding functions for Φ (cid:48) and y = 0 , q = 1(Figure 3.8). We see that these functions restricted to the lattice are stillequal to δ Λ0 , under the condition that − < r <
1. But when | r | >
1, then0 is not any more on the support, so the restriction cannot be δ Λ0 .For y = 0, Theorem 3.10 is equivalent to our description of the localpieces m c s of the functions m s , in [3]. We will show this point in Theorem7.4. 4. Rational functions on U and functions on T We consider a system Φ spanning V , and contained in a lattice Λ withdual lattice Γ. We follow notations of Subsection 3.8. We denote by H thespace of holomorphic functions on U × C N .Recall the definition of V (Definition 3.5). Definition 4.1.
We denote by ˆ V the reciproc image of V in U .The subset ˆ V of U contains 2 π Γ and is a finite union of cosets of 2 π Γ.Remark also that ˆ V depends only of the list Φ and not of the lattice Λcontaining the list Φ. If Φ is unimodular, then ˆ V = 2 π Γ.Denote by u the image of u ∈ U in U/ π Γ. Thus Φ( u ) = { α k , e i (cid:104) α k ,u (cid:105) = 1 } . Definition 4.2.
DefineΘ( r , x, y ) = e i ( (cid:104) r,x (cid:105) + (cid:104) r , y (cid:105) ) (cid:81) Nk =1 ( e i ( (cid:104) α k ,x (cid:105) + y k ) −
1) = N (cid:89) k =1 e ir k ( (cid:104) α k ,x (cid:105) + y k ) e i ( (cid:104) α k ,x (cid:105) + y k ) − . The function Θ( r , x, y ) satisfies the following covariance properties. • If γ ∈ Γ, then Θ( r , x + 2 πγ, y ) = e i (cid:104) r, πγ (cid:105) Θ( r , x, y ) . OISSON SUMMATION AND BOX SPLINE 19 • If r = r (cid:48) + [ a , a , . . . , a N ], thenΘ( r , x, y ) = e i ( (cid:104) a,x (cid:105) + (cid:104) a , y (cid:105) ) Θ( r (cid:48) , x, y )with a = (cid:80) Nk =1 a k α k .For w ∈ V , define Z ( q, r , w )( x, y ) = q | Φ( w ) | e i ( q − (cid:104) r,x − w (cid:105) + (cid:104) r , y (cid:105) ) N (cid:89) k =1 e i ( (cid:104) α k ,x (cid:105) + y k ) − e iq ( (cid:104) α k ,x − w (cid:105) + y k ) e i (cid:104) α k ,w (cid:105) − . If q = 1, Z ( q, r , w ) = 1.Let Z ([ q ] , r , w ) be the Taylor series of Z ( q, r , w ). This is a series of holo-morphic functions of w, x, y , and we write Z ([ q ] , r , w )( x, y ) = (cid:80) ∞ a =0 q a z a ( w, x, y ).When ( x, y ) varies in a compact subset of U C × C N , the functions z a ( w, x, y )are of at most polynomial growth on w , on each coset of 2 π Γ (this will beproven in Lemma 6.1). We can then define the following sum in the senseof generalized functions.
Definition 4.3. B ([ q ] , r )( v )( x, y ) = (cid:88) w ∈ ˆ V Z ([ q ] , r , w )( x, y ) e i (cid:104) v,w (cid:105) . This sum has a meaning as a generalized function of v with coefficientsin H .Then, we have Theorem 4.4. • For v ∈ V reg,r , the function ( v, x, y ) (cid:55)→ B ([ q ] , r )( v )( x, y ) is a formal series of analytic functions of ( v, x, y ) . • Let y be small enough. If (cid:15) is a generic vector, then lim t> ,t → B ([ q ] , r )( t(cid:15) )( x, y ) is a series of analytic functions in ( x, y ) convergent for q = 1 .Furthermore, if r is in the zonotope Z (Φ) and if (cid:15) belongs to thecone Cone ( r, Φ) tangent at r to Z (Φ) , then lim t> ,t → B ([ q ] , r )( t(cid:15) )( x, y ) | q =1 = 1 . As we will see, this theorem is equivalent to Theorem 3.10.We reformulate this theorem by using meromorphic functions. Let Q ( q, r , w, x, y ) = (cid:32) q | Φ( w ) | e iq ( (cid:104) r,x − w (cid:105) + (cid:104) r , y (cid:105) ) e i (cid:104) r,w (cid:105) (cid:81) Nk =1 ( e iq ( (cid:104) α k ,x − w (cid:105) + y k ) e i (cid:104) α k ,w (cid:105) − (cid:33) , that is(10) Q ( q, r , w, x, y ) = Θ( r, qx + (1 − q ) w, q y ) . We have Q ( q, r , w, x, y ) = Z ( q, r , w, x, y )Θ( r , x, y ) . We consider the series T ([ q ] , r )( v )( x, y ) = (cid:88) w ∈ ˆ V Q ([ q ] , r , w, x, y ) e i (cid:104) v,w (cid:105) . Then equivalently,
Theorem 4.5.
Let y be small enough. If (cid:15) is a generic vector, then lim t> ,t → T ([ q ] , r )( t(cid:15) )( x, y ) is a series of meromorphic functions, convergent for q = 1 . Furthermore,if r is in the zonotope Z (Φ) and if (cid:15) belongs to the cone Cone ( r, Φ) tangent at r to Z (Φ) , then lim t> ,t → T ([ q ] , r )( t(cid:15) )( x, y ) | q =1 = Θ( r , x, y ) . Let us explain the philosophy of this theorem and, for this purpose, it issufficient to consider the case y = r = 0, and the unimodular case whereˆ V = 2 π Γ. Consider the functionΘ( x ) = N (cid:89) k =1 e i (cid:104) α k ,x (cid:105) − . It is a function of x invariant by translation by an element of 2 π Γ: Θ( x +2 πγ ) = Θ( x ). Consider the Laurent series T ( x ) of Θ( x ) at x = 0, up to order N −
1. This is a rational function of x decreasing at ∞ . If we sum T ( x ) overthe coset x − π Γ, we reobtain a periodic function of x , with same Taylorseries at x = 0 than Θ( x ) up to order N −
1. However, the summation isnot absolutely convergent. Thus we introduce the oscillatory term e iπ (cid:104) v,γ (cid:105) and consider instead T ([ q ])( v )( x ) = (cid:88) γ ∈ Γ q N N (cid:89) k =1 (cid:18) e i [ q ] (cid:104) α k ,x − πγ (cid:105) − (cid:19) e iπ (cid:104) v,γ (cid:105) a formal sum of rational functions of x , converging in the distribution sense(in v ). Theorem ?? asserts that when q = 1 and v tends to 0 in appropriatedirections, we recover Θ( x ).It is enlightening, and needed for our proof by induction, to give the fullproof of Theorem 3.10 in the simplest case Φ = [ ω ] in V = R ω . The followingwell known lemma is the heart of the proof.Let v ∈ R and let [ v ] denotes the integral part of v . Then the function { v } = v − [ v ] is a periodic function of v , and { v } = v when 0 < v < Lemma 4.6.
We have the equality of L -functions of v ∈ R / Z : (cid:88) n ∈ Z e ix − i ( x − πn ) e iπnv = e i { v } x . OISSON SUMMATION AND BOX SPLINE 21
Proof.
Indeed, let us compute the L -expansion of the periodic function v (cid:55)→ e i { v } x on R / Z . By definition, this is (cid:88) n ∈ Z (cid:18)(cid:90) e i { v } x e − iπnv dv (cid:19) e iπnv = (cid:88) n ∈ Z (cid:18)(cid:90) e iv ( x − πn ) dv (cid:19) e iπnv = (cid:88) n ∈ Z e i ( x − πn ) − i ( x − πn ) e iπnv = (cid:88) n ∈ Z e ix − i ( x − πn ) e iπnv . (cid:3) Thus consider V = R ω , Φ = [ ω ], y = [ y ] and r = [ r ] and letΘ( r , x, y ) = e ir ( x + y ) e i ( x + y ) − ,Q ( q, r, w, x, y ) = e iqr ( x − w + y ) e irw qe iq ( x − w + y ) e iw − . We have ˆ V = 2 π Z . We see that the Taylor series Q ([ q ] , r, πn, x, y ) = (cid:80) ∞ a =0 q a z a ( x, y , n ) of Q ( q, r, πn, x, y ) is of the form e iπnr (cid:32) i ( x − πn + y ) + ∞ (cid:88) a =1 q a p a ( n, x, y ) (cid:33) where p a ( n, x, y ) is polynomial in n, x, y . So the sum (cid:80) n ∈ Z e iπnr p a ( n, x, y ) e iπnv is a distribution of v supported on v + r ∈ Z . Thus, by Lemma 4.6, the series (cid:80) n ∈ Z Q ([ q ] , r, πn, x, y ) e iπnv restricted to v + r / ∈ Z is equal to e i { v + r } ( x + y ) e i ( x + y ) − . If r is not an integer, we obtain when v tends to 0 that the limit is e i ( { r } )( x + y ) e i ( x + y ) − . If 0 < r < r , x, y ).If r = 0, the right limit of { v + r } when v tends to 0 is r .If r = 1, the left limit of { v + r } when v tends to 0 is r .Thus Theorem 4.5 holds, if and only if r, (cid:15) verifies the conditions statedin the theorem. Part Proofs
Our strategy to prove Theorem 3.10 is to apply Poisson formula, and seethat Theorem 3.10 is equivalent to Theorem 4.4. Then we prove Theorem4.4 by induction on the number of elements of Φ. Poisson formula for derivatives of splines
Let f be a smooth function on V with compact support. By Poissonformula, we have the equality for ( x, v ) ∈ U × V :(11) (cid:88) λ ∈ Λ f ( λ + v ) e i (cid:104) x,λ (cid:105) = (cid:88) γ ∈ Γ ˆ f ( x − πγ ) e − i (cid:104) v,x − πγ (cid:105) . As ˆ f is rapidly decreasing on U , the series of the second member is absolutelyconvergent and defines a smooth function of v .Let b ∈ S y r , then b is a generalized function on V with compact support(a derivative of the piecewise analytic function b r (Φ , y )). Thus b can beevaluated on V reg,r . Let v ∈ V reg,r . The point λ + v is again regular, and wecan form (cid:88) λ ∈ Λ b ( λ + v ) e i (cid:104) x,λ (cid:105) . Consider ˆ b ( x ) = (cid:82) V e i (cid:104) x,t (cid:105) b ( t ) dt , an analytic function on U . This time ˆ b is not rapidly decreasing, but it is a function of x with at most polynomialgrowth. We may thus consider the series (cid:88) γ ∈ Γ ˆ b ( x − πγ ) e − i (cid:104) v,x − πγ (cid:105) in the sense of generalized function of v .We have the following theorem. Theorem 5.1.
Let b ∈ S y r . On U × V reg,r , we have the following equalityof analytic functions of ( x, v ) : (12) (cid:88) λ ∈ Λ b ( λ + v ) e i (cid:104) x,λ (cid:105) = (cid:88) γ ∈ Γ ˆ b ( x − πγ ) e − i (cid:104) v,x − πγ (cid:105) . (The second series is defined in the sense of generalized functions of v .)Proof. We first prove the wanted formula for the function b ( v ) := b r (Φ , y )( v )itself. Consider F ( x, v ) = (cid:88) λ ∈ Λ b ( v + λ ) e i (cid:104) x,v + λ (cid:105) . The function F ( x, v ) is analytic in x and is a function of v modulo Λ.Furthermore this function of v is piecewise analytic (in P W ωr ), as it is a sumof a finite number of translates of b ( v ) e i (cid:104) x,v (cid:105) . It thus defines an L functionon V /
Λ. We form its Fourier series (in v ) and obtain in L ( V /
Λ) the equality F ( x, v ) = (cid:88) γ ∈ Γ a γ e iπ (cid:104) v,γ (cid:105) . The coefficient a γ is (cid:82) v ∈ V/ Λ F ( x, v ) e − iπ (cid:104) v,γ (cid:105) . Rewriting F as a sum over Λ,we obtain a γ = ˆ b ( x − iπγ ). Thus we obtain the wanted formula as an OISSON SUMMATION AND BOX SPLINE 23 equality of L functions of v . As the first member is analytic on V reg,r , weobtain our equality everywhere on V reg,r .We can now derivate this equality on V reg,r × U with respect to constantcoefficient operators. We obtain Theorem 5.1.Let us apply Theorem 5.1 to obtain an equivalent formulation of Theorem3.10. We follow the notations of Theorem 3.10.Let v ∈ V reg,r . We compute W ([ q ] , v, x, y ) = (cid:88) λ ∈ Λ ( (cid:88) s ∈V s − λ m s ([ q ] , r , y )( λ + v )) e i (cid:104) λ,x (cid:105) . Theorem 3.10 is equivalent to the fact that for r in the zonotope, any (cid:15) generic in the cone C (Φ , r ), lim t> ,t → W ([1] , t(cid:15), x, y ) is identically equal to1. For each s ∈ V , we choose a representative S ∈ U . We denote this set ofrepresentatives still by V . Then W ([ q ] , v, x, y ) = (cid:88) S ∈V (cid:88) λ ∈ Λ m s ([ q ] , r , y )( λ + v ) e i (cid:104) λ,x − S (cid:105) . The series m s ([ q ] , r , y ) ∈ S y r [[ q ]] and the Fourier transform of m s ([ q ] , r , y )is F s ([ q ] , r , x, y ) . We can apply Poisson formula (Theorem 5.1) for each co-efficient of q a in the series m s ([ q ] , r , y ). We obtain W ([ q ] , v, x, y ) = (cid:88) S ∈V (cid:88) γ ∈ Γ F s ([ q ] , r , x − S − πγ, y ) e − i (cid:104) v,x − S − πγ (cid:105) . Now F s ([ q ] , r , x − S − πγ, y ) is equal to q | Φ( s ) | e i ([ q ] − (cid:104) r,x − S − πγ (cid:105) + (cid:104) r , y (cid:105) ) (cid:32) N (cid:89) k =1 e i ( (cid:104) α k ,x − S − πγ (cid:105) + y k ) s α k − e i [ q ]( (cid:104) α k ,x − S − πγ (cid:105) + y k ) s α k − (cid:33) = q | Φ( s ) | e i ([ q ] − (cid:104) r,x − S − πγ (cid:105) + (cid:104) r , y (cid:105) ) (cid:32) N (cid:89) k =1 e i ( (cid:104) α k ,x (cid:105) + y k ) − e i [ q ]( (cid:104) α k ,x − S − πγ (cid:105) + y k ) s α k − (cid:33) = Z s ([ q ] , r , S + 2 πγ )( x, y ) . Here we used the fact that e − i (cid:104) α k ,S +2 πγ (cid:105) = s − α k .Now, the set { S + 2 πγ, γ ∈ Γ , S ∈ V} is exactly the set ˆ V . We obtain for v regular, W ([ q ] , v, x, y ) = (cid:88) w ∈ ˆ V Z ([ q ] , r , w )( y , x ) e i (cid:104) v,w (cid:105) e − i (cid:104) v,x (cid:105) = e − i (cid:104) v,x (cid:105) B ([ q ] , r )( v )( x, y ) . Thus Theorem 4.4 implies Theorem 3.10. (cid:3) A delicate formula
Let β ∈ V . Denote θ ( β )( x, y ) = 1 e i ( (cid:104) β,x (cid:105) + y ) − , a meromorphic funtion of ( x, y ) ∈ U C × C . For our proof by induction, wewill use the following formula which can be immediately verified.Let β , . . . , β p be elements of V such that (cid:80) pi =1 β i = 0. Then, we havethe identity of meromorphic functions of ( x, y , . . . , y p ) ∈ U C × C p ,(13)( e (cid:80) pj =1 y j − p (cid:89) k =1 θ ( β k )( x, y k ) = p (cid:88) h =1 ( − h − (cid:89) ≤ j Let c ∈ U and W c = c + 2 π Γ be a coset of π Γ . Then for ( x, y ) varying in a compact subset of U C × C N , the function z a ( w, x, y ) is ofat most polynomial growth in w ∈ W c .Proof. Write w = c + 2 πn which n ∈ Γ. Then u k = e i (cid:104) α k ,w (cid:105) = e i (cid:104) α k ,c (cid:105) doesnot depend on w when w varies in W c .We write Z (Φ , q, r , w )( x, y ) = e i ( q − (cid:104) r,x − w (cid:105) + (cid:104) r , y (cid:105) ) × (cid:89) k,e i (cid:104) αk,w (cid:105) (cid:54) =1 e i ( (cid:104) α k ,x (cid:105) + y k ) − e iq ( (cid:104) α k ,x − w (cid:105) + y k ) e i (cid:104) α k ,w (cid:105) − (cid:89) k,e i (cid:104) αk,w (cid:105) =1 qe i ( (cid:104) α k ,x (cid:105) + y k ) − e iq ( (cid:104) α k ,x − w (cid:105) + y k ) − . We analyze the dependance in w of each factor. The expansion e i ([ q ] − (cid:104) r,x − w (cid:105) + (cid:104) r , y (cid:105) ) of the first factor is of the form e i (cid:104) r,w (cid:105) (cid:80) a q a p a ( w, x, y ) where p a ( w, x, y ) ispolynomial in w and analytic in x, y . Thus its growth is polynomial in w ,for ( x, y ) varying in a compact subset of U C × C N .Consider the factor f k ( q, w, x, y ) = e i ( (cid:104) α k ,x (cid:105) + y k ) − e iq ( (cid:104) α k ,x − w (cid:105) + y k ) e i (cid:104) α k ,w (cid:105) − OISSON SUMMATION AND BOX SPLINE 25 associated to α k with u k = e i (cid:104) α k ,w (cid:105) (cid:54) = 1.We define, if u (cid:54) = 1, constants β ( a, u ) so that we have Taylor expansion1 e z u − ∞ (cid:88) (cid:96) =0 β ( (cid:96), u ) z (cid:96) /(cid:96) ! . Then the Taylor series of f k ( q, w, x, y ) is( e i ( (cid:104) α k ,x (cid:105) + y k ) − ∞ (cid:88) (cid:96) =0 β ( (cid:96), u k )( i ( (cid:104) α k , x − w (cid:105) + y k )) (cid:96) q (cid:96) /(cid:96) !and the coefficient of q (cid:96) gives rise to a function of ( x, y, w ), polynomial in w and analytic in ( x, y ).If e i (cid:104) α k ,w (cid:105) = 1, then we write the factor as (cid:32) e i ( (cid:104) α k ,x − w (cid:105) + y k ) − i ( (cid:104) α k , x − w (cid:105) + y k ) (cid:33) (cid:18) iq ( (cid:104) α k , x − w (cid:105) + y k ) e iq ( (cid:104) α k ,x − w (cid:105) + y k ) − (cid:19) . If E ( z ) = e iz − iz , the coefficient of q (cid:96) is E ( (cid:104) α k , x − w (cid:105) + y k ) b ( (cid:96) )( i ( (cid:104) α k , x − w (cid:105) + y k )) (cid:96) /(cid:96) ! . As E ( z ) = (cid:82) e itz dt , we can bound uniformly E ( (cid:104) α k , x − w (cid:105) + y k ) when ( x, y )varies in a compact subset of U C × C N and w varies in U . The other termsare polynomials in ( x, y , w ). (cid:3) Thus if W is a finite union of cosets of 2 π Γ, we can consider the series (cid:88) w ∈ W Z (Φ , [ q ] , r , w )( x, y ) e i (cid:104) v,w (cid:105) . This is a series of generalized function of v , depending holomorphicallyon ( x, y ). Lemma 6.2. Let c ∈ U and W c = c + 2 π Γ be a coset of π Γ . If c / ∈ ˆ V (Φ) ,the generalized function v (cid:55)→ (cid:88) w ∈ W c Z (Φ , [ q ] , r , w ) e i (cid:104) v,w (cid:105) vanishes on V reg,r (Φ) .Proof. Write w = c + 2 πn , with n ∈ Γ. If we look back to the proof ofLemma 6.1, we obtain that z a ( w, x, y ) is equal to a function of f ( n, x, y )polynomial in n and holomorphic in ( x, y ), multiplied by e i (cid:104) r,w (cid:105) (cid:89) k,e i (cid:104) αk,c (cid:105) =1 E ( (cid:104) α k , x − w (cid:105) + y k ) . Our assumption is that the corresponding α k span a proper subspace V of V . This subspace is contained in a hyperplane generated by elements of Φ.Write Γ = Γ ∩ V ⊥ . We perform the summation over W of z a ( w, x, y ) e i (cid:104) v,w (cid:105) by summing on cosets of Γ , then on Γ / Γ . When w = c + 2 πn + 2 πn , the functions E ( (cid:104) α k , x − c − πn (cid:105) + y k ) do not depend on n , thus our sumis (cid:80) n ∈ Γ / Γ (cid:80) n ∈ Γ p ( n , n , x, y ) e i (cid:104) v + r,c +2 iπn +2 iπn (cid:105) , where p ( n , n , x, y )is polynomial in n . Thus the corresponding generalized function of v issupported on V + Λ − r , which is contained on translated affine walls. (cid:3) Corollary 6.3. Let W be a finite union of cosets of π Γ containing ˆ V (Φ) .Then on V reg,r (Φ) , we have B (Φ , [ q ] , r )( v ) = (cid:88) w ∈ W Z (Φ , [ q ] , r , w ) e i (cid:104) v,w (cid:105) . Proof. By definition (Definition 4.3), B (Φ , [ q ] , r )( v ) is the sum over ˆ V (Φ).By Lemma 6.2, the restriction to V reg,r (Φ) of the sum over the other cosetsin W \ ˆ V (Φ) vanishes on V reg,r (Φ). (cid:3) We now prove Theorem 4.5.Recall that Q (Φ , q, r , w, x, y ) = q | Φ( w ) | e iq ( (cid:104) r,x − w (cid:105) + (cid:104) r , y (cid:105) ) e i (cid:104) r,w (cid:105) (cid:81) Nk =1 ( e iq ( (cid:104) α k ,x − w (cid:105) + y k ) e i (cid:104) α k ,w (cid:105) − q | Φ( w ) | Θ(Φ , r , qx + (1 − q ) w, q y ) . We compute T (Φ , [ q ])( v )( x, y ) = (cid:80) w ∈ ˆ V (Φ) Q (Φ , [ q ] , w, x, y ) e i (cid:104) v,w (cid:105) .First, it is equivalent to prove Theorem 4.5 for Φ = [ α , α , . . . , α N ] orfor Φ u = [ u α , u α , . . . , u N α N ] with u i = ± (cid:48) = [ − α , α , . . . , α N ]. If r ∈ Z (Φ), then r (cid:48) = r − α ∈ Z (Φ (cid:48) ), and Cone ( r, Φ) = Cone ( r (cid:48) , Φ (cid:48) ). Indeed if r + t(cid:15) ∈ Z (Φ) for t small,then r − α + t(cid:15) ∈ Z (Φ) − α = Z (Φ (cid:48) ). The sets ˆ V (Φ) and ˆ V (Φ (cid:48) ) are equal.The sets V reg,r (Φ) and V reg,r (cid:48) (Φ (cid:48) ) are equal.Consider a sequence r of N real numbers, which is a Φ-representationof r , that is r = r α + · · · + r N α N . Then r (cid:48) = [1 − r , r , . . . , r N ] isa Φ (cid:48) -representation of r (cid:48) = r − α . Let y = [ y , y , . . . , y N ] and y (cid:48) =[ − y , y , . . . , y N ]. Then (cid:104) r , y (cid:105) = (cid:104) r (cid:48) , y (cid:48) (cid:105) + y .Using the relation e z e z − = − e − z − , we see thatΘ(Φ , r , x, y ) = − Θ(Φ (cid:48) , r (cid:48) , x, y (cid:48) )and consequently Q (Φ , q, r , w, x, y ) = − Q (Φ (cid:48) , q, r (cid:48) , w, x, y (cid:48) ) , T (Φ , [ q ] , r )( v )( x, y ) = −T (Φ (cid:48) , [ q ] , r (cid:48) )( v )( x, y (cid:48) ) . Thus, if Theorem 4.5 is true for Φ (cid:48) , we obtain if r ∈ Z (Φ) and (cid:15) ∈ Cone ( r, Φ),lim t> ,t → T (Φ , q, r )( t(cid:15) )( x, y ) | q =1 = − Θ(Φ (cid:48) , r (cid:48) , x, y (cid:48) ) = Θ(Φ , r , x, y ) . Let M = [ M , M , . . . , M N ] be a sequence of positive integers, and letΦ M = [ M α , M α , . . . , M N α N ] . OISSON SUMMATION AND BOX SPLINE 27 Similarly, let us see that if Theorem 4.5 is true for Φ M , then Theorem 4.5is true for Φ. For example, letΦ M = [ M α , α , . . . , α N ] . Let y M = [ M y , y , . . . , y N ]. If r = [ r , r , . . . , r N ] is a Φ-representation of r for Φ, then r d = [( r + d ) /M, r , . . . , r N ] is a Φ M -representation of r + dα .We use 1 e z − e z + · · · + e ( M − z e Mz − . Then(14) Θ(Φ , r , x, y ) = M − (cid:88) d =0 Θ(Φ M , r d , x, y M ) . From Equation (14), Q (Φ , q, r , w, x, y ) = M − (cid:88) d =0 Q (Φ M , q, r d , w, x, y M ) . Assume that r ∈ Z (Φ), (cid:15) ∈ Cone ( r, Φ). Then for d = 0 , . . . , M − r + dα ∈ Z (Φ M ) and (cid:15) ∈ Cone ( r + dα , Φ M ). By Corollary 6.3, we can computethe series T (Φ , [ q ] , r ) by summing over the set ˆ V (Φ M ) which contains ˆ V (Φ).Then we obtain on V reg,r (Φ) = V reg,r (Φ M ) T (Φ , [ q ] , r )( v )( x, y ) = M − (cid:88) d =0 T (Φ M , [ q ] , r d )( v )( x, y M ) . Taking limits and using Equation (14) , we obtain Theorem 4.5 for Φ, ifwe have proven Theorem 4.5 for Φ M .We are now ready to proceed on our induction. If the number N ofelements of Φ is equal to the dimension of V , then α , α , . . . , α N form abasis of V . We may take as lattice containing the elements α k , the latticewith basis α k , and we are reduced to the calculation in dimension 1 that wehave already done at the end of Section 4.If N > dim( V ), let us consider a relation between elements of Φ. We mayassume after eventual relabeling and changing signs that the relation is ofthe form (cid:80) pk =1 M k α k = 0, with M k positive number.It is sufficient to prove Theorem 4.5 for Φ M = [ M α , . . . , M p α p , α p +1 , . . . , α N ].Renaming the list, we are reduced to prove Theorem 4.5 for a systemΨ = [ β , β , . . . , β N ] with a relation (cid:80) pk =1 β k = 0.We will prove the identity of Theorem 4.4 for Ψ, when y is small andoutside the hyperplane (cid:80) pk =1 y k = 0 in C N . As, after multiplying by (cid:81) Nk =1 ( e i ( (cid:104) β k ,x (cid:105) + y k ) − x, y ), thiswill be sufficient.We assume r in the zonotope Z (Ψ) and we choose a representation r = (cid:80) Nk =1 r k β k with 0 ≤ r k ≤ 1. Similarly we can represent (cid:15) ∈ Cone ( r, Φ) as (cid:15) = (cid:80) Nk =1 s k β k with s k ≥ 0, if r k = 0, and s k ≤ r k = 1. In this case thecurve r + t(cid:15) stays in Z (Φ) when t > p elements of Ψ so that the sequence [ r + ts , . . . , r p + ts p ] is weakly increasing when t is small and positive. That is r ≤ r ≤ · · · ≤ r p and if r u = r u +1 , we take an order so that s u ≤ s u +1 .Define Ψ = [ β , . . . , β p , β p +1 , . . . , β N ]Ψ = [ − β , β , . . . , β p , β p +1 , . . . , β N ] · · · Ψ p = [ − β , − β , . . . , − β p − , β p +1 , . . . , β N ] . Systems Ψ i have N − ≤ h ≤ p , r h = [( r h − r ) , . . . , ( r h − r h − ) , ( r h +1 − r h ) , . . . , ( r p − r h ) , r p +1 , . . . , r N ] , y h = [ − y , . . . , − y h − , y h +1 , . . . , y p , y p +1 , . . . , y N ]Then r h is a Ψ h -representation of r .The following is the crucial proposition. It is taken from ([5], Lemma1.8). Proposition 6.4. Let r ∈ Z (Ψ) and (cid:15) ∈ Cone ( r, Ψ) . Then • The vector (cid:15) is generic for Ψ h , r ∈ Z (Ψ h ) and (cid:15) ∈ Cone ( r, Ψ h ) for ≤ h ≤ p . • (15) Θ(Ψ , r , x, y ) = p (cid:88) h =1 c h ( r , y )Θ(Ψ h , r h )( x, y h ) with c h ( y , r ) = ( − h +1 e irh ( (cid:80) pj =1 ,j (cid:54) = h yj ) e i ( (cid:80) pj =1 yj ) − . Proof. It is clear that if (cid:15) is generic for Ψ, it is generic for the smaller systemΨ h .We have r = (cid:80) pk =1 ,k (cid:54) = h ( r h − r k )( − β k ) + (cid:80) pk = h +1 ( r k − r h )( β k ). We have0 ≤ ( r h − r k ) ≤ k < h , and similarly 0 ≤ ( r k − r h ) ≤ h < k ≤ p .Thus r belongs to the zonotope Z (Ψ h ). Similarly, our choice of order impliesthat the curve r ( t ) = r + t(cid:15) stays in Z (Ψ h ).Equality (15) follows from Equality(13) which is Equality (15) for r = 0.We just multiply by e i ( (cid:104) r,x (cid:105) + (cid:104) r , y (cid:105) ) and remark that (cid:88) j (cid:54) = h y j ( r j − r h ) = (cid:104) y , r (cid:105) − r h (cid:88) j (cid:54) = h y j . (cid:3) OISSON SUMMATION AND BOX SPLINE 29 Equation (15) implies that Q (Ψ , q, r , w, x, y ) = p (cid:88) h =1 c h ( r , q y ) Q (Ψ h , q, r h , w, x, y h ) . Remark that the functions y → c h ( r , q y ) are defined if | q | < y sufficiently small and outside the hyperplane (cid:80) pk =1 y k = 0 . Taking Taylor series, we obtain Q (Ψ , [ q ] , r , w, x, y ) = p + q (cid:88) h =1 c h ( r , [ q ] y ) Q (Φ h , [ q ] , r h , w, x, y h ) . We sum over the set ˆ V (Ψ) which contains the sets ˆ V (Ψ h ). So we obtainover V reg,r (Ψ) (contained in V reg,r (Ψ h )): T (Ψ , [ q ] , r )( v )( x, y ) = p (cid:88) h =1 c h ( r , [ q ] y ) T (Ψ h , [ q ] , r h )( v )( x, y h ) . So for (cid:15) generic, we obtainlim t> ,t → T (Ψ , [ q ] , r )( t(cid:15) )( x, y ) = p (cid:88) h =1 c h ( r , [ q ] y ) lim t> ,t → T (Ψ h , [ q ] , r h )( t(cid:15) )( x, y h ) . When y is sufficiently small, the Taylor series of c h ( r , [ q ] y ) converges for q = 1 to c h ( r , y ). By induction hypothesis, if r ∈ Z (Ψ) and (cid:15) ∈ Cone ( r, Ψ),and our crucial proposition, the limit converges for q = 1 to p (cid:88) h =1 c h ( r , y )Θ(Ψ h , r h , x, y h ) = Θ(Ψ , r , x, y ) . This is the end of the proof of Theorem 4.5. Part Applications Deconvolution formula for the Box spline with parameters As we discuss in the introduction, we can apply Theorem 3.6 to invert thesemi-discrete convolution by the Box spline with parameters. We assumethat Φ = [ α , α , . . . , α N ]spans V and is contained in a lattice Λ.Let f ∈ C (Λ) be a function on Λ. Then P ( f, y )( v ) = (cid:88) λ ∈ Λ f ( λ ) b (Φ , y )( v − λ )is a piecewise analytic function on V . In other words, P ( f, y )( v ) dv is theconvolution of the discrete measure (cid:80) λ ∈ Λ f ( λ ) δ Vλ with the distribution withcompact support B (Φ , y ). When Φ is unimodular, the map f → P ( f, y ) is injective, and Dahmen-Micchelli deconvolution formula computes the inverse map. We now givea general deconvolution formula for the semi-discrete convolution with theBox spline with parameters.Let V be the vertex set for the system Φ. Let ∂ α k be the differentiationin the direction α k . Let s ∈ V . We divide the list y in sublists [ y , y ], y corresponding to the indices k with α k ∈ Φ( s ), and y to the indices not inΦ( s ).Consider the locally analytic function b (Φ , s, y ) = (cid:89) k,s αk (cid:54) =1 ( s α k e iy k δ Vα k − ∗ b (Φ( s ) , y )a sum of translated of the Box spline of the system Φ( s ) . Then define the locally analytic function P ( s, y , f ) by P ( s, y , f )( v ) = (cid:88) ξ ∈ Λ s ξ f ( ξ ) b (Φ , s, y )( v − ξ ) . We can recover the value of f at the point λ ∈ Λ from the knowledge, inthe neighborhood of λ , of the functions P ( s, y , f )( v ), for all s ∈ V .Define the series of differential operators T odd ([ q ] , s, y )( ∂ ) = (cid:89) k,s αk =1 q ( − ∂ α k + iy k ) e [ q ]( − ∂ αk + iy k ) − (cid:89) k,s αk (cid:54) =1 e [ q ]( − ∂ αk + iy k ) s α k − . Theorem 7.1. Let f ∈ C (Λ) . Let y small. For (cid:15) generic, and λ ∈ Λ , theseries lim t> ,t → ( T odd ([ q ] , s, y )( ∂ ) P ( s, y , f ))( λ + t(cid:15) ) is convergent at q = 1 .Furthermore, for (cid:15) generic in the cone generated by Φ , (16) f ( λ ) = (cid:88) s ∈V s − λ lim t> ,t → ( T odd ([ q ] , s, y )( ∂ ) P ( s, y , f ))( λ + t(cid:15) ) | q =1 . Proof. Let us see that this is just a reformulation of Theorem 3.6. Bylinearity, we need to prove the formula for δ functions at any point of thelattice Λ.We use the following formula.(17) F s ( q, x, y ) = D s ( q, x, y ) (cid:89) k,s αk (cid:54) =1 ( e i ( (cid:104) α k ,x (cid:105) + y k ) s α k − (cid:89) k,s αk =1 e i ( (cid:104) α k ,x (cid:105) + y k ) − i ( (cid:104) α k , x (cid:105) + y k ) where D s ( q, x, y ) = (cid:89) k,s αk (cid:54) =1 e iq ( (cid:104) α k ,x (cid:105) + y k ) s α k − (cid:89) k,s αk =1 iq ( (cid:104) α k , x (cid:105) + y k ) e iq ( (cid:104) α k ,x (cid:105) + y k ) − . OISSON SUMMATION AND BOX SPLINE 31 If f = δ Λ0 is the delta function at 0 on the lattice Λ, then taking Taylorexpansions and Fourier transforms, we see that T odd ([ q ] , s, y )( ∂ ) P ( s, y , f )is the series m s ([ q ] , y ), and the theorem is equivalent to Theorem 3.6.Otherwise, for f = δ Λ κ , we see that P ( s, y , f )( v ) = s κ P ( s, y , f )( v − κ ).Thus (cid:88) s ∈V s − λ D ([ q ] , s, y ) P ( s, y , f )( λ + t(cid:15) ) = (cid:88) s ∈V s − ( λ − κ ) D ([ q ] , s, y ) P ( s, y , f )( λ − κ + t(cid:15) ) . By the preceding computation, this is f ( λ − κ ) = f ( λ ). (cid:3) We can as well obtain a deconvolution formula for the translated boxspline. Let r ∈ Z (Φ). Let f ∈ C (Λ) be a function on Λ. Then define P ( f, r , y )( v ) = (cid:88) λ ∈ Λ f ( λ ) b r (Φ , y )( v − λ ) . Then P ( f, r , y ) ∈ P W ωr . Let s ∈ V . We divide our list r in sublists [ r , r ]corresponding to the indices in Φ( s ), and not in Φ( s ). Define B (Φ , s, r , y )( v ) = e − i (cid:104) r , y (cid:105) δ − r ∗ (cid:89) k,s αk (cid:54) =1 ( s α k e iy k δ Vα k − ∗ B (Φ( s ) , r , y )a sum of translates of the Box spline of the system Φ( s ). Remark that B (Φ , s, r , y ) is supported on Z (Φ) − r .Then define the locally analytic function P ( s, r , y , f ) by P ( s, r , y , f )( v ) = (cid:88) ξ ∈ Λ s ξ f ( ξ ) B (Φ , s, r , y )( v − ξ ) . Define the series of differential operators T odd (Φ , [ q ] , s, r , y )( ∂ ) = e [ q ]( − ∂ r + i (cid:104) y , r (cid:105) ) (cid:89) k,s αk =1 q ( − ∂ α k + iy k ) e [ q ]( − ∂ αk + iy k ) − (cid:89) k,s αk (cid:54) =1 e [ q ]( − ∂ αk + iy k ) s α k − . Then, the following theorem is just the reformulation of Theorem 3.10. Theorem 7.2. Let r be in the zonotope. Let f ∈ C (Λ) . Let y small. For (cid:15) generic, and λ ∈ Λ , the series lim t> ,t → ( T odd (Φ , [ q ] , s, r , y )( ∂ ) P ( s, r , y , f ))( λ + t(cid:15) ) is convergent at q = 1 .Furthermore, for (cid:15) generic in the cone Cone ( r, Φ) , (18) f ( λ ) = (cid:88) s ∈V s − λ lim t> ,t → ( T odd ([ q ] , s, r , y )( ∂ ) P ( s, r , y , f ))( λ + t(cid:15) ) | q =1 . In particular, if r = ρ is the center of the zontope, we can take limits in anydirections, as C ( ρ, Φ) = V . This will be important to define multiplicitiesformulae for the Dirac operators.We reformulate the deconvolution formulae using the local pieces of thebox spline. Let c be an alcove. Consider ∆ = ( c − Z (Φ)) ∩ Λ . For any f ∈ C (Λ), we can reconstruct f on ∆, using the functions P ( s, y , f ) . Let P ( s, y , f, c ) be the analytic function on V , coinciding with P ( s, y , f ) on c . Theorem 7.3. Let c be an alcove and let λ ∈ ( c − Z (Φ)) ∩ Λ , then f ( λ ) = (cid:88) s ∈V s − λ ( T odd ([ q ] , s, y )( ∂ ) P ( s, y , f, c ))( λ ) | q =1 . Proof. Let λ ∈ ( c − Z (Φ)) ∩ Λ. We choose r ∈ Z (Φ) such that λ + r ∈ c .Thus λ ∈ V reg,r and belongs to the translated alcove c − r . Thus function P ( s, y , r , f ) near λ is just P ( s, y , f, c )( v + r ). The deconvolution formula forthe translated Box spline asserts that f ( λ ) = (cid:88) s ∈V s − λ ( T odd ([ q ] , s, r , y )( ∂ ) P ( s, y , r , f ))( λ ) | q =1 . As ( T odd ([ q ] , s, r , y )( ∂ ) = e (cid:104) r , y (cid:105) e − [ q ] ∂ r ( T odd ([ q ] , s, y )( ∂ ), we obtain our for-mula. (cid:3) This theorem shows that if τ is a union of alcoves c i , such that for any s ∈ ˆ V the analytic function P ( s, y , f, c i ) coincides, we obtain a reconstructionformula for f on τ − Z (Φ). In the next section, we apply this to Kostantpartition function with parameters.We can use this theorem for y = 0 and f = δ Λ0 . We use notations of [3].The Dahmen-Micchelli space DM (Φ) is a space of Z -valued functions on Λsatisfying some difference equations. Notations and definitions are as in [3].It is possible to define a space DM (Φ , y ) with value in the ring Z [ e iy , e iy , · · · , e iy N ],consisting of the functions f satisfying the equation (cid:81) α k ∈ C ( ∇ k − e iy k ) f = 0for all cocircuits and to compute the structure of DM (Φ , y ). However, wedo not undertake this task for the moment. We just take y = 0, and relateour theorem on translated Box splines to results of [3].We return to the notations of Subsection 3.4. We denote our series m s ([ q ] , 0) simply by m s ([ q ]). Let c an alcove contained in Z (Φ). Considerthe polynomial function m s ( c ) on V such that m s ([1]) coincide with m s ( c )on c . It is a function in the Dahmen-Micchelli space of polynomials D (Φ( s )).The restriction of m s ( c ) to Λ is a polynomial function on Λ. Define Q ( c ) tobe the quasi polynomial on Λ: Q ( c ) = (cid:88) s ∈V ˆ s − m s ( c ) | Λ . Then Q ( c ) belongs to DM (Φ). Theorem 7.3 for f = δ Λ0 and y = 0 gives thefollowing result, proved in [3]. Theorem 7.4. Q ( c ) is the unique Dahmen-Micchelli quasipolynomial suchthat Q ( c )( ν ) = 1 if ν = 0 , and Q ( c )( ν ) = 0 if ν ∈ ( c − Z (Φ)) ∩ Λ . OISSON SUMMATION AND BOX SPLINE 33 Kostant Partition functions with parameters Let Φ be a series of non zero elements of a lattice Λ ⊂ V , and assumethat Φ generates a salient cone.Consider the group T with character group Λ. Let M Φ = ⊕ Nk =1 L α k thelinear representation space of T . Here L α k = C e α k , and te α k = t α k e α k .Consider the action of the diagonal group D N = { e i y = [ e iy , e iy , . . . , e iy N ] } on M Φ acting by e iy k on L α k . Consider the symmetric algebra S ( M Φ ) of M Φ . It decomposes as a T -module as S ( M Φ ) = ⊕ ν ∈ Λ S ν . The space S ν is finite dimensional. The group D N acts on S ν , and f ( y )( ν ) = Trace S ν ( e i y )is a function on Λ.If y = 0, then f (0)( ν ) = dim S ν is the value of the partition function at ν , that is the cardinal of the set P (Φ , ν ) of sequence p = [ p , p , . . . , p N ] ofnon negative integers p k such that (cid:80) Nk =1 p k α k = ν : P (Φ , ν ) = { p ≥ , N (cid:88) k =1 p k α k = ν } . For any y , we have the formulaTrace S ν ( e i y ) (cid:88) p ∈ P (Φ ,ν ) e ip k y k . Now consider the following multispline distributions T (Φ , y ) on V suchthat, for a continuous function F on V , (cid:104) T (Φ , y ) , F (cid:105) = (cid:90) ∞ · · · (cid:90) ∞ e i ( (cid:80) Nk =1 t k y k ) F ( N (cid:88) k =1 t k α k ) dt · · · dt N . Consider the cones generated by subsets of Φ, and consider V reg,plus asthe complement of the union of the boundaries of these cones. A chamber τ is defined as a connected component of V reg,plus . Then it is easy to see thatfor each chamber τ , there exists an analytic function T τ (Φ , y )( v ) of ( v, y )so that T (Φ , y )( v ) = T τ (Φ , y )( v ) when v ∈ τ .Writing the quadrant R N + as the union of the translates of the hypercube { t k , ≤ t k ≤ } , we can write T (Φ , y ) as a sum of translated of B (Φ , y ).(19) T (Φ , y ) = ∞ (cid:88) p =0 · · · ∞ (cid:88) p N =0 e i ( (cid:80) Nk =1 p k y k ) B (Φ , y )( v − N (cid:88) k =1 p k α k ) . Consider the zonotope Z (Φ) = (cid:80) Nk =1 [0 , α k generated by Φ. The following formula was proved in Brion-Vergne, via cone decompo-sitions. We gave also a proof in Szenes-Vergne, via residues. Here is yetanother proof. Theorem 8.1. Let τ be a chamber, and y small. Then for ν ∈ ( τ − Z (Φ)) ∩ Λ , Trace S ( ν ) e i y = (cid:88) s ∈V s − ν ( T odd ([ q ] , y , s )( ∂ ) T τ (Φ , y ))( ν ) | q =1 . Thus the function Trace S ( ν ) e i y is given by an analytic function of ν, y oneach enlarged chamber τ − Z (Φ). Proof. This is a direct consequence of Theorem 7.1. Indeed, we just need toverify that the functions P ( s, f, y ) coincide with an analytic function on τ .But we see that P ( s, f, y ) is just equal to T (Φ( s ) , y ). The chamber τ for Φis smaller than the chamber for Φ( s ) conaining τ , so T (Φ( s ) , y ) is analyticon τ . (cid:3) References [1] Brion M., Vergne M., Residues formulae, vector partition functions and lattice pointsin rational polytopes , Journal of the American Mathematical Society (1997), 797–833.[2] Dahmen W., Micchelli C., On the solution of certain systems of partial differenceequations and linear dependence of translates of box splines , Trans. Amer. Math.Soc. , 1985, 305–320.[3] De Concini C., Procesi C., Vergne M. Box splines and the equivariant index theorem.To appear in Journal of the Institute of Mathematics of Jussieu. arXiv :1012.1049[4] Duflo M., Vergne M., Kirillov’s formula and Guillemin-Sternberg conjecture , Comptes-Rendus de l’Acad´emie des Sciences (2011) 1213–1217. arXiv:1110.0987[5] Szenes, A. and Vergne M., Residue formulae for vector partitions and Euler-MacLaurin sums. Proceedings of FPSAC-01 : Advances in Applied Mathematics , ,2003, 295–342. arXiv :math/0202253[6] Vergne M., Euler-MacLaurin formula for the multiplicities of the index of transver-sally elliptic operators. arXiv:1211.5547 Universit´e Denis-Diderot-Paris 7, Institut de Math´ematiques de Jussieu,C.P. 7012, 2 place Jussieu, F-75251 Paris cedex 05 E-mail address ::