A STRONG VERSION OF POISSON SUMMATION

We establish a generalized version of the classical Poisson summation formula. This formula incorporates a special feature called "compression", whereby, at the same time that the formula equates a series to its Fourier dual, the compressive feature serves to enable both sides of the equation to converge.

Throughout this article, all functions f:d C are to be understood as Lebesgue measurable, and defined almost everywhere in d.Given x,y E d, let xy denote the dot product x.y, and let x denote x. x.A function f:IR dC is said to be ezponentially bounded if or some M > 0, k > 0, and almost all x E [Rd: f x < Me kllxll (1.1) Evidently, if f: d is exponentially bounded, then the function x f(x)exp(-r$x) belongs to L(d) for every > 0. If, moreover, the lit I lim [ f(x)edx exists and is finite, then we denote this limit by the symbol I f(x)dz and call it the compressed integral of f over e.Similarly, if c, n Ze, is a sequence of complex numbers that grows no faster than an exponential, and if the limit S lim ce -0+ exists and is finite, then we denote this limit by the symbol S= Cn nE_.and call it the compressed series of c.
Provided the implied limit exists, a compressed version of the Fourier transform is defined by '*(f)(t) df(x)e-'xt dx   (1.2) The local averaging operator .4,defined on measurable functions f: IRg--+ g, acts by the formula: A(f) (zo) lime -/ [ f(z + zo)e -/ dz (1.3) --.o+ allzll<," If f is essentially bounded in a neighborhood of xo, then this limit, if it exists, is independent of r > 0 (proof below).If the limit exists, then f is said to be averageable at x0.
In terms of the above symbolism, the formula that we wish to establish is nE nEZ This formula will be shown to be valid for the class of everywhere averageable compressible functions (see Section 4 for the definition of "compressible").
Given xo E [R d and r > 0, let Br(x0) denote the open ball of radius r centered at x0.A Lebesgue measureable function f: [Rd-+C is said to be averageable at xo 6 R d provided that for some r > 0 the function f is essentially bounded in Br(xo) and provided that the limit A(I) (x0) lira -a/= f f(x + xo)e -2/ dx (2.1) -,0+ Jll:ll< exists.By "essentially bounded" we mean bounded relative to the L-norm on B(zo), so that f may be averageable at zo even if f(zo) is undefined.
In this section we state and prove some basic facts about averageable functions nd the local averaging operator 4.These facts will be used later.LEMMA 2.1.Suppose f: [Ra--,C is essentially bounded in a neighborhood V of zo.Let V0 V zo denote the corresponding neighborhood of the origin in IRa.If B(0 Choose M > 0 such that I/(z)l < M for almost all x 6 V. Then IIl_< e-":2/'dz,< Me-a/= e-/dz= M which tends to zero as e+O because exp(-rz=) is integrable over a.
e -=2 dx COROLLARY 2.2.Suppose f is averageable at zo and essentially bounded in a neighborhood V of zo.Let Vo V zo denote the corresponding neighborhood of the origin in [Rd.Then .A(f) (xo) lira -d/ [ f(x + Xo)e -'x:/ dx PROPOSITION 2.3.Suppose 9 [Rd--C is continuous at Xo and f:[RdC is averageable at Xo. Then the product f9 is averageable at xo, and A(fg)(o) g(xo).A(f)(xo) PROOF.Shifting f and g by :Co, we may suppose that z0 0.Moreover, by treating the real and imaginary parts of f and 9 separately, we may suppose that f and g are both real.Choose rl > 0 such that f(z) is essentially bounded in ]lz]l < r, say by M > 0. Let A > 0. Since 9 is continuous at 0 we can find ro < ra such that ]9(x)-9(0)] < A for all Itz]l < r0.Thus, for almost all IlxII < so, we have The next proposition asserts that the uniform limit of a family of averageable functions is itself an averageable function.
The final proposition of this section asserts that the local average of a sum of averageable functions equals the sum of their local averages (i.e.J[ jr), provided that the convergence of the sum is sufficiently well-controlled.
PROPOSITION 2.5.Let f d n d be a family of functions each of which is averageable at x0 and essentially bounded in a neighborhood V of x0.Let M esssupev [f(x)[, and suppose that M converges.Let () s() Then F is averageable at xo, and PROOF.Let Vo V zo (i.e. the translation of V to the origin ), and let ,a()(o) -z ./F( + o)d By hypothesis, the series in the integrand is uniformly convergent and bounded in Vo.Thus we may reverse the order of integration and summation to obtain Now we want to let 0, and we want to be able to push this limit through the last sum.To do this we have to show that this sum converges uniformly in .Consider the absolute value of the terms I(A)(o)l e-a/=/vo A(= / =o) -=/ d= <-M'e-el2 /e" e-'=:21 dx M, By the Weierstrass Comparison Test, the sum (2.2) converges uniformly in , so that, as --,0, we get M(F)(xo) E .A(f,)(xo) 3. POLYGONALLY CONTINUOUS FUNCTIONS.By Proposition 2.3 we know that if f: R-is continuous then f is averageable, and M(f) f.We turn now to the construction of a basic class of discontinous averageable functions.Although still very small relative to the class of all averageable functions, this class of functions, called "polygonally continuous", will be large enough to meet all our requirements.
Given x0 [R and r > 0, let B(x0) denote the open ball of radius r centered at x0.The open set S formed by the intersection of B(x0) with a finite number of open half-spaces each tangent to x0 is referred to as an open polygonal cone or polygonal sector of radius r centered at .T O By definition, the content a(S) of a polygonal sector 5' is the ratio of the volume of S to the volume of the ball B(x0) in which it resides.If the ball B(x0) is partitioned (except for a set of measure zero) as a finite disjoint union of polygonal sectors, say then by definition of a(S) we have Br(xo) '1 (-J (.J N where r is the radius of S. Let x0 R d A function g: iZ4 will be called polygonally continuous at xo provided that for some r > 0 the ball Br(x0) is decomposible (except for a set of measure zero) as a finite disjoint union of polygonal sectors Sa such that for each j the restriction of g to Sa admits an extension ga to S: U {x0) which is continuous at x0.Thus, whenever x-*xo from within the sector S, we have lim ga(x) g(xo) (3.3) '--*0 PROPOSITION 3.2.If g IRd--* is polygonally continuous at x0, then g is averageable at x0. Suppose B(x0) SI U... U SN (decomposition into polygonal sectors) and let g be as in formula (3.3).Then where R(e, x) e-a/: Z e-'q"+')'/ (4.3) no Let IE g [-, 1/2) denote the central unit hypercube in IR Note that if x IE d and n e Z n-0 then [[n/x][> ][nl[, so that IR(, x)l _< -a/= -=/,, ( [ + + [ + Since F is averageable at :co, the limit as e0 of the first term on the right exists and equals A(F) (z0); furthermore (4.5) together with the boundedness of F and the compactness of E immediately imply that the second term vanishes as e--.0.
5. COMPRESSIBLE FUNCTIONS.Our aim in this section is to gain some aquaintance with a class of functions f:[Rell for which the formula As the reader may recall, a function f: IRd--,l is said to be exponentially bounded if the condition If(x)l < Me kllll (5.2) is satisfied for some M > 0, k > 0, and almost all x E IR Given an exponentially bounded function f" IRd e, > 0, and z e ;e [__}, 1/2)e, consider the sum S(I) (x) f(n + x)e -'('+,) (5.3) Me-,e(,',+:)2 except possibly on aset of measure0.On the other hand, if x E E d and n Id, then, by the inequality llnll < I1 + :11 _< I1,11 + (5.4) we have If(n + :c)le-e(+) < Me-+1111+ @ M (5.5) so that the series (5.3) is absolutely dominated almost everywhere by the convergent series M.
In particular, for every fixed 8 > O, the series defining &(f)(x) is essentially bounded in E d and uniformly convergent outside of a set of measure zero.
Of special interest to us at this point is the behavior of the limit So(f)(x) S,(f)(x) (5.6) which, for a typical exponentially bounded f, may or may not exist.
An exponentially bounded function f:Rd will be called compressible provided that the following special condition is satisfied by the sums Se(f) Compressibility condition: There exists a function S0(f) essentially bounded on E d such that lim S(f) So(f)1[ 0 0 The formula (5.1) will be shown to be valid for the class of compressible everywhere averageable functions f:e.This will be proved in the next section.For now, let us collect some basic facts about compressible functions.
Z fE f(n + n67/a ,E S6(f)(x) O(g, x) dx (6.2) where the reversal of order of integration and summation is justified by the uniform convergence (outside a set of measure zero) of the sum representing S6(f)(:r.).
For x e E [-7, g), the critical sum S(f) is given by n>-x As 50, the family {Se(/)} converges in L(E1) to the bounded function Thus, f is compressible.Next, we must check that f is averageable everywhere in R. The compressed Fourier transform of f is given by (f) (t) which is the Mittag-Leffter expansion of cot(ra).
CLOSING REMARKS AND ACKNOWLEDGEMENTS.This paper grew out of an effort to de- velop an explanatory framework to tie together a number of isolated examples of theta functions attached to indefinite quadratic forms.These highly suggestive examples, as in [1 ], [2 ], [3 ], were first brought to my attention jointly by Don Zagier and Stephen Kudla, to both of whom am deeply indebted.
This paper is the first in a series of three, the last of which will demonstrate a certain method by means of which to construct theta functions attached to indefinite quadratic forms.The method makes use of compressed Poisson summation acting upon eigenfunctions of the compressed Fourier transform, and closely resembles the classical construction of theta functions attached to positive definite quadratic forms.
wish to express my gratitude to Don Zagier for the many specific suggestions and corrections he offered along the way.For instance, the main formulation (1.3) of the local averaging operator A appearing in this paper is one of his suggestions, as is its formulation in terms of the principal theta function (Proposition 4.1).
which proves the proposition by letting A-,0.