Fetching the paper…
Reading the bibliography…
We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform).
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis , fourth ed., Cambridge University Press, Cambridge, 1927
1927
Earlier work this paper cites.
W. L. Ferrar, Summation formulae and their relation to Dirichlet’s series II , Compos. Math. 4
1937
Earlier work this paper cites.
A. P. Guinand, Integral modular forms and summation formulae , Proc. Cambridge Phil. Soc. 43
1947
Earlier work this paper cites.
S. Bochner, Some properties of modular relations , Ann. Math. 53
1951
Earlier work this paper cites.
by same author, Harmonic Analysis and the Theory of Probability , University of California Press, Berkeley, 1955
1955
Earlier work this paper cites.
by same author, Concordance and the harmonic analysis of sequences , Acta. Math. 101
1959
Earlier work this paper cites.
A. Sklar, On Some Exact Formulae in Analytic Number Theory , Report of the Institute in the Theory of Numbers (D. C. B. Marsh, ed.), American Mathematical Society, Boulder, CO, 1959, pp. 104–110
1959
Earlier work this paper cites.
R. Bellman, A Brief Introduction to Theta Functions , Holt, Rinehart and Winston, New York, 1961
1961
Earlier work this paper cites.
B. Simon, Distributions and their Hermite expansions , J. Math. Phys. 12
1971
Earlier work this paper cites.
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces , Princeton University Press, Princeton, 1971
1971
Earlier work this paper cites.
T. Kubota, On an analogy to the Poisson summation formula for generalized Fourier transformation , J. Reine Angew. Math. 268/269
1974
Earlier work this paper cites.
C. L. Mallows, A. M. Odlyzko, and N. J. A. Sloane, Upper bounds for modular forms, lattices, and codes , J. Algebra 36
1975
Earlier work this paper cites.
R. A. Rankin, Modular Forms and Functions , Cambridge University Press, Cambridge, 1977
1977
Earlier work this paper cites.
M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis , rev. and enl. ed., Academic Press, New York, 1980
1980
Earlier work this paper cites.
C. Ryavec, The Poisson summation formula , Aequationes Math. 21
1980
Cited alongside, same era.
A. Córdoba, La formule sommatoire de Poisson , C. R. Acad. Sci., Paris, Sér. I 306
1988
Cited alongside, same era.
by same author, Dirac combs , Lett. Math. Phys. 17
1989
Cited alongside, same era.
J. B. Conway, A Course in Functional Analysis , second ed., Springer, New York, 1990
1990
Cited alongside, same era.
S. Lang, Real and Functional Analysis , third ed., Springer, New York, 1993
1993
Cited alongside, same era.
R. V. Moody and J. Patera, Quasicrystals and icosians , J. Phys. A, Math. Gen. 26
1993
Cited alongside, same era.
A. Uribe, Trace formulae , First Summer School in Analysis and Mathematical Physics, Cuernavaca, Mexico (S. Pérez-Esteva and C. Villegas-Blas, eds.), Contemp. Math., vol. 260, American Mathematical Society, 2000, pp. 61–90
2000
Later among the works it cites.
M. Baake and U. Grimm, A note on shelling , Discrete Comput. Geom. 30
2003
Later among the works it cites.
R. Contino and A. Gambassi, On dimensional regularization of sums , J. Math. Phys. 44
2003
Later among the works it cites.
J. Hanke, Some recent results about (ternary) quadratic forms , Number Theory (H. Kisilevsky and E. Z. Goren, eds.), CRM Proc. Lecture Notes, vol. 36, American Mathematical Society, Providence, RI, 2004, pp. 147–164
2004
Later among the works it cites.
H. Iwaniec and E. Kowalski, Analytic Number Theory , American Mathematical Society, Providence, RI, 2004
2004
alphaXiv searches the wider corpus for related work and actual follow-ups.
alphaXiv is searching for related work…
R. V. Moody and A. Weiss, On shelling E 8 E_{8} quasicrystals , J. Number Theory 47
1994
Cited alongside, same era.
M. N. Huxley, Area, Lattice Points, and Exponential Sums , Oxford University Press, New York, 1996
1996
Cited alongside, same era.
J. Morita and K. Sakamoto, Octagonal quasicrystals and a formula for shelling , J. Phys. A, Math. Gen. 31
1998
Cited alongside, same era.
L. Schwartz, Théorie des Distributions , third ed., Hermann, Paris, 1998
1998
Cited alongside, same era.
G. E. Andrews, R. Askey, and R. Roy, Special Functions , Encyclopedia of Mathematics and Its Applications, vol. 71, Cambridge University Press, Cambridge, 1999
1999
Cited alongside, same era.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups , third ed., Springer, New York, 1999
1999
Cited alongside, same era.
Later among the works it cites.
S. D. Miller and W. Schmid, Summation formulas, from Poisson and Voronoi to the present , Noncommutative Harmonic Analysis: In honor of Jacques Carmona (P. Delorme and M. Vergne, eds.), Birkhäuser, Boston, 2004, pp. 419–440
2004
Later among the works it cites.
A. Unterberger, A spectral analysis of automorphic distributions and Poisson summation formulas , Ann. Inst. Fourier 54
2004
Later among the works it cites.
C. Pache, Shells of selfdual lattices viewed as spherical designs , Int. J. Algebra Comput. 15
2005
Later among the works it cites.
M. Baake, D. Frettlöh, and U. Grimm, A radial analogue of Poisson’s summation formula with applications to powder diffraction and pinwheel patterns , J. Geom. Phys. 57
2007
Later among the works it cites.
H. Cohen, Number Theory, Volume II: Analytic and Modern Tools , Springer, New York, 2007
2007
Later among the works it cites.
R. E. Johnson and S. Ranganathan, Generalized approach to Ewald sums , Phys. Rev. E 75
2007
Later among the works it cites.
P. Jenkins and J. Rouse, Bounds for coefficients of cusp forms and extremal lattices , Bull. London Math. Soc. 43
2011
Closest in time.
Wolfram Research, The Wolfram Functions Site , 2011, http://functions.wolfram.com
2011
Closest in time.