A remark on the convolution with the box spline

Abstract

The semi-discrete convolution with the box spline is an important tool in approximation theory. We give a formula for the difference between semi-discrete convolution and convolution with the box spline. This formula involves multiple Bernoulli polynomials.

  • [BV] A. Boysal and M. Vergne, Multiple Bernoulli series, an Euler-MacLaurin formula and wall crossings.
    @misc{BV,
      author={Boysal, A. and Vergne, M.},
      TITLE={Multiple {B}ernoulli series, an {E}uler-{M}ac{L}aurin formula and wall crossings},
      NOTE={{\it Annales de l'Institut Fourier},
      to appear},
      ARXIV={1008.0263},
      }
  • [brver2] Go to document M. Brion and M. Vergne, "Arrangement of hyperplanes. II. The Szenes formula and Eisenstein series," Duke Math. J., vol. 103, iss. 2, pp. 279-302, 2000.
    @article {brver2, MRKEY = {1760629},
      AUTHOR = {Brion, Michel and Vergne, Mich{è}le},
      TITLE = {Arrangement of hyperplanes. {II}. {T}he {S}zenes formula and {E}isenstein series},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {103},
      YEAR = {2000},
      NUMBER = {2},
      PAGES = {279--302},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {32S22 (11B68 11M36)},
      MRNUMBER = {1760629},
      MRREVIEWER = {Hiroaki Terao},
      DOI = {10.1215/S0012-7094-00-10325-0},
      ZBLNUMBER = {0968.32016},
      }
  • [DM] Go to document W. Dahmen and C. A. Micchelli, "Translates of multivariate splines," Linear Algebra Appl., vol. 52/53, pp. 217-234, 1983.
    @article {DM, MRKEY = {0709352},
      AUTHOR = {Dahmen, Wolfgang and Micchelli, Charles A.},
      TITLE = {Translates of multivariate splines},
      JOURNAL = {Linear Algebra Appl.},
      FJOURNAL = {Linear Algebra and its Applications},
      VOLUME = {52/53},
      YEAR = {1983},
      PAGES = {217--234},
      ISSN = {0024-3795},
      CODEN = {LAAPAW},
      MRCLASS = {41A63 (41A15 65D07)},
      MRNUMBER = {0709352},
      MRREVIEWER = {Klaus H{ö}llig},
      DOI = {10.1016/0024-3795(83)80015-9},
      ZBLNUMBER = {0522.41009},
      }
  • [DM1] Go to document W. Dahmen and C. A. Micchelli, "On the solution of certain systems of partial difference equations and linear dependence of translates of box splines," Trans. Amer. Math. Soc., vol. 292, iss. 1, pp. 305-320, 1985.
    @article {DM1, MRKEY = {0805964},
      AUTHOR = {Dahmen, Wolfgang and Micchelli, Charles A.},
      TITLE = {On the solution of certain systems of partial difference equations and linear dependence of translates of box splines},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {292},
      YEAR = {1985},
      NUMBER = {1},
      PAGES = {305--320},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {41A15 (39A10 47B39)},
      MRNUMBER = {0805964},
      MRREVIEWER = {Jean Tzimbalario},
      DOI = {10.2307/2000181},
      ZBLNUMBER = {0637.41012},
      }
  • [deboor] C. De Boor, K. Höllig, and S. Riemenschneider, Box Splines, New York: Springer-Verlag, 1993, vol. 98.
    @book {deboor, KEY={5},
      MRKEY = {1243635},
      AUTHOR = {De Boor, C. and H{ö}llig, K. and Riemenschneider, S.},
      TITLE = {Box Splines},
      SERIES = {Appl. Math. Sci.},
      VOLUME = {98},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1993},
      PAGES = {xviii+200},
      ISBN = {0-387-94101-0},
      MRCLASS = {65-02 (65D07)},
      MRNUMBER = {1243635},
      MRREVIEWER = {Charles K. Chui},
      ZBLNUMBER = {0814.41012},
      }
  • [dp1] Go to document C. De Concini and C. Procesi, Topics in Hyperplane Arrangements, Polytopes and Box-splines, New York: Springer-Verlag, 2011.
    @book {dp1, MRKEY = {2722776},
      KEY={6},
      AUTHOR = {De Concini, Corrado and Procesi, Claudio},
      TITLE = {Topics in Hyperplane Arrangements, Polytopes and Box-splines},
      SERIES = {Universitext},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2011},
      PAGES = {xx+384},
      ISBN = {978-0-387-78962-0},
      MRCLASS = {32S22 (52-01 52C35)},
      MRNUMBER = {2722776},
      ZBLNUMBER = {05288674},
      URL={www.mat.uniroma1.it/~procesi/dida.html},
      }
  • [sze1] Go to document A. Szenes, "Iterated residues and multiple Bernoulli polynomials," Internat. Math. Res. Notices, iss. 18, pp. 937-956, 1998.
    @article {sze1, MRKEY = {1653791},
      AUTHOR = {Szenes, Andr{á}s},
      TITLE = {Iterated residues and multiple {B}ernoulli polynomials},
      JOURNAL = {Internat. Math. Res. Notices},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {1998},
      NUMBER = {18},
      PAGES = {937--956},
      ISSN = {1073-7928},
      MRCLASS = {11B68 (11L99 14D20 52C35)},
      MRNUMBER = {1653791},
      MRREVIEWER = {Hiroaki Terao},
      DOI = {10.1155/S1073792898000567},
      ZBLNUMBER = {1653791},
      }

Authors

Michèle Vergne

Institut de Mathématiques de Jussieu
175 rue du Chevaleret
75013 Paris
France