Two Theorems on Lattice Expansions

Journal Article (Journal Article)

It is shown that there is a trade-off between the smoothness and decay properties of the dual functions, occurring in the lattice expansion problem. More precisely, it is shown that if g and [formula Omitted] are dual, then 1) at least one of H1/2 g and [formula Omitted] is not in [formula Omitted] at least one of Hg and [formula Omitted] is not in [formula Omitted]. Here, H is the operator-1/(4π2 )d2 /(dt2 ) + t2 . The first result is a generalization of a theorem first stated by Balian and independently by Low, which was recently rigorously proved by Coifman and Semmes; a new, much shorter proof was very recently given by Battle. Battle suggests a theorem of type (i), but our result is stronger in the sense that certain implicit assumptions made by Battle are removed. Result 2) is new and relies heavily on the fact that, when G E W2,2 (S) with [formula Omited] and G(0) = 0, then [formula Omitted]. The latter result was not known to us and may be of independent interest. © 1993 IEEE

Full Text

Duke Authors

Cited Authors

  • Daubechies, I

Published Date

  • January 1, 1993

Published In

Volume / Issue

  • 39 / 1

Start / End Page

  • 3 - 6

Electronic International Standard Serial Number (EISSN)

  • 1557-9654

International Standard Serial Number (ISSN)

  • 0018-9448

Digital Object Identifier (DOI)

  • 10.1109/18.179336

Citation Source

  • Scopus