Completeness of integer translates in function spaces on ℝ

A. Atzmon*, A. Olevskiǐ

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We show that each of the Banach spaces C0(ℝ) and Lp(ℝ), 2 < p < ∞, contains a function whose integer translates are complete. This function can also be chosen so that one of the following additional conditions hold: (1) Its non-negative integer translates are already complete. (2) Its integer translates form an orthonormal system in L2(ℝ). (3) Its integer translates form a minimal system. A similar result holds for the corresponding Sobolev space, for certain weighted L2 spaces, and in the multivariate setting. We also prove some results in the opposite direction.

Original languageEnglish
Pages (from-to)291-327
Number of pages37
JournalJournal of Approximation Theory
Issue number3
StatePublished - Dec 1996


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