Type and cotype with respect to arbitrary orthonormal systems

Stefan Geiss, Marius Junge

Research output: Contribution to journalArticlepeer-review


Let Φ = (φk)k ∈ N be an orthonormal system on some σ-finite measure space (Ω, p). We study the notion of cotype with respect to Φ for an operator T between two Banach spaces X and Y, defined by cΦ(T): = inf c such that where (gk)(k ∈ N) is the sequence of independent and normalized gaussian variables. It is shown that this Φ-cotype coincides with the usual notion of cotype 2 iff cΦ(Iln ∼ uniformly in n iff there is a positive η > 0 such that for all n ∈ ℕ one can find an orthonormal Ψ = (ψl)n1 ⊂ span (φk|k ∈ ℕ) and a sequence of disjoint measurable sets (Al)n1 ⊂ Ω with ∫All|2dp ≥ η for all l = 1,., n. A similar result holds for the type situation. The study of type and cotype with respect to orthonormal systems of a given length provides the appropriate approach to this result. We intend to give a quite complete picture for orthonormal systems in measure space with few atoms.

Original languageEnglish (US)
Pages (from-to)399-433
Number of pages35
JournalJournal of Approximation Theory
Issue number3
StatePublished - Sep 1995
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Mathematics(all)
  • Applied Mathematics


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