## Abstract

Let Φ = (φ_{k})_{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 c_{Φ}(T): = inf c such that where (g_{k})(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 c_{Φ}(I_{ln ∞} ∼ uniformly in n iff there is a positive η > 0 such that for all n ∈ ℕ one can find an orthonormal Ψ = (ψ_{l})^{n}_{1} ⊂ span (φ_{k}|k ∈ ℕ) and a sequence of disjoint measurable sets (A_{l})^{n}_{1} ⊂ Ω with ∫_{Al}|ψ_{l}|^{2}dp ≥ 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 language | English (US) |
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Pages (from-to) | 399-433 |

Number of pages | 35 |

Journal | Journal of Approximation Theory |

Volume | 82 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1995 |

Externally published | Yes |

## ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Mathematics(all)
- Applied Mathematics