# Ultrasymmetric sequence spaces in approximation theory.

Collectanea Mathematica (2006)

• Volume: 57, Issue: 3, page 257-277
• ISSN: 0010-0757

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## Abstract

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Let φ(t) be a positive increasing function and let Ê be an arbitrary sequence space, rearrangement-invariant with respect to the atomic measure µ(n) = 1/n. Let {an*} mean the decreasing rearrangement of a sequence {|an|}. A sequence space lφ,E with symmetric (quasi)norm || {φ(n)an*} ||Ê is called ultrasymmetric, because it is not only intermediate but also interpolation between the corresponding Lorentz and Marcinkiewicz spaces Λφ and Mφ. We study properties of the spaces lφ,E for all admissible parameters φ, E and use them for the definition of ultrasymmetric approximation spaces Xφ,E, which essentially generalize most of classical approximation spaces. At the same time we show that the spaces Xφ,E possess almost all properties of classical prototypes, such as equivalent norms, representation, reiteration, embeddings, transformation etc. Special attention is paid to interpolation properties of these spaces. At last, we apply our results to ultrasymmetric operator ideals.

## How to cite

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Pustylnik, Evgeniy. "Ultrasymmetric sequence spaces in approximation theory.." Collectanea Mathematica 57.3 (2006): 257-277. <http://eudml.org/doc/41776>.

@article{Pustylnik2006,
abstract = {Let φ(t) be a positive increasing function and let Ê be an arbitrary sequence space, rearrangement-invariant with respect to the atomic measure µ(n) = 1/n. Let \{an*\} mean the decreasing rearrangement of a sequence \{|an|\}. A sequence space lφ,E with symmetric (quasi)norm || \{φ(n)an*\} ||Ê is called ultrasymmetric, because it is not only intermediate but also interpolation between the corresponding Lorentz and Marcinkiewicz spaces Λφ and Mφ. We study properties of the spaces lφ,E for all admissible parameters φ, E and use them for the definition of ultrasymmetric approximation spaces Xφ,E, which essentially generalize most of classical approximation spaces. At the same time we show that the spaces Xφ,E possess almost all properties of classical prototypes, such as equivalent norms, representation, reiteration, embeddings, transformation etc. Special attention is paid to interpolation properties of these spaces. At last, we apply our results to ultrasymmetric operator ideals.},
author = {Pustylnik, Evgeniy},
journal = {Collectanea Mathematica},
keywords = {Teoría de la aproximación; Espacio de sucesiones; Espacios de interpolación; Ideal de operadores; Espacios lineales normados; interpolation; approximation; ultrasymmetric spaces; operator ideals},
language = {eng},
number = {3},
pages = {257-277},
title = {Ultrasymmetric sequence spaces in approximation theory.},
url = {http://eudml.org/doc/41776},
volume = {57},
year = {2006},
}

TY - JOUR
AU - Pustylnik, Evgeniy
TI - Ultrasymmetric sequence spaces in approximation theory.
JO - Collectanea Mathematica
PY - 2006
VL - 57
IS - 3
SP - 257
EP - 277
AB - Let φ(t) be a positive increasing function and let Ê be an arbitrary sequence space, rearrangement-invariant with respect to the atomic measure µ(n) = 1/n. Let {an*} mean the decreasing rearrangement of a sequence {|an|}. A sequence space lφ,E with symmetric (quasi)norm || {φ(n)an*} ||Ê is called ultrasymmetric, because it is not only intermediate but also interpolation between the corresponding Lorentz and Marcinkiewicz spaces Λφ and Mφ. We study properties of the spaces lφ,E for all admissible parameters φ, E and use them for the definition of ultrasymmetric approximation spaces Xφ,E, which essentially generalize most of classical approximation spaces. At the same time we show that the spaces Xφ,E possess almost all properties of classical prototypes, such as equivalent norms, representation, reiteration, embeddings, transformation etc. Special attention is paid to interpolation properties of these spaces. At last, we apply our results to ultrasymmetric operator ideals.
LA - eng
KW - Teoría de la aproximación; Espacio de sucesiones; Espacios de interpolación; Ideal de operadores; Espacios lineales normados; interpolation; approximation; ultrasymmetric spaces; operator ideals
UR - http://eudml.org/doc/41776
ER -

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