Quasinorm

Quasinorm

Limit Theorems and B est Constants in Approximation Theory

Published in George Anastassiou, Handbook of Analytic-Computational Methods in Applied Mathematics, 2019

We consider first some properties of these spaces. For all f ∈ F and h ∈ F, () ‖f+h‖F≤D(‖f‖F+‖h‖F),where 0 < D < 21/μ–1.(11.71) easily follows from (11.70). This property shows that every μ-QRIS is a quasi-normed space. On the other hand, it is known [16, p. 20] that for every quasi-normed space F there exists an equivalent μ-quasi-norm ‖·‖F.F is a φ-space withφF=‖⋅‖Fμ.A normed rearrangement-invariant space (NRIS) is a 1-QRIS.For example, Lp(ℝm), 1 ≤ p ≤ ∞, Orlitz, Marcinkiewicz, and Lorentz spaces are NRISes. For all this, see [15, 16, 42].Lp(ℝm), 0 < p < 1, is a p-QRIS.F contains the characteristic functions of all measurable sets of finite measure.

Sufficient conditions on stable reconstruction of weighted problem

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Journal Information

Published in Applicable Analysis, 2022

Jing Bu, Liejun Xie, Jinping Wang

This is to recover a weighted sparse vector via the weighted minimization, it is more general than classic sparse recovery problems such as recover standard sparse vector via the minimization with in [7] and the weighted minimization in [2]. Moreover, it is a weaker condition that we use the weighted quasinorm to state and estimate error bounds instead of the weighted norm or the standard quasinorm. We notice that the weighted norm reproduces the norm when for all and the weighted norm equals to the norm.

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Limit Theorems and B est Constants in Approximation Theory

Sufficient conditions on stable reconstruction of weighted problem