Suppose \(\mathbb{K}\) is either \(\mathbb{C}\) or \(\mathbb{R}\), \(x\in \mathbb{K}^N\), \(s\le N\) and \(0s-term approximation error by

\[ \sigma_s(x)_q\le \frac{\lVert x\rVert_p}{s^{1/p-1/q}}. \]

[1, Satz 1.6]

Additionally, there is another estimate in terms of the weak \(l^p\)-norm , i.e.

\[ \sigma_s(x)_q \le \biggl(\frac{p}{q-p}\biggr)^{1/q} \frac{\lVert x\rVert_{p,\infty }}{s^{1/p-1/q}}. \]

[1, Satz 1.13]

Remarks
  • There are sharper estimates by adding some multiplicative constant in the front of the right hand side.

References Link to heading

  1. I. Veselic. Class Lecture, Topic: Compressive Sensing. Technische Universität Dortmund, 2017.