Suppose \(\mathbb{K}\) is either \(\mathbb{C}\) or \(\mathbb{R}\), \(x\in \mathbb{K}^N\), \(s\le N\) and \(p\in (0,\infty]\). The \(s\)-term approximation error of \(x\) is given by
\[ \sigma_s(x)_p=\inf\{\lVert x-y\rVert_p\mid y\in \mathbb{K}^N_s\}, \]where \(\mathbb{K}^N_s\) the space of \(s\)-sparse vectors .
Remarks
- Obviously, \(\sigma_s(x)_p=\lVert x-z\rVert_p\) where \(z\) is the best s-term approximation of \(x\).
- \((k-s)x_k^*\le \lVert x-y\rVert_1+\sigma_s(x)_1\)
- The \(s\)-term approximation error is continuous (for \(p=1\)).