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  "body": "\n## Definition\n\nLet $X$ be a real or complex [[normed vector space|03J5RAKB30]] with norm $\\norm{\\cdot}$, and let $\\mathcal{D} \\subseteq X$ be a set, called the *dictionary*.\nLet $n \\ge 1$ be an integer.\nThe set of *$n$-term approximants* is\n$$\\Sigma_n(\\mathcal{D}) = \\set{\\textstyle\\sum_{k=1}^{n} c_k g_k : c_k \\text{ scalars}, \\ g_k \\in \\mathcal{D}} .$$\nThe *error of best $n$-term approximation* of $f \\in X$ is\n$$\\sigma_n(f, \\mathcal{D}) = \\inf\\set{\\norm{f - g} : g \\in \\Sigma_n(\\mathcal{D})} .$$\n*$n$-term approximation* is the approximation of elements $f \\in X$ by elements of $\\Sigma_n(\\mathcal{D})$, measured by $\\sigma_n(f, \\mathcal{D})$.\n\n## Remarks\n\nThis is the [[error of best approximation|YDN0T4MEX0]] from the set $\\Sigma_n(\\mathcal{D})$, which is a union of subspaces and not a subspace.\nIndeed $\\Sigma_n + \\Sigma_n \\subseteq \\Sigma_{2n}$, but in general $\\Sigma_n + \\Sigma_n \\ne \\Sigma_n$.\nExample: let $X$ be a Hilbert space with orthonormal basis $\\mathcal{D} = (e_j)$ and $f = \\sum_j a_j e_j$.\nKeeping the $n$ coefficients $a_j$ of largest absolute value gives a best approximant, and $\\sigma_n(f, \\mathcal{D})^2$ is the sum of $\\abs{a_j}^2$ over the remaining indices.\nCheck: for $g = \\sum_{j \\in S} c_j e_j$ with $\\abs{S} \\le n$, $\\norm{f - g}^2 = \\sum_{j \\in S} \\abs{a_j - c_j}^2 + \\sum_{j \\notin S} \\abs{a_j}^2$, which is smallest when $c_j = a_j$ and $S$ carries the largest $\\abs{a_j}$.\nNon-example: in $\\R^3$ with the standard basis, $3e_1 + e_2 + 2e_3$ is not in $\\Sigma_2$, since it has three nonzero coordinates.\nWhen $\\mathcal{D}$ is redundant, the best approximant may be hard to compute, and greedy algorithms are used instead.\n",
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