Let $H$ be a [Hilbert space](/page/Hilbert%20Space) over $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$ with [orthonormal basis](/page/Orthonormal%20Basis) $(e_k)_{k \in I}$. Let $f \in H$, define the Fourier coefficients $a_k := (f,e_k)_H$ for $k \in I$, and suppose that
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\begin{align*}
f = \sum_{k \in I} a_k e_k
\end{align*}
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with convergence in $H$. Let $(a_m^*)_{m \geq 1}$ denote the nonincreasing rearrangement of the coefficient magnitudes $(|a_k|)_{k \in I}$. Let $\mathcal{D} := \{e_k : k \in I\}$, and for $n \in \mathbb{N}$ define the best $n$-term approximation error by