Approximations and expansions Definition

Nonlinear approximation

Builds on Normed vector space Error of best approximation Orthonormal basis

Definition

Let $X$ be a normed space and let $\Sigma_0 \subseteq \Sigma_1 \subseteq \Sigma_2 \subseteq \cdots$ be subsets of $X$ , which in general are not linear subspaces. Usually there is a constant $c$ with $\Sigma_n + \Sigma_n \subseteq \Sigma_{cn}$ . Nonlinear approximation is the approximation of elements $f \in X$ by elements of $\Sigma_n$ , measured by the error of best approximation $\sigma_n(f) = \inf_{s \in \Sigma_n} \norm{f - s}_X.$ The main example is $n$ -term approximation from a dictionary $\mathcal{D} \subseteq X$ , with $\Sigma_n = \set{\sum_{k=1}^{n} c_k g_k : c_k \in \R,\ g_k \in \mathcal{D}}$ .

Remarks

Example: let $X$ be a separable Hilbert space with orthonormal basis $(e_j)$ , and let $\Sigma_n$ be the vectors with at most $n$ nonzero coordinates. The set $\Sigma_1$ is not a subspace: $e_1, e_2 \in \Sigma_1$ but $e_1 + e_2 \notin \Sigma_1$ . For $f = \sum_j c_j e_j$ , the best element of $\Sigma_n$ keeps the $n$ coefficients of largest absolute value, so $\sigma_n(f)^2$ is the sum of $\abs{c_j}^2$ over all other $j$ .

Non-example: $\Sigma_n$ the polynomials of degree at most $n$ in $C[0, 1]$ is a linear subspace, so approximation from it is linear approximation.

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Tag	`D94626DTTW`
URI	`m2://D94626DTTW`
Kind	definition
Writing	W0 stub
MSC	41A46, 41A25
Defines	Nonlinear approximation
Uses	Normed vector space, Error of best approximation, Orthonormal basis

Nonlinear approximation

Definition

Remarks

Reading path

Further reading

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