On the approximation of vector-valued functions by samples
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Given a Hilbert space ℋ and a finite measure space Ω, the approximation of a vector-valued function f: Ω→ℋ by a k-dimensional subspace 𝒰⊂ℋ plays an important role in dimension reduction techniques, such as reduced basis methods for solving parameter-dependent partial differential equations. For functions in the Lebesgue–Bochner space L^2(Ω;ℋ), the best possible subspace approximation error d_k^(2) is characterized by the singular values of f. However, for practical reasons, 𝒰 is often restricted to be spanned by point samples of f. We show that this restriction only has a mild impact on the attainable error; there always exist k samples such that the resulting error is not larger than √(k+1)· d_k^(2). Our work extends existing results by Binev at al. (SIAM J. Math. Anal., 43(3):1457–1472, 2011) on approximation in supremum norm and by Deshpande et al. (Theory Comput., 2:225–247, 2006) on column subset selection for matrices.
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