Approximation Methods in Function Spaces

Approximation methods in function spaces characterise how well complex functions can be represented or recovered using limited information such as function values or linear measurements. Central to this theory are metrics like approximation numbers, which quantify the minimal error of best-possible linear reconstructions, and sampling numbers, which measure performance when only pointwise evaluations are available. Classic frameworks include reproducing kernel Hilbert spaces, Sobolev and Besov spaces with varying smoothness, and Banach spaces more generally. Techniques range from projection onto finite-dimensional subspaces, sparse-grid constructions and orthogonal series expansions to greedy algorithms, least-squares regression and random sampling strategies. Recent advances have clarified the interplay between deterministic and randomized schemes, showing for many spaces that function values can achieve near-optimal convergence rates traditionally believed to require broader linear information. These insights have broad implications across numerical integration, data assimilation, machine learning and the solution of partial differential equations, where efficient representation and recovery underpin both theoretical guarantees and practical algorithms.

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Recent studies have established precise upper and lower bounds for sampling numbers in various function spaces. A 2023 investigation demonstrated a universal bound relating sampling numbers to approximation numbers in reproducing kernel Hilbert spaces, confirming that the decay of worst-case sampling errors can mirror that of optimal linear schemes up to a constant factor. In parallel, another 2023 contribution showed that independent random sampling points on convex domains yield asymptotically optimal approximation for Sobolev spaces, matching the performance of carefully designed point sets and unifying geometric and probabilistic perspectives. A further advance in 2022 addressed pointwise reconstruction in L₂ spaces by introducing a weighted least-squares approach combined with a novel sparsification strategy. This method achieves near-best projection error using only a slightly enlarged sampling budget, closing the logarithmic gap and enhancing computational feasibility of random sampling schemes.

Approximation Methods in Function Spaces publication trend

The graph below shows the total number of articles in approximation methods in function spaces across all publications each year (not limited to Nature Index journals).

Technical terms

Sampling number: The minimal worst-case approximation error achievable using a specified number of pointwise function evaluations.

Approximation number: The minimal worst-case error achievable when arbitrary linear information of fixed dimension is available.

Kolmogorov width: A measure of how well a function class can be approximated by n-dimensional subspaces in a normed space.

Reproducing kernel Hilbert space: A Hilbert space of functions in which evaluation at any point is realised by an inner product with a kernel function.

References

1. Function Values Are Enough for L2-Approximation. Foundations of Computational Mathematics (2020).
2. Random points are optimal for the approximation of Sobolev functions. IMA Journal of Numerical Analysis (2023).
3. Worst-case Recovery Guarantees for Least Squares Approximation Using Random Samples. Constructive Approximation (2021).