A survey of sampling discretization of integral and uniform norms
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908861477158912 |
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| author | Dai, F. Kosov, E. Temlyakov, V. |
| author_facet | Dai, F. Kosov, E. Temlyakov, V. |
| contents | This paper surveys recent developments in the sampling discretization of integral and uniform norms for functions in general finite-dimensional spaces. These results generalize the classical Marcinkiewicz-Zygmund inequalities for trigonometric and algebraic polynomials, which play a crucial role in Fourier analysis, interpolation, and approximation theory. We focus on the problem in the broad context of finite-dimensional subspaces, where norms defined by general probability measures are approximated by their discrete counterparts.
The primary emphasis is on results closely related to the authors' recent research. A key objective is to highlight the main ideas and techniques that form the foundation of the proofs in this area. This survey serves as a complement to three recently published survey papers on sampling discretization \cite{DPTT, KKLT, LMT}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_02459 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A survey of sampling discretization of integral and uniform norms Dai, F. Kosov, E. Temlyakov, V. Numerical Analysis Functional Analysis This paper surveys recent developments in the sampling discretization of integral and uniform norms for functions in general finite-dimensional spaces. These results generalize the classical Marcinkiewicz-Zygmund inequalities for trigonometric and algebraic polynomials, which play a crucial role in Fourier analysis, interpolation, and approximation theory. We focus on the problem in the broad context of finite-dimensional subspaces, where norms defined by general probability measures are approximated by their discrete counterparts. The primary emphasis is on results closely related to the authors' recent research. A key objective is to highlight the main ideas and techniques that form the foundation of the proofs in this area. This survey serves as a complement to three recently published survey papers on sampling discretization \cite{DPTT, KKLT, LMT}. |
| title | A survey of sampling discretization of integral and uniform norms |
| topic | Numerical Analysis Functional Analysis |
| url | https://arxiv.org/abs/2603.02459 |