Local Bernstein theory, and lower bounds for Lebesgue constants
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908985351733248 |
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| author | Tao, Terence |
| author_facet | Tao, Terence |
| contents | Classical (or ``global'') Bernstein theory establishes sharp control on entire functions of exponential type that are bounded and real-valued on the real axis. We localize some of this theory to rectangular regions $\{ x+iy: x \in I, 0 \leq y \leq y_0 \}$, showing that Bernstein-type bounds with acceptable errors can continue to hold for functions holomorphic in such rectangles, bounded and real-valued on the lower edge of the rectangle, at most exponentially large on the upper edge, and at most double exponentially large on the vertical sides.
As a consequence of these bounds, we are able to localize the Erdős lower bound $\sup_{x \in [-1,1]} λ(x) \geq \frac{2}π \log n - O(1)$ on the Lebesgue constant of interpolation on $C([-1,1])$ to shorter intervals $I$ than $[-1,1]$, answering a question of Erdős and Turán. By using suitably weighted versions of the residue theorem, we also obtain the asymptotically sharp lower bound $\int_I λ(x)\ dx \geq \frac{4|I|}{π^2} \log n - o(\log n)$ for integral variants of such constants, answering a further question of Erdős. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21453 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local Bernstein theory, and lower bounds for Lebesgue constants Tao, Terence Classical Analysis and ODEs 41A05, 42A15, 30D15 Classical (or ``global'') Bernstein theory establishes sharp control on entire functions of exponential type that are bounded and real-valued on the real axis. We localize some of this theory to rectangular regions $\{ x+iy: x \in I, 0 \leq y \leq y_0 \}$, showing that Bernstein-type bounds with acceptable errors can continue to hold for functions holomorphic in such rectangles, bounded and real-valued on the lower edge of the rectangle, at most exponentially large on the upper edge, and at most double exponentially large on the vertical sides. As a consequence of these bounds, we are able to localize the Erdős lower bound $\sup_{x \in [-1,1]} λ(x) \geq \frac{2}π \log n - O(1)$ on the Lebesgue constant of interpolation on $C([-1,1])$ to shorter intervals $I$ than $[-1,1]$, answering a question of Erdős and Turán. By using suitably weighted versions of the residue theorem, we also obtain the asymptotically sharp lower bound $\int_I λ(x)\ dx \geq \frac{4|I|}{π^2} \log n - o(\log n)$ for integral variants of such constants, answering a further question of Erdős. |
| title | Local Bernstein theory, and lower bounds for Lebesgue constants |
| topic | Classical Analysis and ODEs 41A05, 42A15, 30D15 |
| url | https://arxiv.org/abs/2603.21453 |