A dimension-free discrete Remez-type inequality on the polytorus
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| Format: | Preprint |
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2023
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| _version_ | 1866915286724116480 |
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| author | Slote, Joseph Volberg, Alexander Zhang, Haonan |
| author_facet | Slote, Joseph Volberg, Alexander Zhang, Haonan |
| contents | Consider $f:Ω^n_K \to \mathbf{C}$ a function from the $n$-fold product of multiplicative cyclic groups of order $K$. Any such $f$ may be extended via its Fourier expansion to an analytic polynomial on the polytorus $\mathbf{T}^n$, and the set of such polynomials coincides with the set of all analytic polynomials on $\mathbf{T}^n$ of individual degree at most $K-1$.
In this setting it is natural to ask how the supremum norms of $f$ over $\mathbf{T}^n$ and over $Ω_K^n$ compare. We prove the following \emph{discretization of the uniform norm} for low-degree polynomials: if $f$ has degree at most $d$ as an analytic polynomial, then $\|f\|_{\mathbf{T}^n}\leq C(d,K)\|f\|_{Ω_K^n}$ with $C(d,K)$ independent of dimension $n$. As a consequence we also obtain a new proof of the Bohnenblust--Hille inequality for functions on products of cyclic groups.
Key to our argument is a special class of Fourier multipliers on $Ω_K^n$ which are $L^\infty\to L^\infty$ bounded independent of dimension when restricted to low-degree polynomials. This class includes projections onto the $k$-homogeneous parts of low-degree polynomials as well as projections of much finer granularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_10828 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A dimension-free discrete Remez-type inequality on the polytorus Slote, Joseph Volberg, Alexander Zhang, Haonan Classical Analysis and ODEs Analysis of PDEs Complex Variables 46B10, 46B09, 46B07, 60E15 F.2.2 Consider $f:Ω^n_K \to \mathbf{C}$ a function from the $n$-fold product of multiplicative cyclic groups of order $K$. Any such $f$ may be extended via its Fourier expansion to an analytic polynomial on the polytorus $\mathbf{T}^n$, and the set of such polynomials coincides with the set of all analytic polynomials on $\mathbf{T}^n$ of individual degree at most $K-1$. In this setting it is natural to ask how the supremum norms of $f$ over $\mathbf{T}^n$ and over $Ω_K^n$ compare. We prove the following \emph{discretization of the uniform norm} for low-degree polynomials: if $f$ has degree at most $d$ as an analytic polynomial, then $\|f\|_{\mathbf{T}^n}\leq C(d,K)\|f\|_{Ω_K^n}$ with $C(d,K)$ independent of dimension $n$. As a consequence we also obtain a new proof of the Bohnenblust--Hille inequality for functions on products of cyclic groups. Key to our argument is a special class of Fourier multipliers on $Ω_K^n$ which are $L^\infty\to L^\infty$ bounded independent of dimension when restricted to low-degree polynomials. This class includes projections onto the $k$-homogeneous parts of low-degree polynomials as well as projections of much finer granularity. |
| title | A dimension-free discrete Remez-type inequality on the polytorus |
| topic | Classical Analysis and ODEs Analysis of PDEs Complex Variables 46B10, 46B09, 46B07, 60E15 F.2.2 |
| url | https://arxiv.org/abs/2305.10828 |